Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
958
E Rudolf Beyl JLirgen Tappe
Group Extensions, Rep resentations, and the Schur Multiplicator
Springer-Verla£ Berlin Heidelberg New York 1982
Authors
E Rudolf Beyl Mathematisches Institut der Universit~t Im Neuenheimer Feld 288, 6900 Heidelberg, Germany Ji]rgen Tappe Lehrstuhl f~r Mathematik Rhein.-Westf. Technische Hochschule Aachen Templergraben 55, 5100 Aachen, Germany
AMS Subject Classifications (1980): 2 0 C 2 5 , 20E22, 20J05, 2 0 C 2 0 , 20E10, 2 0 J 0 6 ISBN 3-540-11954-X Springer-Verlag Berlin Heidelberg New York ISBN 0-387-11954-X Springer-Verlag New York Heidelberg Berlin This work is subject to copyright.All rights are reserved,whetherthe whole or part of the material is concerned,specificallythose of translation,reprinting, re-use of illustrations,broadcasting, reproduction by photocopyingmachineor similar means,and storage in data banks. Under § 54 of the GermanCopyright Law where copies are madefor other than private use, a fee is payableto "VerwertungsgesellschaftWort", Munich. © by Springer-VerlagBerlin Heidelberg1982 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210
TABLE OF CONTENTS
Introduction
Chapter I.
Group Extensions with Abelian Kernel
I. The Calculus of Induced Extensions
5
2. The Exact Sequence for Opext
19
3. The Schur Multlplicator and the Universal Coefficient Theorem
28
4. The Ganea Map of Central Extensions
4O
5. Compatibility with Other Approaches
47
6. Corestrictlon
58
Chapter II.
(Transfer)
Schur's Theory of Projective Representations
I. Projective Representations
67
2. The Problem of Lifting Homomorphisms
77
3. Representation Groups
91
4. Representation Groups of Free and Direct Products
101
5. The Covering Theory of Perfect Groups
113
Chapter III.
Isoclinism
I. Isoclinic Groups and Central Extensions
123
2. Isoclinism and the Schur Multlplicator
137
3. The Isomorphism Classes of Isoclinic Central Extensions and the Hall Formulae
144
4. On Presentations of Isoclinlc Groups
155
5. Representations
169
of Isoclinic Groups
IV Chapter IV.
Other Group-Theoretic Applications of the Schur Multipllcator
I. Deficiency of Finitely Presented Groups
179
2. Metacycllc Groups
193
3. The Precise Center of an Extension Group and Capable Groups
204
4. Examples of the Computation of Z*(G)
213
5. Preliminaries on Group Varieties
227
6. Central Extensions and Varieties
233
7. Schur-Baer Multipllcators and Isologlsm
244
Bibliography
261
Index of Special Symbols
271
Subject Index
274
INTRODUCTION
The aim of these notes is a unified treatment of various grouptheoretic topics for which, as it turns out, the Schur multiplicator is the key.
At the beginning of this century, classical projective
geometry was at its peak, while representation theory was growing in the hands of Frobenius and Burnside.
In this climate our subject
started with the two important papers of Schur ~I~,~2~ on the proJective representations of finite groups.
But it was only in the light of the much more recent (co)homology theory of groups that the true nature of Schur's "Multlplicator" and its impact on group theory was fully realized;
the papers by GREEN
EI~, YAMAZAKI EI~, STALLINGS ~I~, and STAMMBACH ~I~ have been most influential in this regard.
The first chapter provides the setting for these notes.
We start
out with the concepts of group extension (handled in terms of diagrams) and Schur multiplicator (here defined by the Schur-Hopf Formula)
to obtain a group-theoretlc version of the Universal
Coefficient Theorem.
All these concepts and the Ganea map have a
homological flavor, but are here developed in a rather elementary group-theoretic
fashion; the (co)homology theory of groups is not
a prerequisite for reading most of these notes.
The first chapter
also includes a full translation from our approach to the usual group (co)homology for the reader's convenience. presentation
(We feel that our
is very suited for the applications to follow, but this
view is to some extent a matter of taste.)
In the second chapter we consider projective which can be regarded as homomorphisms Schur
showed that the projective
into projective
representations
Q over the complex field C can be described representations
of certain central
vented the "Darstellungsgruppen
representations, groups.
of the finite group
in terms of the (linear)
extensions
by Q, and thus In-
yon Q" or, in English,
sentation
groups of Q.
variations
of Schur's
theme are discussed.
(The problem
lifting homomorphisms
where the projective
representations
replaced
by more general homomorphisms.)
on Schur's
theory,
finite groups.
representation
groups
provided
condidate
sentation
groups
one carefully
distinguishes
For example, treatment related
aspects
the final section of Chapter
of the covering
e.g. to finite
ticular attention
groups,
for its relevance
and the
in many parts of
of representation
theory.
II gives a comprehensive
theory of perfect
simple groups,
between
We find that repre-
groups are important
group theory beyond the original
are
is seen to be unneces-
H2(Q,C *) ~ Hom(M(Q),t*)
of arbitrary
is that of
etc. are defined only for
as the common kernel of all representation
competing
several
In most of the literature
In these notes this attitude
sarily restrictive, M(Q),
In the course of this chapter,
the repre-
groups.
This theory is
but has recently gained par-
to Milnor's
K 2 functor - where
Q is an infinite matrix group.
In the third chapter we study the notion P. Hall introduced of groups. Hall's
in his GGttingen
[I],[2],[3],[4];
to central
in "Crelle's
summaries
Journal",
it may be due to these circumstances
further details. extensions;
which
on the classification
In spite of World War II having begun,
lectures were published
not publish
lectures
of isoclinism,
The notion of isoclinism
of
cf. P. HALL that Hall did is extended
this step is more or less technical,
but
provides for clarity and enables us to apply the machinery of Chapter I.
The first major result of this chapter is a description
of isoclinism classes in terms of the subgroups of the Schur multipllcator°
We then study a refinement of the isoclinism concept and
prove formulae of P. Hall ~3S.
Our treatment in terms of central
extensions differs from Hall's, which employs free presentations. In any case, Chapter III brings out some connections between both views.
In the final section of Chapter III, we work out implications of the isocllnism relation for the ordinary and the modular representations of finite groups.
The last chapter contains further group-theoretic applications of the Schur multiplicator, ~3S.
in some aspects it supplements STAMMBACH
We first resume the question of group deficiency, a concept
grown out of the desire to present a group with as few relators as possible. pretation.
Our emphasis is on worked-out examples and their interAmong other results, we give rather elementary treat-
ments of (i) Swan's examples of finite groups with trivial multiplicator and large deficiency;
(ii) an interesting representation
group of the non-abelian group of order p3 and exponent p, for p an odd prime;
(iii) metacyclic groups and their multiplicators.
The
next topic is a group invariant Z*(G), the central subgroup that measures how much G deviates from being a group of inner automorphisms.
(This concept is unrelated to Glauberman's Z*, any
serious confusion seems to be unlikely.)
We then obtain rather
explicit results on the question whether a central group extension lies in a given variety of exponent zero.
These sections again
show the importance of Schur's "hinrelchend erg~nzte Gruppen",
i.e.
central extensions having the lifting property for complex projective representations;
they are called "generalized representation
groups" in these notes.
The chapter ends with a development of
isologism, a related concept of P. Hall, in analogy with our treatment of isoclinism in Chapter III. to study LEEDHAM-GREEN/McKAY
The reader will now be prepared
~I] and other papers on varietal
cohomology.
These notes partly present results from our "Habilitationsleistungen" at "Ruprecht-Karls-Universit~t
Heidelberg" and "Rheinisch-West-
f~lische Technische Hochschule Aachen",
respectively.
We thankfully
acknowledge the support we received from our institutions,
as well
as partial support from the "Deutsche Forschungsgemeinschaft the "Forschungslnstitut
f~r Mathematik
(DFG)",
(ETH ZGrich)", and the
"Gesellschaft yon Freunden der Aachener Hochschule
(FAHO)".
We remember with pleasure that we greatly profited from the feedback various seminar audiences gave us, in particular from discussions with the late R. Baer, with P. Hilton, C.R. Leedham-Green, R. Laue, J. NeubUser, J. Wiegold.
J. Ritter, U. Stammbach, R. Strebel, and
C H A P T E R I.
GROUP EXTENSIONS WITH ABELIAN KERNEL
T h e core of t h i s c h a p t e r c o n s i s t s
of S e c t i o n s
3 and 4.
1. The C a l c u l u s of I n d u c e d E x t e n s i o n s
T h i s s e c t i o n is p r e p a r a t o r y .
We i n t r o d u c e
f o r w a r d and b a c k w a r d
i n d u c e d g r o u p e x t e n s i o n s and d i s c u s s the r e l a t i o n s h i p sions a n d f a c t o r systems.
b e t w e e n exten-
A n e x t e n s i o n of the g r o u p N by the g r o u p Q
is a short exact s e q u e n c e (1.1)
e =
(~,~)
or, e q u i v a l e n t l y , e
:
O--~N
of groups.
:
N~
0 =
~ •G
w ~Q
The a r r o w s ;
~11
~Q
an exact s e q u e n c e
jectlve h o m o m o r p h i s m s and
~ G
~0
> and
~
(monomorphlsms
d e n o t e i n j e c t i v e and sur-
and e p i m o r p h i s m s ) ,
stands for the g r o u p of one element.
respectively,
(This t e r m i n o l o g y
a c c o r d s w i t h c a t e g o r y t h e o r y and is e a s y to w o r k with.)
In m o s t of the c a s e s t r e a t e d here, e x t e n s i o n e g i v e s rise to a Q - m o d u l e d e f i n e d by
qn = ~ - 1 ( g . ~ n . g - 1 )
is any e l e m e n t w i t h Q-module.
wg = q .
N will be abelian.
s t r u c t u r e on N, w h i c h is w e l l -
, where
q ~ Q , n e N
N o w let
(N,
W e call e a Q - e x t e n s i o n of
congruent,
Two e x t e n s i o n s
if t h e r e
eI
and
e2
exists an i s o m o r p h i s m
f o l l o w i n g d i a g r a m is c o m m u t a t i v e ;
and
~: Q - A u t ( N ) )
(N,~)
of N by Q w h i c h i n d u c e s the g i v e n Q - m o d u l e m e t h o d above.
T h e n the
g ~ G be a
if e is an e x t e n s i o n
structure
on N by the
of N by Q are c a l l e d ~: G 1 ~ G 2
we t h e n w r i t e
such t h a t the
eft ~ e 2 .
6
eI
:
N >
~
e2 :
N ;
G1
-'~ Q
II t
m Q
Congruence
is an e q u i v a l e n c e
relation.
extensions
define the same Q - m o d u l e
(1.2)
0pext(Q,N,~)
denote
sions of
(N,~)
.
(1.3)
e o : N~
11
; G2
W h e n N is abelian,
structure
the set of c o n g r u e n c e
The
existence
Let
on N.
classes
congruent
[e]
of Q - e x t e n -
of the e x t e n s i o n
KO
implies
that
semidirect formula
~a= 1Q .
Opext(Q,N,m)
product
~o(n)
A morphism
and
e of
-
Here
The maps = q . -
Ko
is the
~o
are
The e x t e n s i o n
e as
is a h o m o m o r p h i s m
(N,m)
N~Q
N w i t h the m u l t i p l i c a t i o n
Vo(n,q)
split if there
The Q - e x t e n s i o n to
is not empty.
= (n.qnl,q-ql)
= (n,1)
is c a l l e d
is c o n g r u e n t
o ;;Q
of Q by the Q - m o d u l e
(n,q).(nl,ql)
d e f i n e d by in (1.1)
; N~ Q
and
a: Q - G
is split p r e c i s e l y
with
w h e n it
eo .
(e,e,~):
eI - e2
of e x t e n s i o n s
is a c o m m u t a t i v e
diagram
eI
:
NI
'~ G1
~ Q1
N2 ;
; G2
11~ Q2
(1.4) e2 : This
is c a l l e d an i s o m o r p h i s m
are group isomorphisms. mi: Qi " Aut(Ni) If a m o r p h l s m (1.5)
~(qn)
Now assume
be the structual
(~,8,~): = ~(q)~(n)
or, e q u i v a l e n t l y ,
of e x t e n s i o n s
eI - e2 for all
Ni
if ~ and ~ (hence also
abelian
and let
maps d e f i n e d by
exists,
5: (NI,~ I) - (N2,~2~)
e i , for
then
n e NI ,
~)
q e Q1
is Q 1 - h o m o m o r p h i c .
i:=1,2.
If the e x t e n s i o n center
Z(G)
trivial.
e as in (1.1)
of G, then the a s s o c i a t e d
The c o n v e r s e
homomorphism
is also true.
.
set
of c o n g r u e n c e
Ext(Q;N)
If m o r e o v e r
as a subset.
that of MAC LANE[2;
i.e.
map
Cext(Q,N)
of a b e l i a n
This i n t e r p r e t a t i o n
of
extensions,
extensions Ext(Q;N)
with N abelian I Uq ~ G
I ~(Uq) = q e Q
morphism, factor
f(r,s)
the n o t i o n
-I = Ur.Us.Ur.s law
f(r,s).f(r.s,t)
f'=f+sg:
(1.7)
N e x t we c o n s i d e r factor
sets.
~N
lql
al-
systems and (1.1)
, choose a t r a n s v e r s a l
Q - G
a system of coset will not be a homo-
is m e a s u r e d
by the a s s o c i a t e d
, this set map b e i n g d e f i n e d by for
r,s E Q .
= f(r,s.t)
, where
all f u n c t i o n s
implies
rf(s,t) ~u~l
g =
the formula
r,s,t ~ Q .
, then the new factor
~q~
f which
For every f u n c t i o n &g
to con-
systems,
w i t h factor
)~-1(u~.u~1)):
:= g ( r ) - r g ( s ) . g ( r - s ) -I
by (1.7); we note that
extensions
in this context.
of factor
in G, i.e.
~Uql:
transversal
Q × Q " A
(~g)(r,s)
to
(UrUs)U t = Ur(UsUt)
If we choose a n o t h e r is
e: Q - A u t N
The d e v i a t i o n
f: Q x Q - A
The a s s o c i a t i v e (1.6)
I
The map
in general.
system
agrees with
If you are given the e x t e n s i o n
and o p e r a t i o n
representatives.
the
(i.e. G
ourselves
extensions
t h o u g h we try to a v o i d d i r e c t c o m p u t a t i o n s as far as possible.
using
we allow
classes with representing
Here we shortly review
cocycles
contains
III,~1].
yields the same for c o n g r u e n t
1.1.
is
for
Cext(Q,N)
Once we h a v e shown that a c e r t a i n c o n s t r u c t i o n
fuse c o n g r u e n c e
lies in the
~: Q - A u t N
Q is abelian, classes
~(N)
We w r i t e 0 for the t r i v i a l
b e t w e e n any two g r o u p s and
0pext(Q,N,~=0)
abelian)
is central,
Q - A
and
r,s E Q .
satisfy
g: Q - A
system
(1.6)
we define
and call them ~g: Q x Q - A
is a factor set and call it a p r i n c i p a l
f a c t o r set or t r a n s f o r m a t i o n
set.
For computational
ing this s u b s e c t i o n c o n s u l t MAC L A N E
[2; IV,§4].
The f a c t o r sets f o r m an a d d i t i v e a b e l i a n group, d e f i n e d by p o i n t w i s e m u l t i p l i c a t i o n , are a s u b g r o u p thereof. as in (1.2), u (2) q = 9Uq
If we are g i v e n a c o n g r u e n c e
eI
:=
and
of
e2 .
(N,e)
A l l told,
Ifactor s e t s } / ~ p r i n c i p a l
Then
f(q,1)
= f(1,q)
a f a c t o r set n o r m a l i z e d . f(1,q)
= f(1,1)
f - 5g , w h e r e f(1,1) in
; then
H(Q,A,¢)
g(1)
= 1 .
e = eI ~ e2 i.e.
f a c t o r sets 1
~Uq} = I
Condition
f - ~g
, we are free to r e q u i r e
for all (1.6)
q ~ Q ; we call
implies
; hence
f(q,1) = qf(1,1) f by
is the c o n s t a n t f u n c t i o n w i t h value
For a n y 5g
We conclude that
L e t an e l e m e n t (N,¢)
such
is n o r m a l i z e d a n d d e t e r m i n e s the same e l e m e n t
as f does.
G - e x t e n s l o n e of
c l a s s of
e l e m e n t in the g r o u p
g: G - A
, formula
is a n o r m a l i z e d H(Q,N,~)
X ~ H(Q,N,~)
(1.7) g i v e s
f a c t o r set e x a c t l y if
is i s o m o r p h i c
~normalized factor sets}/~normallzed principal
1.2.
f a c t o r sets
for any f a c t o r set; ~e can r e p l a c e
g: Q ~ A
g(1) = (5g)(1,1)
every congruence
determines a unique
W h e n c h o o s i n g the t r a n s v e r s a l
and
and the p r i n c i p a l
t h e n we may c h o o s e the t r a n s v e r s a l s c o m p a t i b l e ,
Q-extensions
uI = 1
with addition
, a n d c o n c l u d e t h a t t h e s e c h o i c e s y i e l d the same f a c t o r
s y s t e m for
H(Q,N,¢)
details regard-
to
f a c t o r sets}
be given,
is t h e r e some
y i e l d i n g t h i s X by the p r o c e d u r e
If we s p e c i f y a f a c t o r set f in X,
is it p o s s i b l e
of 1.1?
to o b t a i n f as a
f a c t o r s y s t e m for e a n d a s u i t a b l e c h o i c e of c o s e t r e p r e s e n t a t i v e s ? The a n s w e r is "Yes"
in b o t h cases,
as the f o l l o w i n g c o n s t r u c t i o n
shows. A n y f a c t o r set in X has the f o r m generality, Q - A
f is a n o r m a l i z e d
f + 5g
where,
w i t h o u t loss of
f a c t o r set and g is some f u n c t i o n
(The n o r m a l i z a t i o n a s s u m p t i o n
on f is for c o n v e n i e n c e
only,
the given product
formula also works without
G with underlying
set
(n,q).(nl,ql)
N x Q
Moreover
(1,1)
~: N - G
and
O.
)N
~
~Q
,O
{u~ = (g(q),q)l
factor system is NORTHCOTT
[I;
f + 5g .
classes
y:=q-1
~(n) = (n,1)
is a Q-extension
of
(N,~)
.
If we
then the associated
For computational
details
consult
e.g.
~10.I0].
LANE
[I],
to the cohomology
[2] discovered
of group extensions
theory of groups,
is isomorphic
to the second cohomology
(N,@)
at a particular
viz. the normalized
resolution,
form,
when
that the set of congruence
group of G in the Q-module
inhomogeneous
with
such that
as a transversal,
It was a great stimulus EILENBERG/MAC
from (1.6), the neutral
~: G - Q , defined by
~(n,q) = q , are homomorphisms ~G
follows
(n,q) -I = (Y(n-1)-f(y,q)-1,y)
and
choose
a group
and multiplication
associativlty
, and
the maps
~
Construct
= (n.qnl.f(q,ql) , q.ql )
This is indeed a group: element is
it.)
By computing
cohomology
groups
bar resolution
they were led to the functional
equations
in (1.6)
and (1.7) and thus found the second cohomology
group also given by
{normalized
factor
factor
setsl/{normalized
Now the idea of factor been used by HOLDER and systematized
however.
[1] in 1893.
the treatment
groups N allowed) cohomology
systems
via factor
principal
is near at hand and had already Later
SCHREIER
of group extensions
systems.
group via a particular
1.3 DEFINITION.
without
resolution
7:Q1
continued
(also non-abelian to the second
looks artificial, a reasonably
complete
resolutions.
Given a Q-extension
and a group homomorphism
[I],[2]
The transition
In this chapter we try to present
theory of group extensions
sets~.
" Q •
e of the Q-module
(N,~)
We define a Q1-extension
e~
10
of
(N,~)
as the top row of the c o m m u t a t i v e
e?
:
N>
o
e
:
N;~
; G~
o
diagram
:;QI
(1.s)
Here
G7 =
~ (g,q)
~o = [ ( g ' q ) '
~
;G
~ G x Q1
)q[
[ ~ g = Yq
' ?" = ~ ( g ' q ) '
This c o n s t r u c t i o n
for a r b i t r a r y N, too).
with associated for
q ~ QI
The reader
systems:
factor
f(? × ?): QI x QI " N
If
product
e.g.
induced
that
f: Q x Q - N for
extension
of groups
e?
)Uq e G
e?
The group
GAQI
>(~n,1) I
~n!
easily v e r i f i e s
ql
system
.
~o =
in the c a t e g o r y
is a t r a n s v e r s a l
as W i e l a n d t ' s
and
,
;gl
in the above definition,
In terms of factor
[
is c a l l e d the b a c k w a r d
it makes use of the p u l l - b a c k
implied
,~Q
~
G~
e~
(and is valid
the a s s e r t i o n s
is an extension. is a t r a n s v e r s a l
,
,
then
w i t h factor
q~
for e
~(u q,q)
system
may be known to the reader
, the latter being d e f i n e d w h e n
? is
epimorphic.
1.4 P R O P O S I T I O N . (a,~,?):
eI - e
el Here If
from an e x t e n s i o n
(a,~,1))
I: Q1 " QI e ~ e'
, then
Opext(Q,N,w)
PROOF.
e?
(I'¥°'?)
and
e? ~ e'?
of D e f i n i t i o n
e ~ e'
yields
1.3, any m o r p h i s m
can be f a c t o r e d u n i q u e l y
as
.
denote the i d e n t i t y h o m o m o r p h l s m s . ?* =
lie] i
~[e?] I :
is w e l l - d e f i n e d .
for the f a c t o r i z a t i o n
~ (~g,~lg) l: G 1 - G ? .
the c o n g r u e n c e
(1,~o?,,?): e? - e'
eI
Hence
) Opext(QI,N,~?)
~g~
>e
I: N - N
The map ~ r e q u i r e d
and has to be, affords
In the s i t u a t i o n
If
of
(1,8,1):
, then the f a c t o r i z a t i o n
e? ~ e'?
[]
eI - e e - e' of
is,
11
1.5 DEFINITION. and a Q - l i n e a r of
(N2,~2)
Given a Q-extension
map
a: N - (N2,~2)
e of the Q - m o d u l e
We define a Q - e x t e n s i o n
as the b o t t o m row of the c o m m u t a t i v e e :
N)
~
(N,m)
~G
w
~Q
~Go
~o
~Q
ae
diagram
(1.9) ~e Here
:
N2~
o
G o = ((N2,~2.) ~ G ) / S
~O =
{n|
) (n,1)S}
, ~o =
In terms of factor and f the a s s o c i a t e d for
q ~ Q
with
S =
i (an-l,~n)
{(n2'g)S:
systems: factor
If
, a. =
~Uq
is a t r a n s v e r s a l
q:
for
ae
with
~g|
}
~gl
system as above,
is a t r a n s v e r s a l
I n e N
then
factor
qm
and
>(1,g)S}
.
for e
~(1,Uq)S
system
of: Q x Q - N 2
We call
ae
a forward
induced
v e r i f y the a s s e r t i o n s
implied
a normal
N2~ G
subgroup
of
is a n o r m a l m o n o m o r p h l s m . since all
(on,l) n ~ N
sion,
.
and
T h u s the e x t e n s i o n on
ae
N 2 ; in this case
1.6 P R O P O S I T I O N . (o,S,~):
e - e2
uniquely
as
in
N 2~G
operation
The r e a d e r
definition.
In l
are c o n g r u e n t
of Q on
is central p r e c i s e l y
should Indeed,
,(~n-1,~n)~:
of (1.9)
is w e l l - d e f i n e d ,
~e N2
S is
N - N 2 ~G
is c o m m u t a t i v e modulo S, for
is indeed an extenis given by
m2
when Q acts t r i v i a l l y
G a = (N 2 x G)/S
into a Q 2 - e x t e n s i o n
of D e f i n i t i o n of
(N2,m 2)
(1,~,~) ~ ~e
e' ~ e , then
Opext(Q,N,~)
since even
In the s i t u a t i o n
(o,o.,1) e If
~o
and the a s s o c i a t e d
in the above
The left square
(1,xn)
The map
extension.
oe' ~ ~e
)e 2 .
~ 0pext(Q,N2,~2)
Hence
o. =
l[e]l
is w e l l - d e f i n e d .
1.5, any m o r p h i s m
can be f a c t o r e d
12 PROOF. be,
The map
}(n2,g)Si
~ required
)~2n2.sg}
}(n2,g)l
) ~2n2.eg}:
lates S.
If
1.7 REMARK. nition
of
nition
a = 1
(I,~,~):
eI - e
e~
of
(~,e,1):
.
ae
has
e - e2
is ( c o n g r u e n t ~
e
(1,~',~):
e~ - e
eI
ee' ~ ~e .
[]
u s e d in the defithat any
as ( c o n g r u e n t e - ae
to the i n d u c e d
u s e d in the defi-
let a m o r p h i s m
with a b e l l a n
(1.5),
e' ~ e , then
1.4 implies
Conversely
to the i n d u c e d
yields
(~,a.,1):
of e x t e n s i o n s from
e' - ~e
exhibits
which annihi-
the c o n g r u e n c e
Proposition
~ = 1
kernel
and P r o p o s i t i o n
extension)
ee
.
be given.
Then
1.6 implies that
e2
Obviously
le
~
1.8 EXAMPLES. an i n d u c e d ing,
.8,I):
since
is a h o m o m o r p h i s m
affords
The m o r p h i s m
is Q - h o m o m o r p h l c
el
(,,~
has
extension)
;G 2
e' - e
of
is, and has to
T h i s is w e l l - d e f l n e d
The m o r p h i s m
e~
morphlsm
.
N2~ G
(1,~,1):
the f a c t o r i z a t l o n
for the f a c t o r l z a t i o n
The p r e c e d i n g
extension without
let e as in (1.1)
If P is a subgroup
remark
invoking
often allows
the definition.
be a Q - e x t e n s i o n
of Q and
one to d e s c r i b e In the follow-
of the Q - m o d u l e
i: P C
Q
denotes
(P)
.~-~ P
(N,~)
.
the inclusion,
look at eIP
N ~
:
~
N; where
~'
and
commutative. and
~'
natural
) G - - - - - ~ -~ Q
are the
(unique)maps
T h e n the top row
[elP ] = i*[e]
Next, l e t
~
(N2,e 2)
inclusion
tative d i a g r a m
.
Sometimes be a n o t h e r
elP i*
i.[e]
of
(N,~oi)
is called res for "restriction".
Q-module
= [e 2]
w h i c h r e n d e r this d i a g r a m
is a P - e x t e n s l o n
as the first summand.
exhibits
,
and
i: N ;
~N x N 2
Then the f o l l o w i n g ,
be the commu-
13
e
•
N x N2 ~
e2 : where
~2 =
}(n'n2);
>(n2,"n)~
The use of induced "folklore" extension usually
implicit
was the t r e a t m e n t
effort
to exhibit
that MAC LANE
1.9 DEFINITION.
tive a b e l i a n
A derivation d: Q - A
If
homomorphlsm
Der(Q,A,~)
)Der(G,A,m~)
~q:
=
for modules. for groups
in
= d(x).Xd(y)
~(d(q),q)l:
Der(Q,A,O)
= Hom(Q,A)
for all
of Q in A forms an addi-
}dl
d: Q - A
In the case of trivial
morphisms
extensions
+
defined
by point-
is a group homomorphism,
Der(,)
morphism.
Given
of G.
for this section
constructions
with addition
the set map if
for induced
d(x.y)
~: G ~ Q
is an induced
exactly,
the p r o p e r t i e s
it
of the group Q in the Q-module
with
Der(Q,A,m)
wise multiplication.
Alternatively,
if the
set as in 1.2,
~I] had given
The set of all d e r i v a t i o n s group
is
of his book.
is a set map
x and y in Q.
, and
systems
However,
The model
[2; III
&~g}
factor
constructions
in this regard.
version
~(n2'g):
from a factor
Mac Lane had used the diagram
a preliminary
(A,e)
~2 =
in computations.
the diagram
are much more helpful
~ Q
via induced
group G is c o n s t r u c t e d
takes a tedious
Actually
'
extensions
and often
On the other hand,
(A,~)
) (N 2,~2 ~ ) ~ G
~do-I
:
is a d e r i v a t i o n
Q - (A,~)~ Q
operation,
there
of Q in
is a group homo-
obviously
.
Qi-extensions
ei
~: N 1 - N 2
and
of
(Ni,~ i)
~: QI " Q2
for "
i=1,2
A basic
and group homo-
technical
problem
14
is: W h e n can a h o m o m o r p h i s m following
diagram
el
NI~
) G1
~ Q1
!
1
e2 :
be found as to make the
commutative?
:
(1.1o)
S: G 1 - G 2
'
1
$
N2~
~ G2
~Q2
We then say: The d i a g r a m can be solved for ~. a necessary
condition
1.10 THEOREM. satisfying ~:
is that e and ~ s a t i s f y
thus
~e I
(NI,~ 1) - (N2,~2~)
bijective
being defined
ae I ~ e2~
In this case,
correspondence
of i n d u c e d
PROOF.
If
with
,e I - e2~
(a,B,~):
Propositions lence
set b i J e c t i v e l y
d: Q - A
e I - e2
~ is u n i q u e l y
~
d'
the
used hereafter.
(~,.,~)
Conversely,
We factor
(,,8,~)
let a
using
determined
of 1.4 and 1.6
by B-
H e n c e the s o l u t i o n
to the set of c o n g r u e n c e s
~ ~Q then
be a Q - e x t e n s l o n ~d=IgJ
(I,~d,I):
e - e
given a s e l f - c o n g r u e n c e
of G in
parts
(1,.,I):
to the set of the s e l f - c o n g r u e n c e s
> G
such that
checks that the rule vation
constitute
then the c o m p o s i t i o n
The u n i q u e n e s s
is a derivation,
Conversely,
be given.
corresponds
e: A >
homomorphism
set is in
and will be f r e q u e n t l y
is a map of type
or, likewise,
Let
three p r o p o s i t i o n s
is a congruence,
se I - e2~
imply that the map
~e I
the s o l u t i o n
~ if,
1.4 and 1.6 in either order and thus o b t a i n an equiva-
(I,~,1):
oe I - e2~
for the Q 1 - h o m o m o r p h i s m
Der(QI,N2,m2V)
extensions
e I - se I - e2~ - e 2 morphism
(1.5).
T h e n d i a g r a m (1.10) can be solved for
This t h e o r e m and the f o l l o w i n g calculus
out above,
G i v e n the data of d i a g r a m (1.10) w i t h ~ and
(1.5),
and only if,
As p o i n t e d
~d'(g) (A,~)
= ~(g).g .
Since
d'
(A,~)
~d(~g)'gl:
G - G
is a m o r p h i s m
(I,~,1): -1
of
for
e - e g E G
is c o n s t a n t
of e.g. .
If
is a
of extensions. The r e a d e r
d e f i n e s a derion cosets
15 g.~A
, d'
determines
d' = d ~ o ~ inverse
.
Clearly
a derivation
d~
of Q in
the a s s i g n m e n t s
to each other.
1.11 PROPOSITION.
d ~
(A,m)
~d
with
and
~
;d~
are
[]
Let
(N,~)
be a Q-module
and
e a Q-extension
of N. (a)
If
morphisms,
ci: N - (NI,~1)
then
(b)
If
~I: QI " Q
If
Q1-module
PROOF.
then
c(e~)
(NI,~I?)
and
Apply
"only
the
with
~
~2:Q2
is split
Given
with
~ (ce)~
N1
and
then
(a), to
(e~1)~ 2 - e~1 - e
e and
treated
a
as the
.
of Theorem
(c).
~: Q1 " Q
being
if"-part
1.10 to in case
(b),
[]
~ as in D e f i n i t i o n
if, and only if, there
If
c: Q1 " G~
is a splitting
fies
~(8~os)
= ~
Then
~=lq!
~(~q,q) l: QI " G~
Conversely,
1.13 PROPOSITION. ae
are homomorphisms,
1.3.
The exten-
exists a lifting
6: QI " G
= ~ .
PROOF.
sion
" QI
is a Q - h o m o m o r p h i s m
e(e~)
in case
1.12 PROPOSITION. e~
.
is defined
in case
e~ - e - ~e
sion
are Q-homo-
•
e - ~I e - ~2(aI e) and to
and
~: N - (N1,ml)
homomorphlsm,
o2: N I - (N2,m 2)
a2(cle ) ~ (e2c1)e
(e~l)V 2 ~ e(7172) (c)
and
is split
d: G - (N2,~2~)
Given
, then
8 with
is a splitting
of
e and e as in D e f i n i t i o n exists
8~.o
~6=~
e~
1.5.
satis-
be given. []
The exten-
a derivation
do~ = c
Included is the w e l l - k n o w n when a derivation
ev
let a lifting
if, and only if, there
with
of
d: G - N
fact that e itself with
do~ = 1N
is split p r e c i s e l y
exists
(splitting
16
derivation).
In case of c e n t r a l
for a h o m o m o r p h i s m
d: G - N 2
extensions,
the p r o p o s i t i o n
e x t e n d i n g e.
calls
On the o t h e r extreme,
our p r o o f e a s i l y a d a p t s to e x t e n s i o n s w i t h n o n - a b e l i a n kernel. this case, satisfy
the map
d(x.y)
PROOF.
If
(1.11) then (iii)
Ig~
~d(g) -1}
r a t h e r t h a n d is r e q u i r e d to
= d(x).Xd(y)
d: G - N 2
,.(g) (i)
d =
and
Z: G - G
= ,od(g).£(g)
Wo Z = ~ ,
(ii)
d is a d e r i v a t i o n
are set m a p s
for all
d~ = o
g e G
satisfying
,
if a n d only if
Zx = 0
, and
if and only if ~ is a n o m o m o r p h i s m .
p r o o f s of (i) and (il) are immediate. (N2,~2~)
In
The a c t i o n of
can also be g i v e n by c o n j u g a t i o n w i t h
g e G
~.(g)
in
The on
G°
U s i n g this, we o b t a i n ~od(g.gl).~(g-gl ) = ~o[gd(gl).d(g)].zg-Zgl Since
N2
is a b e l i a n ,
(iii)
Now to our a s s e r t i o n . = o~
with
d~ = e .
Define
= o~ .
The map
1.14 P R O P O S I T I O N . Q-module,
let
m o r p h l s m and
q=~x:
It f o l l o w s from Conversely,
Z by (1.11). £~ = 0
c: Q - G O
e: Q - G
T h i s is p o s s i b l e
d~ = ~
2 is a h o m o m o r p h i s m w i t h
a
Given a splitting
~ K e r ~o = Im ~o
d is a d e r i v a t i o n w i t h
g'gl E G
.
follows.
and d e f i n e d by (1.11).
(~,g).(Zg)-I
for all
Then
, hence
splits
o
of
oe
since (li) and
(iii) t h a t
let d be a d e r i v a t i o n
(ii) a n d
(iii) give t h a t
Z f a c t o r s u n i q u e l y as
oe
, since
~ o C ~ = 1o~ .
Let Q be a f i n i t e g r o u p of o r d e r q and ~xq~: N - N
9.: O p e x t ( Q , N , ~ )
.
, set
(N,~)
T h e n ~ is a Q - m o d u l e h o m o -
- Opext(Q,N,w)
is the c o n s t a n t map
onto the c l a s s of the split e x t e n s i o n s .
PROOF. of 1.13, d~ = ~ .
Let e as in (1.1) be any Q - e x t e n s l o n it s u f f i c e s to find a d e r i v a t i o n Choose any transversal
~Uq}
of
(N,~)
d: G - (N,w-)
and d e f i n e d by
In v i e w with
17
~d(g)
=
~ C(g,r) r~Q
T h i s is w e l l - d e f i n e d ,
,
C(g,r)
since
:= g . U r . U
D(g,r)
~ Ker
-1 (g). r
~ = Im ~ .
One e a s i l y
verifies (1.12)
C(g.h,r)
-- g C ( h , r ) g - 1 - C ( g , ~ ( h ) r )
Fix g and h for the m o m e n t r ~ Q .
for
g,h ~ G
and take t~e p r o d u c t
Since N is an a b e l i a n
of (1.12)
group and the value
not depend on the m u l t i p l i c a t i o n
index,
= w(g)d(h).d(g)
d is a derivation.
C(~(n),r)
follows.
= ~(n)
for
1.15 D E F I N I T I O N . morphisms (?,a):
and
(1.13)
Let
a copair
is G 2 - h o m o m o r p h i c .
for all
for
i=1,2
define
(~,c)
~:(A1,~1~)
(~2,a2)o(~1,al)
which
and
the c a t e g o r y implies
Show that
of c o p a i r s
that
Opext
- (A2,~ 2)
Opext(~,c)
- Opext(G2,A2,@2 )
form a c a t e g o r y w i t h c o m p o s i t i o n
1.16 EXERCISE.
W e call
g ~ G2 , a E A I
For a c o p a i r
~IO72
.
be group homo-
is satisfied,
if
is d e f i n e d w h e n e v e r
=
= nq .
is a c o p a i r exactly,
0pext(GI,A1,~l ) - Opext(G2,A1,~1~)
does
Since
a: A I - A 2
if (1.13)
for all
d(g.h)
and
be G i - m o d u l e s
= go(a)
(~,a)
d~(n)
r ~ O .
of a p r o d u c t
the formula
, clearly
F: G 2 - G I
- (G2,A 2)
e(~(g)a)
Actually,
n ~ N
(Ai,~i)
(GI,AI)
Hence
and
Opext
a2°~ 1
,
are defined.
of sets.
can be made a f u n c t o r
The c o p a i r s
= (71o~2,a2o~I)
is a c o v a r i a n t
to the c a t e g o r y
= a.~*:
functor
from
( T h e o r e m 2.4 below
to the c a t e g o r y
of a b e l i a n
groups).
1.17 EXAMPLE. h ~ Q and
and let l=~a:
copalr.
G i v e n a group Q and a Q - m o d u l e i=~q~
~ha}:
For every
A - A
~h-lqh}:
Q - Q
The r e a d e r
e ~ Opext(Q,A,~)
(A,~)
.
F i x an
be the inner a u t o m o r p h i s m verifies
that
(i,1)
we have a c o m m u t a t i v e
is a diagram
18
where
e :
Ar
~G
))Q
e
A ;
)G
>.~Q
J=~g~
By Theorem Thus
:
~ k-lgkl:
1.10 t h i s
Opext(i,1)
G - G
implies
for
some
l~1[e]
is t h e i d e n t i t y
fixed
= i*[e]
map.
k ~ G in
with
wk = h
Opext(Q,A,~i)
.
.
Ig 2. The Exact Sequence
The main purpose sequence
(2.3),
of this section
associated
and a G-module A. cations.
for Opext
with a group extension
it was obtainable
for cohomology
had derived with spectral
LANE
Later sequence functors, [1; Vl
[2; Thm.
appli-
However,
of
[I]
in the spe-
this sequence was known to H2(Q,A)
just from the axioms
for derived
final stage being reached by HILTON/STAMMBACH
§10]
by group extensions. mology groups,
In Section
cohomology
application
sequence
above.
If A and B are subgroups
b e B .
particular, quotient
of G.
our presentation
with the
enables the
published
In particular,
else-
it allows
for
the Opext groups.
of a group G, let
If A and B are normal and
These notes
This transition
by all commutators
[G,G] ~ G
1.
(2.3) with applications
means of computing
generated
is
and the proof of exact-
of Section
5 we shall compare
in terms of group cohomology.
subgroup
(2.3)
can be used to obtain group theoretical
version mentioned
reader to connect
additional
to this sequence
all maps are very explicit,
will show how this sequence results.
approach
Then the terms are Opext rather than cono-
ness is a straightforward
and
important
13.1] and used to describe
We feel that the most natural
where
~Q
only as the specialization
techniques.
(2.3) was constructed
a certain
2 G
groups that HOCHSCHILD/SERRE
sequence
cial case that G is a free group, EILENBERG/MAC
N~
This sequence has found numerous
At first,
an exact sequence
is to derive the 5-term exact
Gab
[A,B]
denote
[a,b] = aba-lb -I
in G, then so is
:= G/[G,G]
the
with
[A,B]
is the commutator
aeA In
20
2.1 D E F I N I T I O N . ab(e)
For an e x t e n s i o n
and the c e n t r a l i z a t i o n e
:
N~
e
define
the a b e l i a n i z a t i o n
c(e) by the f o l l o w i n g
x
> G
lab
~
diagram
:~G
[l
lnat
(2.1) c(e) where
N/~-I[G,xN]~
the v e r t i c a l
horizontal An place
~-I[G,~N]
The e x t e n s i o n
sion
maps are n a t u r a l
ab(e)
the a c t i o n
c(e)
over
kernel;
e - eI
even central.
2.2 LEMMA. i.e.
factors u n i q u e l y
ab(el)
F is a free group and
is a Q - h o m o m o r p h i s m lently,
[G,N]
(~,p,?):
- ab(e2)
~F
if
as a
The extenof e x t e n s i o n s
eI
has an a b e l i a n
e - c(e)
if
eI
(ac,-,~)
: If
and
c(e) ~ c(e')
~Q
be a free p r e s e n t a t i o n
for every Q - e x t e n s i o n such that
Then
is
as in (1.4)
c(e) ~ c(ab(e))
R = Ker(F - Q)
f : Rab - N
in G.
eI - e2
and
Nab
e - eI
over
1.6 implies
eo: R ~
property:
exhibits
as in (2.1)
ab(e) ~ ab(e')
Let
the f o l l o w i n g
of G,
eo = ab(eo)
e of a n y e ~ f(eo )
(N,~) or,
has there
equiva-
[e] = f.[ab(eo) ]
PROOF. Then
e - ab(e)
Proposition
, clearly
kernel
C l e a r l y any m o r p h i s m
(~ab,.,~):
c(e 1) - c(e 2)
(We u s u a l l y write
from c o n j u g a t i o n
H e n c e any m o r p h i s m
i n d u c e s maps
;~ Q
, if the c o n t e x t permits.)
with a b e l i a n
is central.
the map
[G,~N]
is induced
factors u n i q u e l y
e ~ e'
or
N
maps onto factor g r o u p s and the
maps are i n d u c e d by × and ~.
of
Q-module,
x " ~ G/[G,xN]
Let
F - G
e: N ~
can be lifted over
obtain a m o r p h i s m this m o r p h i s m commutative
~ G ---@Q
(f',.,1):
can be f a c t o r e d
diagram
be any e x t e n s i o n w i t h N abelian. G--@Q
e° - e over
by the freeness
of extensions. e o - ab(eo)=e o
of F; we
As N is abelian, and yields a
21
eo :
e Then eo
Rab~
N;
by (1.5)
group
~(a,b)~
and
of
: Q - Q x Q
e ~ f(eo )
a O-module,
~a.b}:
N x N - N
that
[~
then the diagonal
is a group homomorphism.
resp.
Note
by 1.7.
e o , but not on e.
If O is a group,
)(q,q)l
abelian
~Q
only on the choice
2.3 NOTATION.
X7 =
"G
f is Q - h o m o m o r p h i c
~ql
O
I, 111
:
depends
A =
.~ F/[R,R] ~
If N is an
then the codiagonal is a h o m o m o r p h l s m
resp.
a Q-llnear
map.
Let Q be abelian extensions 7"
~e
for the moment.
and
e?
Ext(Q,N)
on the subset
notions
addition
our d e f i n i t i o n s
in 1.5 and 1.3 and of induced
in 1.6 and 1.4 agree
the c o r r e s p o n d i n g
Then
customarily
is defined
of abelian given
for
of induced
maps
s.
extensions Ext(Q,N)
and
with
.
In
by
[e I] + [e 2] = [~TN(e I ~ e2)~ Q] and makes
Ext
an additive
noncommutatlve for
i=1,2
again.
, let
abelian
G i v e n any
e I x e2
, G I × G2---@
(2.2)
e 1 + e 2 = ~ N ( ( e I x e2)AQ)
(N,e)
sense
for O - e x t e n s l o n s
and is again
2.4 THEOREM. Opext(Q,N,~) fined by
"
ei
[el] + [e2]
= [e I + e2]
i
extension
of one and the same Q-module
Let O be a group and an additive
) Gi---~O
Then
such an extension:
becomes
el: Ni:
the obvious
N 1 x N2~
makes
From now on, Q may be
extensions
denote
Q I x Q2
group.
it is called
(N,~)
abelian
a Q-module.
group
The class
the Baer
if a d d i t i o n [eo]
sum.
Then is de-
of the split
22 extension map
is the neutral
O: N - N
Q-linear
and all e.
maps
¥: QI " Q
element,
C,el,a2
the maps
and
The inverse
a. =
l[e]1
~* =
~[e]~
Opext(QI,N,e?)
are homomorphisms,
+
~2"
In~-~(n)-1}.
~
~Q
Then
where
obviously
allows
fi: Q x Q - N
the factor
system
as one of its factor
extensions
is induced
PROOF.
-
(~I + a2)* =
(N,e)
as in ( 1 . 1 ) ,
the h o m o m o r p h i s m (-1N,IG,IQ):
[2; Thm.
III.2.1,
We adapt
it to the present
e2 ~ e2
clearly
Thus
e - e
is a
(a I x ~2,.,I):
[eli
(~I x a2)(e
x e) and
~1 e + ~2 e = ~ ' ( ( a l e
~N(fl
and
al,a2:
literally.
el ~ el
.
x a2e)~ ) ~ ~ ( ~ ( ~ 1
and
' whence
N ~ N1
e - a2e
of extensions.
he ~ (e x e)~
systems.
the r e a s o n i n g
applies
To this
gives a m o r p h i s m
of extensions.
and
of
is well-deflned. for
is defined
after
x f2)~x Q =
of factor
e I x e 2 ~ el x e2
e - ale
, to
e I x e2
the a d d i t i o n
Congruences
+ [e2]
is a m o r p h i s m
Then
extensions,
2nd proof p.70]
e x e - (al e) x (a2e)
e - e x e
x (~2 e)
of
addition
i=1,2
By our remarks
has
Hence
of central
situation.
for
correspond.
systems.
~I e + a2e ~ (~I+~2)e
the direct p r o d u c t
(N,e)
e I + e2
imply a congruence
el + e2 ~ el + e2 Claim:
1.5,
case
of
fl x f2 "
by the p o i n t w i s e
In the special
of MAC LANE
(ale)
, because
systems
= fl + f2
that
of
denotes
be Q - e x t e n s i o n s
1.3 and D e f i n i t i o n
(~,~,~):
For
-
0pext(Q,N,~)
and m o r e o v e r
ei
Definition
end,
~ [e~]}:
-~
[e] = -[e]
Let
the factor
(a)
[(-IN)e ] .
of extensions.
On the sum. which
is
and group h o m o m o r p h l s m s
If e is a Q - e x t e n s i o n
~: N ~ -~ ~G
morphism
[e]
for the zero-
"
On the inverse. let
= [eo]
)[~e] I: Opext(Q,N,e)
and
el*
of
: N - (N1,el)
Opext(Q,N1,~1)
=
0.[e]
Moreover
By 1.7, we conclude
(~1 x a2)(e x e) Next P r o p o s i t i o n
x a 2 ) ( e x e)~A)
1.11
gives
23 ('~(~1 =
(~1
x ~2))((e × e)~)
+ ~2 )e
(b)
O.e ~ eo .
is commutative e
:
N)
eo:
of
~ (%~'(~1 x e 2 ) ~ ) e
~[0
N ~
(N,~)
fi: Rab ~ N
With
fl(g)
= (1,~(g))
and gives the assertion
)
(c) Commutative e2
× e2 ) ) ( ~ e )
G
~ N~Q
~
,
such that
The associative (d) Claim: let
~0
by 1.6:
~) Q
laws.
Let Q-extenslons
By 2.2, there are e i ~ fi(eo)
law can be proved
eo
eI
and
and homomorphisms
Hence by step (a),
e I + e2
~ "'" ~ e2 + el
in the same spirit.
~(e I + e2) ~ ~e I + ~e 2
e i ~ fi(eo)
following
~Q
and associative
be given.
, the
fl(eo ) + f2(eo ) ~ (fl + f2)(eo ) = (f2 + fl)(eo)
Again,
=
"
Claim:
diagram
~ (~(el
by 2.2.
and
(e I + e2)~ ~ el~ + e2~
Then by 1.11 and step a) above,
ael + ~e2 ~ ~fl(eo ) + ~f2(eo ) ~ (~fl + ~f2)eo = (°(fl + f2))eo ~ °((fl
+ f2)eo ) ~ ~(el + e2)
and
el~ + e2~ = (fl~o)~ + (f2eo)~
f1(eo ~) + f2(eo ~) ~ (fl + f2)(eo ~) ~ ((fl + f2)eo )~ ~ (el + e2)~ Steps without
(c) and (d) can also be proved by a suitable invoking
[2; Thm.
Lemma
III.2.1,
with caution
2.2.
2rid proof]
When adapting from
for the following
holds whenever
~e
is defined,
Ext
reason.
Opext
~(e~)
is defined and
situation
of formula
trivially
on N.
[]
~e
i.e. the quotient
is not;
e.g.
(21.2) cannot be defined
of MAC LANE
, one must proceed
The congruence
ates on the range of ~ such that ~ is Q-linear. that
diagram play,
the arguments
to
"
~(e~) ~ (ae)~
group Q in e operBut it may well be
~(e unless
× e)
in the
Q operates
24
2.5 COROLLARY.
Given a direct
il
P2
N1 (
~, N (
~N 2
Pl of Q-modules,
i2 i.e.
Plil=
Pli2 = 0 = P2il
, and
N e N I x N2)
Then
1
on
is a g a i n a direct
we have whence
sum diagram,
From Definition
P'
(N,~)
where
Pl* = 0 p e x t ( I Q ' P l )
etc.
1.11
2.&, and P r o p o s i t i o n
(co~). = ~.oP.
of order n.
N = N 1 x N2 . and
N2
Q-modules, Proposition
Let P be the set of p r i m e s
to
Since
N1
N 2 , actually
of 2.5.
and a u n i q u e Hall P ' - s u b g r o u p
N = N1 x N2
be the q-th p o w e r map. hence
so is
1.1& gives
every Q - e x t e n s i o n
of
of N must send as Q-modules.
q. = 0 , we c o n c l u d e
(N,@)
by E x a m p l e
in the n o t a t i o n
factor
Then
G i v e n an e x t e n s i o n
to
of
= 0 .
1.8 and c o n c l u d e
has a s e m i - d i r e c t
N2 ,
Let
that
isomorphic
N2 .
2.7 THEOREM.
q,
Since
0pext(Q,N2)
is an isomorphism,
ii.
NI
T h e n ~ is an a u t o m o r p h i s m
9.: 0 p e x t ( Q , N 2) - 0 p e x t ( Q , N 2)
We now i n t e r p r e t
dividing
T h e n N as an a b e l i a n group
every a u t o m o r p h i s m
11.: O p e x t ( Q , N 1) - O p e x t ( Q , N )
(A, ~)
,
G i v e n a finite group Q of order q and a finite
the set of the other primes.
q: N 2 - N 2
to
(This implies
:P2-~_Opext(Q,N2 ) i2-
1.5, T h e o r e m
has a u n i q u e Hall P - s u b g r o u p
N1
N2 ,
the assertion.
Q-module
and
on N.
on
1. = I , 0. = 0 , (~ + ~). = ~. + ~. , and
2.6 EXAMPLE.
and
N I , P2i2 = I
ilP 1 + i2P 2 = I
ii. Opext(Q,N1)(-----~0pext(Q,N) PI*
PROOF.
sum d i a g r a m
e as in (I. I) and a Q - m o d u l e
Thus
25 ~Der(Q,A,~)Der
N ~Der(G,A,eN)
o(e,A) ~ HomQ(Nab,A )
(2.3)
e*(e,A) is a natural where
Opext(Q,A,e)
exact sequence
p = p(e,A)
Q-linear map
~*
of abelian groups and homomorphisms,
is such that
In.[N,N]~
~ Opext(G,A,@~)
p(d)
for
~ d~(n) l , and
What does n a t u r a l i t y mean?
map
h: (A2,~2~)
from the data the data
le2,A 2}
ing term of assertion
- (AI,~1)
, say
is natural"
diagram.
If~
is the
~ f.[ab(e)]}.
(a,~,~):
(Ai,~i)
for
e I - e2
i=1,2
and a
Then we have a sequence
s I , and a similar sequence
and maps from each term of
sI ; altogether
"(2.3)
commutative diagram
{e1,A1~
e*(e,A) =
Given a m o r p h i s m
of extensions as in (1.4) and Qi-modules Q1-1inear
d ~ Der(G,A,~)
a laddershaped
s2
s2
(2.3) from
to the correspond-
diagram
s2 - s I .
The
means that this ladder always is a
The interesting
part of the assertion
is that
(2.4) be commutative: e*
*
(el 'A1 ) Opext(Q1 'A1 '~1 ) ~~1 Opext(G1
HOmQ I (Nlab,A1) (2.4)
Hom(~ab,h )
0pext(~,h)
0pext(~,h)
e* ( e 2 ,A 2) HOmQ2(N2ab,A 2 )
PROOF of 2.7. trivially
~=Der(nat):
derivation
N2 ~ 0pext(Q2,A2, • 2 )
Keep in mind that A is abelian and N operates
If
nat: G
G'=G/~[N,N]
Der(G',(A,~w'))
~ is a monomorphism. d: G - A
is constant
on the cosets
d~[N,N]
d(n-g) = d(n).nd(g)
n ~ ~[N,N]
and
is an isomorphism.
It suffices to show that every
Since
~II
is as in 2.1, then
Der(G,(A,~))
in the image of ~. =
; Opext(G2,A2, ~2~ 2 )
on A.
(a) Claim:
Obviously
'A1 ' ~1N1 )
d~: N - A
g~[N,N]
, hence is
is a homomorphism, = d(g)
for
g ~ G
and
26
(b) For the moment, (2.3)
consider
for the e x t e n s i o n
ab(e)
is c l e a r l y w e l l - d e f i n e d morphism.
It follows
d E Der(G',A)
and
from
(c) W e
and
Hom(Nab,A)
d ( g ) . g d ( g -1) = d(1)
g e G'
, that
p'(d)
which
is sequence
Here
0' = p(ab(e),A)
and is a homo= 1
for
is Q - h o m o m o r p h i c .
that the s e q u e n c e
(2.3')
The
is exact at
Der(G',A,~')
observe
that
0 is w e l l - d e f i n e d , Der(Q,A,e)
(2.3')
r a t h e r than e.
as a map to
reader now easily v e r i f i e s Der(Q,A,e)
sequence
and
Der(~)
= ~ Der(~')
and (b) i m p l i e s
Der(G,A,~)
e*(e,A)
and
~*
ness at
HomG(Nab,A )
that
and
sequence
Theorem
are h o m o m o r p h i s m s .
and
0 = p,~-I (2.3)
is exact at
2.4 i m m e d i a t e l y
We will
Opext(Q,A,~)
Hence
g i v e s that
show below the exact-
and the c o m m u t a t i v i t y
of
(2.4). (d) E x a c t n e s s is the class applied
at
of the
= Ker
= Ker
e*(ab(e),A)
~*e*(e,A)f
Let
ab(e)
(f) Claim: given morphism (~ab,-,?):
1.12.
ab(e) Imp
= Im(p'~ -1) = I m p '
as
=
f e H o m Q ( N a b , A ) , then by P r o p o s i t i o n
Thus
W*oe*(e,A)
~)Q
with
1.12 yields a map of
1.11 and
= 0 . Con-
~*[e I] = [e l q
5: G - G 1
with
= 0 ~I 5 = ~ .
5, then we have a m o r p h i s m
and h e n c e by 2.1 also
Now T h e o r e m
1.10 e x h i b i t s
e I ~ aab.ab(e)
.
The left square (~B,~):
1.13, w h e n
, can be p a r a p h r a s e d
If
~G 1
of e x t e n s i o n s
[el] = e * ( e , A ) ¢ a b
Opext(Q,A,~)
Hence Proposition
= f.(ab(e)~*)
(A,~)~
- eI
of
e*(e,A)
be the r e s t r i c t i o n
e - eI
The zero e l e m e n t
Opext(Q,A,~)
Proposition
(~ab,.,1): thus
at
el:
¢: N - A
(a,5,1):
= Ker
Hence
by P r o p o s i t i o n
let
be given.
.
= (f.ab(e))v*)
= O
versely,
extension
e*(ab(e),A)
(e) E x a c t n e s s
.
split extensions.
to the a b e l i a n
Imp'
ab(e)~*
HomQ(Nab,A)
of (2.4)
e I - e2
ab(e 1) - ab(e 2)
is commutative.
F r o m the
we o b t a i n a m o r p h i s m
by 2.1, and T h e o r e m
1.10 yields
,
27 aab(ab(el))
~ ab(e2) ~
e*(el,As)Hom(~ab,h)f h.~*[f ab(e2)]
For
f ~ HomQ2(N2ab,A 2)
thus
= (hf).[~abab(el) ] = (hf).y*[ab(e2) ] =
= 0pext(~,h)e*(e2,A2)f
.
(g) Claim: The right square of (2.4) is commutative. 0pext(~,h).~ *2 = h.~ * ~2* =
w~oOpext(~,h)
[-]
h.(
~2 ~ )* = h * ( ~ l
)*
* = h*~1~
*
Indeed, = ~I* h*7 * =
28 3. The Schur Multiplicator
and the Universal
Now we define the Schur multiplicator nite)
group Q and, likewise,
homomorphism
7-
We define
M(Q) = RO[F,F])/[R,F] tation of Q. (certain HOPF
M(Q)
, where
by the Schur-Hopf
RC
~F
~Q
This formula was discovered
[I] for arbitrary
Q in a quite different
its proof is Just
theorems
of elementary
(3.4) was known to SCHUR
form was first discovered
[I; Satz II,p.31].
spectral
sequence
But it was
cf. STALLINGS
this sequence with the exact Der-Hom-Opext
Theorem
topology.
(in dimension
two),
In the present
maps are given by explicit
se-
version of the Universal
a celebrated
formulation
constructions,
theorem
in
of this theorem,
all
while the proof is quite
and self-contained.
We wish to add some comments the definition
of
systems,
M(Q)
on the use of free presentations
We regard the free presentations
each e determining
For each pair of free presentations coordinate
Part
that its general
and its value appreciated,
(2.3) and prove a group-theoretical
coordinate
group theory.
[1].
We then combine
elementary
) G ~ Q
although
only via the Hochschild-Serre
algebraic
(~,~): N)
context.
is quite powerful,
the isomorphism
Coefficient
(topological)
M(~): M(G) - M ( ~
applying
quence
[2; p. IO1] for
in which
This theorem
and STAMMBACH
Formula
(3.4),
appears.
[1]
of a
of) finite groups Q and obtained by
a 5-term exact sequence
of sequence
M(~)
infi-
is any free presen-
by SCHUR
A first result is that every group extension determines
Theorem
of a (possibly
the Schur multiplicator M(Q)
free presentations
Coefficient
isomorphism
an abellan group
in
e as
M(Q) e .
e and ~, there is a unique
M(Q) e ~ M(Q)~
.
It will be a great advantage
29 that we may choose given problem. any logical
a free p r e s e n t a t i o n
We could have a v o i d e d
difficulties
each group Q, e.g.
the standard
the r e l a t i o n s h i p
between
cohomology treatment
M(Q)
Alternatively,
group.
This
of infinite
transformations
techniques
introduced
3.1LEMMA. to that (1.1)
There
of abelian
the abelian
(~,B,~):
eI " e2
subquotients ally,
PROOF. (a,~,~):
e I - e2
Then for every ~(g)
whence
extensions
morphism lifted
, we obtain
~: QI " G2
to a m o r p h i s m
m(a,S,~) definition
depends
for the
II.3.10.
anyway,
The
with the
]
e as in
of
induced
G i , for than
by B on the
i=1,2
.
Actu-
~.
As for the last assertion,
morphism is an
of extensions n(g) ~ ~2N2
~2N2/[~2N2,G2 ] [~g1,~g2]
of extensions
and to a m o r p h i s m
only on ? rather
Since
after
to the extension
the h o m o m o r p h i s m
g ~ G I , there
the assertion.
Given
assigns
m is a functor.
= ~(g).n(g)
G2/[~2N2,G2]
which
be a n o t h e r
as a certain
however,
m from the category
~iNi0[Gi,Gi]/[~iNi,Gi
Clearly
M(Q)
see
section.
~N0[G,G]/[~N,G]
depends
of groups,
not appropriate,
is a functor
groups,
of Q as in 3.3.
define
are easy to handle,
as in (1.4)
m(a,~,~)
for
Q, cf. the remark
in this
group
one free p r e s e n t a t i o n
and the d e f i c i e n c y
is mostly
coordinate
and
e.g. w h e n we wish to discuss
some authors
groups
suited to the
transformations
free p r e s e n t a t i o n
w o u l d be i n c o n v e n i e n t
IV.I.
coordinate
by d i s t i n g u i s h i n g
Such an a p p r o a c h
Section
particularly
with the same
?.
with
lies in the center
~ [sg1,~g2]
let
modulo
of
[~2N2,G2 ] ,
[]
el: Ni; •
If
~ Gi GI
(~,B,?):
on ? only,
~Qi
for
i=1,2
is a free group, e I - e2
of the Schur m u l t i p l i c a t o r
~ certainly
of extensions.
not on the choice
and a homo-
of ~.
looks artificial,
can be
By Lemma
3.1
The following at first,
a
30 coordinate-free Remark
but more abstract
characterization
is given in
3.10 (a).
3.2 DEFINITION.
If
e: R)
) F---~Q
is a free presentation
of
the group Q, i.e. an extension with a free group F and kernel R, then define the value of the Schur multlplicator abelian group ~: Q - Q' define ~.
M(Q) e = R n [F,F]/[R,F]
and free presentations = m(~,p,~)
M(~)ele,
M(Q)
at e as the
Given a homomorphism
e and e' of Q and Q', respectively,
for any choice
of lifting
(a,~,?)
of
Then
(3.1)
M(1Q)el e = 1
whenever
~1~2
and
M(~l~2)ele,
is defined.
, = M ( ~ l ) e , le,,
Free presentations
e as a coordinate
and the isomorphisms
as coordinate
every homomorphism
~ : Q - Q'
M(~): M(Q) - M(Q') Schur-Hopf
the obvious liftings.
M(~)
transformations.
Thus
homomorphism
for
M(Q)
is called the
The formulas
(3.1)
follow by using
M(1)ele,
The properties
are compatible
(3.1) mean that
of) groups to abelian presentation
groups,
for each group Q.
M(-)
is an isomorphism,
because
(3.1) also imply that the
with the coordinate
transfor-
is a functor
For example, ~: Q - G
let
map
G = N~ Q
(notation
group with operators
Then the composite
from (the category
once you have chosen a fixed free
product with splitting
but N may be a non-abellan Q-module.
M(Q)
[]
Equations
semidirect
The map
is its inverse.
various maps mations.
formula
exist.
Formula.
PROOF of implied properties.
M(1)e, le
system for
induces a well-defined
The defining
,
of Q certainly
We regard the free presentation M(1)ele,
o M(~2)ele,
be a
as in (1.3)),
Q rather than a
31
M(Q)is identity
M(~)
• M(G).
due to
direct product. HAEBICH
[2],
TAHARA
M(%)
M(Q)
~o.~ = IQ , thus
For details
M(G)
~ M(Q)
on the d e c o m p o s i t i o n
in view of P r o p o s i t i o n
x Ker M ( ~ O)
is a
of
see
5.5 also EVENS
M ( N ~ Q) [2] and
[I].
3.3 DEFINITION.
For a group Q, let the standard
free p r e s e n t a t i o n
of Q be
(3.2)
e(Q)
where Xq FQ
FQ
:
RG
PROOF. is free,
There
Also write
~: Q - Q'
induces
a morphlsm
with
e(Xq)
is an abvious
and another
extension
(3.3)
I q ~ Q , q ~ I }
with
abelian
(cf. STALLINGS
e as in (1.1)
M(G) M ( N ~ M ( Q )
then
M(Q)
free p r e s e n t a t i o n
F = Z
XI
for I in
N
= 0 .
with
R = 0
if Q is cyclic.
[I; p.172],
determines
e.(e).
and ~ maps
= X'~q •
If Q is free or cyclic,
3.5 P R O P O S I T I O N Every
I Xq
of ~.
e(Q) - e(Q')
3.4 LEMMA.
on
,
is the kernel
A homomorphism
(-,~,~):
; FQ---T-~Q
is the free group
onto q and .
RQ •
STAMMBACH
an exact
if Q
[]
[I; p.17o]).
sequence
a'~Ga b ~ a ~ Q a b
~0
,
[N,G] natUral
with
duced by
respect
to m o r p h l s m s
~: N - G .)
If
e ~ e'
is free and ~ an inclusion,
M(Q) e = (N Q [ F , F ] ) / [ N , F ] Recall assertion
that
N/[N,G]
implies
being defined
C
, then e.(e)
(Here
8.(e) = 8.(e')
~' .
is inIf
is Just the i n c l u s i o n
G = F map
;N/IN,F]
stands
e.(e~)
in 2.1.
then
of extensions.
for
N/×-I[~N,G]
= e.(e)M(~)
The h o m o m o r p h i s m
and
.
The n a t u r a l i t y
e.(~e) = ace.(e),
8.(e)
is sometimes
with
ec
called
32 the "homology
PROOF.
transgression".
Let
S ~ F
R = Ker(wp) ~ S . e" :
(3.4)
e'
:
e
:
0 ,G
be a free presentation
Thus we have a commutative
S ~
>F
1
0
of G and
diagram
;;G
II F
R :
N-
~
.~Q
~0
~ F
II
~G
in which the rows and the left column are short-exact. llne of the following of the pairs
diagram,
(numerator,
In the top
let all maps be induced by inclusions
denominator),
while the isomorphism
~'
is
induced by ~: S A [F.F]
R D [F,F]
R
IS,F]
[R,F]
S.[R,F]
II
II
M(~)e,,le,
M(G)e"
e.(e)
~ M(Q)e'
The left square
is commutative
to
hi ~ [N,G]
because
of (3.4), we define
as to make the right square commutative of
e.(e)
IF,F]
in case
N (S.[R,F])
Is M(.) = Ker implies
G = F
choices were.
coordinate
system for
by the modular law,
- Gab ) .
of extensions
diagrams
A similar argument
is independent
Since
R D ([F,F]-S)
Im 8.(e) = Ker (N/[N,G]
in e, since every morphism
8.(e)
N S)-[R,F]
Likewise
lation of the associated
The given description
free is immediate.
= (IF,F]
8.(e)
also.
= (R O [F,F]).S
Sequence
(3.3) is natural
can be extended to a trans-
(3.4), whatever
the arbitrary
shows that the definition
from the free presentation M(Q)
8.(e)
The remaining
e'
assertion
of
chosen as a is immediate
33 from naturality.
[]
3.6 ADDENDA.
If e is central,
each other and obtain
the exact
M(~) (3.3'7
M(G)
w i t h respect
lation
of P r o p o s i t i o n
this is not apparent presentation free,
e
for some rality
:
N~
~
suitable
of
of central
3.5 determines at first.
~O
"G
~Q
~ 0
,
extensions.
The formu-
completely,
e: R C
~ F---~Q
although be the free
shall be evaluated.
~ and a, p o s s i b l y
8. , w h e n a p p l i e d
, and N with
Since
F is
diagram
~F
maps
e.(e)
Let
M(Q)
we obtain a c o m m u t a t i v e R ~
, Nab
sequence
to m o r p h l s m s
of Q, at w h i c h
:
N/O
8.(e) ~ab ~ ~ab ~ N ~Uab----*~ab
~ M(Q)
natural
we confuse
to
not
(e,p,1):
surJectlve.
The natu-
e - e , exhibits
8.(e)
as the composite = (R n [F,F])/[R,F] " e . ( e ) ~ R / [ R , F ]
M(Q)~ Here
ec
is induced
3.7 LEMMA. extension.
Let
If
e.e*(e,A)~ Here
PROOF. one has
I qa'a -I
A/[A,G']
when applied
(A,e)
be a Q-module
= AQ
e - ~ ab(e)
= ~ce.(e)
i a ~ A
first that .
and
e: N~
is a Q-homomorphism,
= e.[~ ab(e)]
Note
(~oab,-,fl):
by e as in 2.1.
~: Nab - A
AQ = A /
a c .N/.-I[.N,G]
:
, a e Q
M(Q) I
yields
an
- N/[N,G]
- AQ
.
by definition.
By 2.1 and 1.5 there
to this morphlsm,
~Q
then
for every Q - e x t e n s l o n
of extensions.
~G
(A,m))
)G'
~Q
is a m o r p h i s m
The n a t u r a l i t y 8.[~ ab(e)]
of
= ~ce.(e)
e. , []
34 Recall abellan
that
Cext(Q,A)
extensions
3.8 U N I V E R S A L exact
of the abelian
COEFFICIENT
Ext(Qab,A)~
which
is split. and
and
Ext(Q,A)
the
group A by the group Q.
There
THEOREM.
is a natural
Let
ab*[e]
s: Q - E s = ~oab
• e.(e)~
a) Claim:
.
e: A#
b) Claim: ~F
is again
free abelian
O,
(3.7) with
~Q
an exact
of Q.
short-
N: Rab
e.Z(~)
I.fl2 gives
and since
M(F)
3.5.
2.4,
W is a
extension
= 0
, R/[R,F]
free abelian.
~R/[R,F]
- M(Q) induces
Consider
a lifting
s factors
as
.
B of a free abelian
group
by 3.4, P r o p o s i t i o n
3.5
~B
Thus
with
(3.7)
toe.(eo)
an i s o m o r p h i s m
the h o m o m o r p h i s m
- Cext(Q,A)
T h e n for
= e . [ e * ( e o , A ) ~ t ~ ] = ~te.(eo)
,0 is split by some homo= I ~* :
Hom(R/KR,F],A)
z=e*(eo,A)~*t*
~: M(Q) - A = ~
The natural
by 3.7,
:
we have thus
9.Z = fl
by 3.7 we have !
0.(¢.ab(e o) + ¢. ab(eo))
= (¢+¢')
e.(e o) =
! =
for Lemma
e.($.ab(eo))
+ ~.(~. ab(eo))
¢,¢' E H O m Q ( R a b , A ) 2.2,
e.
such
Fix a free p r e s e n t a t i o n
Since a subgroup
e.(e o)
t: R/[R,F]
Hom(M(Q),A)
be an abelian
epimorphlsm.
the natural
by P r o p o s i t i o n
Since E is abelian,
,M(Q)
HOmQ(Rab,A)
Q - Qab
By T h e o r e m
T: Qab ~ E
,
sequence
B = Im ~'
morphlsm
.
is a split
eo: R f
yields
~E - ~ Q a b
e is split by
e.
ab:
is defined
Then P r o p o s i t i o n
~s = ab
Thus
for
W is a monomorphism.
splits.
with
Hom(M(Q),A)
~Cext(Q,A) W[e] = ab*[e]
1[e],
homomorphism. that
~
Here
O. =
PROOF.
Again
the central
sequence
(3.6)
map,
denotes
Since
is a homomorphlsm.
e*(eo,A)
is surjectlve
by
map -
35
c) Claim:
8.~ = 0 .
extension.
The n a t u r a l l t y
(e)ab - e , gives = K e r ( A - Eab)
Ker
e.(e) = O . A"
eI :
A
is commutative. the exact ~
and
are n a t u r a l
Concerning
6: A - A' and
of
8.
extensions
of
Hom(M(Q),A)
A
by
then
PROOF.
Q
As
ab: Q ~ Qab an a b e l i a n
assertion
, i.e.
"
Q
A
Q
= 0
= ab*e.~b
e
in
be a b e l i a n
for
e e Cext(Q,A) and 1.11.
after Proposition
groups.
.
The
3.5.
All central
if, and only if,
extension
by the a b e l i a n
group
.
let us i d e n t i f y ~
Q
of the t h e o r e m extension.
3.8 e q u i v a l e n t
from the first one,
to
case
M(Q) = 0 .
[7
with
Qab
by
is Oust r e g a r d i n g
Thus all central
are a b e l l a n p r e c i s e l y
implies
for
~aboab = abo~
A
as a central
is by T h e o r e m
Hom(M(Q),M(Q))
= O
T h e n the map
follows
from
~.(e) = O
by 1.10.
= a e.(e) M(~)
and
is abelian,
by
with
to h o m o m o r p h l s m s
e.~*(ab*e)
are even a b e l i a n
M(Q)
extension
of
Q
exact since
with respect
If every central
is abellan,
O.~[e] = O .
[e] = ab*[e I]
8.(~.~*e)
Let
= 0 .
Im 8.(e) =
[el ~ Cext(Q,A)
has been e x p l a i n e d
3.9 COROLLARY.
the l a t t e r
Thus
~ , this follows
naturallty
:
* Qab
(3.3).
and
(1,.,ab)
.~Q
~ab
) Gab
~.
Now Thus
The b o t t o m row is short
e ~ Ext(Qab,A)
tensions
~
sequence
~: Q' ~ Q
Q
Given
to
the d i a g r a m ~ G
'
be an a b e l i a n
= e.(e) M(ab)
8. ~ Im W .
~
x
~E~Qab
since E is abelian.
Clearly
e :
e: A;
of (3.3), w h e n a p p l i e d
e.(¥[e])
= O
d) Claim:
e)
Let
when
~
8. = 0 A
:= M(Q)
ex-
is surJectlve, The second For then
36 3.10 REMARKS. A
.
Then
Coker
into itself. senting
is a functor
vlz.
M(Q)
definition
This feature
of
M(Q)
but would be rather abstract.
is quite interesting.
For this particular
G
Q ^ Q
3.8 as follows.
~Q
denotes
Q
an alternating
= x-1[gl,g2 ]
reap.
the exterior form"
abellan
Using factor
~ el
~~
~
given a central
billnear
form
extension
by
9: Q ^ Q - A
where
Q ® Q / ( q ® q I q ~ Q > •
vanishes
precisely
systems,
is a homomorphlsm
[ , ]
@: Q × Q - A
a homomorphism
Now the "commutator extension.
case, BAER [2] ob-
Then the commutator
square @
3.9 that the case of abellan
Suppose
with abelian
determines
~(~g1,~g2)
of abelian groups
could be used for a coor-
in Corollary
e: A; ~ ; G
=
from the category
(b) We noticed already
tained Theorem
in
W
Q , but vary the abellan group
Theorem 3.8 implies that this functor has a repre-
object,
dinate-free
Q
(a) Fix the group
when
e
is an
BAER [2] showed that
such that the following
sequence
is exact: (3.8)
0
~ Ext(Q,A)
Comparison
Q .
in WARFIELD
[fl], viz.
[I;
of
extension
by
e.(e)
M(Q) ~ Q ^ Q
commutator
form is
Baer's results are readily accessible
Q
is abelian.
group is absolutely
abelian.
.
§5].
Q
abelian groups
~0
We will give a direct proof of Miller's
(c) Let us call a group
Q
absolutely
abellan,
It is a standard
abelian.
if every central
fact that every
Corollary
with trivial multipllcator
3.9 implies:
All
are absolutely
Now a finite abelian group has trivial multipllcator
cisely when it is cyclic. abellan groups, groups.
^ Q,A)
in 4.7 below and show in 4.8 that Baer's
an interpretation
cyclic
? •Hom(Q
implies a theorem by C. MILLER
for abellan groups theorem
> Cext(Q,A)
e.g.
But there are plenty
the rationals
~
On the other hand, MOSKALENKO
pre-
of such infinite
and all divisible [I] has determined
torsion the
37 absolutely
abelian groups without appeal
basing himself on (3.8) instead. groups will be resumed
3.11 EXAMPLE. Coefficient dihedral
This theme of absolutely
in Examples
is not natural.
group of order 8, the inclusion
gives a commutative
Let
U ~ Z/2 map.
lab: Uab " Qab If Theorem
vanishes
3.8 admitted
ting compatible
with
the direct
also vanish.
a natural
i: U - Q
~: G----~Q
~-I(u) = [G,G]
e/U :
e • exhibits
res[e]
U , hence
Moreover, Ext(lab,1)
, then the middle terms Cext would such that the middle map hence
extension
res = i* res
would
e ~ Cext(Q,A)
does not split;
this means
G = Z / 8 ~ I/2
be the dihedral
group of order 16
be induced by an eplmorphism is cyclic
of order four.
~-I(u)
w
•
=
G
= O.
or at least some split-
elU
z/2c
Ker
M(i)* = 0 .
splitting-
We shall exhibit a central
To this end, let
Hom(M(Q),Z/2)
sum of the maps on the sides,
such that its restriction
and
thus
by the choice of
allow direct sum decompositions becomes
l M(i)*
• Cext(Q,Z/2) is cyclic,
A = 2/2
Hom(M(U),Z/2)
res
U
and
rows:
I
since
be the
subgroup,
3.8 with
Cext(U,Z/2)
Ext(Qab,Z/2) ~ M(U) = 0
its commutator
Then Theorem
Ext(lab,1) I
Now
in the Universal
Q = 2 / 4 ~ Z/2
diagram with spilt-exact
Ext(U,Z/2);
abelian
IV.4.8 and IV.6.17(b).
We show that the splitting
Theorem
i: U c---~Q
to the multlplicator,
.~U
w
= [elU] $ O by 1.8.
~
Q
Z/8 - Z/4 .
Thus the diagram
res $ 0 .
Then
38
3.12 D E F I N I T I O N . extension,
An extension
if it is c e n t r a l and
Proposition
e
as in (1.1)
is c a l l e d a stem
~N ~ [G,G]
3.5 i m p l i e s t h a t a c e n t r a l e x t e n s i o n
e.(e)
precisely when
is surOectlve.
e
is s t e m
Stem extensions will repeat-
edly o c c u r in the sequel b e c a u s e t h e y are very u s e f u l
for the c o m p u -
t a t i o n of S c h u r m u l t i p l l c a t o r s .
3.13 P R O P O S I T I O N .
Let
Q
be a g r o u p a n d
E v e r y c e n t r a l e x t e n s i o n c l a s s of a stem e x t e n s i o n
PROOF. class
)G
e.(e) = io~ i: N 1 ~ N
e.(e I) = ~ .
hence
.
Since
By 3.5,
Let
eI
e.(i.el)
= O
The n a t u r a l l t y of
e I - ~(e)
the i d e n t i t y of
U
in
G
extension
f a c t o r s as
an e p l m o r p h l s m
and
be any e x t e n s i o n
exists by T h e o r e m of
= 0 .
3.8 a n d is a
e. , w h e n a p p l i e d to
, the map
W e now invoke 8.
3.8 for
is i s o m o r p h i c a n d
e x t e n s i o n of
N
e.(e I)
Q
is i n d u c e d
e ~ Ext(Qab,N)
of e x t e n s i o n s ,
Since
by
where
e I - V(e)
eI
, we find
is stem.
, gives
is s u r J e c t l v e ,
we c o n c l u d e
e = v-lj.(el ) = O .
3.14 P R O P O S I T I O N .
of
e.(e): M(Q) - N
e. , w h e n a p p l i e d to
O e.(e I) = e.~(e I) = 0 .
U
P i c k any c e n t r a l
T h e n for an a r b i t r a r y
(0,-,1):
(b) Let
Ext(Qab,N)
as desired.
by a stem e x t e n s i o n .
Thus
.
= iT = e.(e)
(b) A s s u m e t h a t e v e r y c e n t r a l
J = O .
is i n d u c e d f o r w a r d from
e I e C e x t ( Q , N I)
Such
Ext(Qab,N)
e = i.e I
a morphlsm
= O
B y the n a t u r a l i t y
e I - i.e I , we h a v e := N
Q
~: M ( Q ) - - - - ~ N O [ G , G ] = N I
an inclusion.
stem extension.
A
Ext(Qab,N)
~ Q .
with
class w i t h
by
an a b e l i a n group.
(with f a c t o r Q) if, and only if,
(a) A s s u m e
e: N~
N
N
(a) E v e r y
M(G)
inner automorphism
G
induces
.
be a s u b g r o u p of the g r o u p , let
of
i: U - G
G
and
be the i n c l u s i o n map.
N
the n o r m a l l z e r T h e n the k e r n e l
3g
of
res=M(1): nm'm-1
M(U) - M(G) I m ~ M(U),
contains
n e N
I ;
in other words,
M(1)
of
is i n d u c e d by c o n j u g a t i o n
N
on
M(U)
PROOF.
(a) Let
M(G)
inner a u t o m o r p h l s m of
r[R,F]
(b) Let G
of
n E N
of
G
M(1) ~M(G)
M(U)
M(l)
m ~ M(U)
at
in
G
.
Here the a c t i o n
, restricted
e: R ~
Now
;F
M(~)
xrx-l[R,F]
~G
.
to
maps the t y p i c a l
= r[R,F] s
of
. U
and
)M(G)
Consequently,
and thus
U
Any
T h e n the d i a g r a m
by the functor p r o p e r t y
step.
- M(U) N
the a u t o m o r p h i s m s
G .
M(U)
the p r e v i o u s
x e F .
onto
determine in
M(U)
can be l i f t e d to an inner a u t o m o r p h l s m
by
M(G) e
by c o n j u g a t i o n
is c o m m u t a t i v e
for
~
over
be e v a l u a t e d
F , say to c o n j u g a t i o n
element
of
factors
nm.m-1E
of
res(nm) Ker(res)
M
.
Now
= resoM(s) []
M(8 ) = I
by
m = res(m)
.
40 4. The Ganea Map of Central
In the case
of a central
quence
can be extended
larged
sequence
theorem
Ganea
A ^ A
one step further
by the Ganea
An immediate
(a)
groups A.
Our d e s c r i p t i o n
commutator
and
[xy,z]
identities
[x,y]
(d)
[Zx,[y,z]].[Yz,[x,y]].[Xy,[z,x]]
(e)
X[z,[x-l,y]].Y[x,[y-l,z]].Z[y,[z-l,x]]
(a) If G
= Xy[x-l,y-1]
G
is g e n e r a t e d
A
and
A
,
, = I , = I .
be subgroups by
of the
due to P. Hall).
= [x,y].[x,z].[[z,x],y]
B
exte-
,
[y,x] -I = [x,y]
and
is a
= xyx-ly -1 = Xy.y-1
(c)
A
en-
it is e l e m e n t a r y
(mainly
= X[y,z].[x,z]
= [x,y].Y[x,z]
This
as the second
[I];
se-
application
for abellan
[x,yz]
Let
map.
M(A)
(b)
A.2 LEMMA.
exact
describes
Xy = xyx-1
4.1LEMMA.
the 5-term
[I] which
on various
Recall
and
B
of the group , then
G .
[A,B]
is normal
. (b) If
respectively, by
extension,
map follows E C K M A N N / H I L T O N / S T A M M B A C H
and rests
in
group
has many uses later.
of C. M I L L E R
rior power
Extensions
~ Is,t]
PROOF. In case gation [s-l,z]
B
are g e n e r a t e d
then the normal I s e S, t e T
Both assertions
(a),
it suffices
by elements = s-1[z,s]s
g E A and
I
closure
by the sets of
as a normal
easily
follow
to show that or
[A,B]
g e B .
S in
and G
T ,
is g e n e r a t e d
subgroup.
from Lemma
[A,B]
is closed
In case
[x,t -1] = t-l[t,x]t
4.1
.
(b), []
(a), under
(b). conju-
one also needs
41 4.3 T H R E E - S U B G R O U P S mal subgroup in
N
of
G .
, then so is
PROOF.
LEMMA. If
[A,[B,C]]
[C,[A,B]]
It is a special
provided
[c,[a,b]]
condition
follows
4.4 THEOREM.
~ N
Let A,B,C
be subgroups
and
[B,[C,A]]
and
N
a nor-
are c o n t a i n e d
.
case of Lemma 4.2
for a]l
[C,[A,B]]
(b) that
The latter
a e A, b e B, c e C
from the a s s u m p t i o n s
For every group
G
by &.l
~ N
(e).
with center
Z(G)
there
is a
homomorphism (4.1)
~G = X(G) : Gab ~ Z(G) - M(G)
called
the Ganea
(i) then
If
M(~).~G (ii)
If
® ~IZ(G))
~ ~ G--/-@Q
induced
by
~Z(G) ~ Z(H)
,
.
is a central
~ , there
x(e) ~ M ( G ) - -
with
is an exact
M(N) ~ M ( Q )
to morphisms
extension
and
sequence
,
of central
extensions.
Here
~(e)
as the composite
Gab ® N
I®i
Sequences which
= ~HO(~ab
with respect
is defined
properties:
is a group h o m o m o r p h i s m
Gab ® N
natural
with the following
~: G - H
e: N~
i: N - Z(G) (4 • 2)
map,
,
Ga b ® Z ( G ) _ X ( G )
(3.3')
and
(4.2)
~ M(G)
combine
has first been d i s c o v e r e d
.
to give a longer
by GANEA
exact
sequence
[I] from t o p o l o g i c a l
considerations.
PROOF x(e) X
(ECKMANN/HILTON/STAMM~CH
directly
is natural
~(G)
= ~(eG)
extension.
for every central with respect
, where Thus
[I]). group
extension
to morphisms.
eG: 2(G)~:
wG
(1) and the formula
Actually,
we first define e
and show that
We then define
natmG/Z(G)
is the obvious
~(e) = ~(G)(1®i)
are immediate.
42
We a g a i n use d i a g r a m [R,F] ~ S
, since
e
Define a set map Since
is central.
S [F,S]
the c o m m u t a t o r
identities
and
c(S × R) = I.[S,F]
= c(F x S)
r ~ R
c .
Cab
c(f,rl-r2)
is a h o m o m o r p h l s m All
told,
c
The i n d u c e d map ~(e)
as the c o m p o s i t e
so that
(4.2)
is exact.
(.,.,?):
presentation
of
(-,.,?)
special
e = ~
from the c o o r d i n a t e
theorem
and
X
g i v e n an a b e l i a n
to it.
This
A ® A
~(A)
x R) = I.[S,F] for each fixed
map
surjective. >M(Q)e, , ,
~ M(G)e, , , f ~ p-1(g),
of e x t e n s i o n s
and
(3.4) a free
some t r a n s l a t i o n ~ = M(?)e,,l~,, that
r~a-ICn).
of d i a g r a m s
• N . M
F r o m the
is i n d e p e n d e n t
group
A
and a p p l y the p r e c e d i n g x(A): A ® A " M(A)
to h o m o m o r p h l s m s
diagram
?: A " B
,
of a b e l i a n
is c o m m u t a t i v e
~M(A)
'®~i i M(') B ®B ~(B) ,MCB) Considering
(4.2)
clude that
M = ~(A)
=
Trivially
~[R,F]/[S,F] ~
yields a h o m o m o r p h l s m
the f o l l o w i n g
c(fl.f2,r)
e"
Suppose
i.e.
F/IS,F]
is given by
, whence
w h i c h is n a t u r a l w i t h r e s p e c t groups,
c([F,F]
? = 1 , we c o n c l u d e
system
become
is c e r t a i n l y
Gab ® N
e , then there exists
over
4.5
(a),(b)
in
, [R,F]/[S,F]
is a m o r p h i s m
(3.4) - (3.~) case
Thus
Now
= [f,r]-[S,F]
is c e n t r a l
a billnear
~ n) = [f,r].[S,F]
e - e
c(f,r)
in the first v a r i a b l e
Define
If
c
Gab @ N - [R,F]/[S,F]
~(g[C,a]
by
Moreover
× R/S
e .
Ker M(,) = [R,F]/[S,F]
of 4.1
of
= c(f,rl).c(f,r2)
determines
× N ~ F/([F,F]S)
C4.3)
Hence
, the image
= c(fl,r).c(f2,r)
since
as a "free p r e s e n t a t i o n "
c: F x R - [R,F]/[S,F]
[F,[F,R]]
Therefore
(3.4)
for the d e g e n e r a t e
extension
is an epimorphism.
If
A = A - 0 , we conM(A)
is e v a l u a t e d
at
43
the free p r e s e n t a t i o n for
f,g ~ F .
denotes
and
A2(A)
4.6 E X A M P L E S . compute
~A
A
but
Results:
(i)
T =
M ( Y / m x Z/n)
In the g r o u p
determinant
@ a
I a ~ A~
Note: W e w r i t e
multiplicatively.
then
A2A ~ M(A) group
~ 0 .
We now
on two g e n e r a -
stem extensions.
~ Z/n
In both cases ~o
ad(i).
Thus
, where
(A @ A ) / ( a
of any a b e l i a n
.
M(Z x 2) ~ Z ,
(ii) allowed.
suitable
a ~ A
M(A)
is cyclic,
the S c h u r m u l t i p l i c a t o r
tors by c o n s i d e r i n g
square
= [f,g].[R,F]
for all
is an isomorphism.
addltively,
If
M(wf@wg)
Xo: A ^ A - M(A)
the e x t e r i o r Mo
, then
x(a ® a) = 0
epimorphism
Our aim is to show that A @ B
~F.
In p a r t i c u l a r ,
induces a n a t u r a l A ^ A = A2A
Rc
whenever
nlm
, and
m = o
is
is an isomorphism.
SL(3,2)
3x3-matrices
with
a subgroup w i t h the m u l t i p l i c a t i o n
rule
One c o n s i d e r
of i n t e g r a l
the subset
(ii)
i T(j,k,1)=
i J,k,l ~ Z
0
This subset (4.4) thus
is a c t u a l l y
T(j,k,l)-T(J',k',l')
T(J,k,l) -I = T ( - J , - k , - l + k j )
(4.5)
[T(j',k',l'),T(J,k,1)]
The a s s i g n m e n t of
= T(j+J',k+k',l+l'+kJ')
T
onto
Moreover,
T(J,k,l)i
= T(O,O,k'j-J'k)
~ (j,k)
Z x 7 , with kernel IT,T]
the e x t e n s i o n
is central
e: IT,T] ~
in
obviously IT,T]
T .
r T --~
.
morphlsm
x Z) - M(Z × Z)
of that
2
2~A2(Z
is an isomorphism,
=
.
defines
~ T(O,O,I)
Applying
are
a homomorphism i 1 ~ Z
Proposition
1
3.5 to
Z x 2 , we o b t a i n an e p i m o r p h i s m
M(2 x 2 ) ~ [ T , T ] ~ 2 ~o:
As
and the c o m m u t a t o r s
;
^2(Z x X) ~ 2 , we have a n o t h e r Since
epi-
every e p i - e n d o m o r p h i s m
~o
is an isomorphism.
e.(e): M(Z x 2) - IT,T]
is an isomorphism.
This also implies
44
ad (ii).
Assume
nlm
.
The
is a normal
subgroup
of
T
, let
T(j,k,1) ~
~ (J+m2,k+n2)
Z/m x Z/n
, the kernel
([T,T].K)/K Now
A2(Z/m
extension
Z/n) ~ Z/n
e':
endomorphism Hence
Mo
.
0.(e')
map
zl
of
T'
onto
is
Applying
.
Proposition
~T'~Z/m
3.5 to the central
x 2/n
, we obtain
x ~/n)
an epi-
~ 2/n
.
are isomorphisms.
(C.MILLER
[1;Thm.3]).
Mo: A ^ A - M(A)
Our proof
I ~,~,~
The assignment
Z/n ~ A2(Z/m x Z/n) ~ M ( Z / m
and
PROOF.
.
~ T(~m,~,n,~n)
an e p i m o r p h i s m
n K) ~ Z/n
([T,T]-K)/K>
4.7 T H E O R E M natural
of which
K =
T' = T/K
induces
~ [T,T]/([T,T]
×
subset
For abelian
, as defined
is elementary,
groups
in 4.5,
though
A
the
is an isomorphism.
a direct
limit a r g u m e n t
r
can be recognized. Mo(A)x = I
in
e: R ~
0~A
~F
such that
Let
M(A)
be a free p r e s e n t a t i o n
p(gi)
= ai
and
r ~ [gi,hi]
F .
0(hi)
means,
in
A2(A)
with
x = 0 .
Let
of
Choose
= bi .
A
Now
that there are
.
~o(A)x = I
rj ~ R
and
gi,hi E F in fj ~ F with
s ~ [r~,fj] !I
=
i=1 in
~ (a i ^ bi) i=I
We wish to show
M(A) e = [F,F]/[R,F] (4.6)
x =
j=l
Let
F1
be the subgroup
gl,hi,fj
which appear
in (4.6).
generated
subgroup
A
of
of
F
Then
generated
by
A 1 := P(F1)
R
and the
is a finitely
, which has
PlF 1 ,A 1 eI :
R c
~ F1
~ A1 r
as a free presentation. than
A2A
.
Then
equation
(4.6)
i: A 1 •
~A
Let
Mo(A1)Y
lle in
denotes
FI
y =
= I
in
~ (a i ^ b i) i=I M(AI)el
But clearly
the inclusion.
Thus
in
A2AI
rather
since all terms
of
A2(i)y = x , where y = 0
implies
x = 0 ;
45 the problem figure
is reduced
A 1 ~ (tl> x ... x (t k)
We may assume whenever A2AI = 0
ti
both orders are finite
and
and
moment.
y = 0 ; done. integers
Let
z := A2(pr)y
we conclude
(See the
Fix
A t n)
Since
in
for all pairs
x ~
A2A
I I
~°(A)
T
^2(1)
(m,n)
Y
A2AI
A2
z ~
A2A2 )
Q
defined
abelian,
Let and
= M(pr)
and hence and
in 3.flO (b).
from 4.7.
PROOF.
Let
M(Q)
in
= I .
by &.6, A2(AI)
x = 0
.
follow.
---~M(A)
[]
~-;M(A2)
~ )G
A2Q - N
Recall
(f.nat,g,1): with
)
:
^2Q
eo: R c
shall
f: Rab - (N,O)
x,y ~ F
~o(A1)Y
~Q
be a central
be the commutator
the i s o m o r p n l s m
Xo(Q):
extension
form of A2(Q)
e , as
- M(Q)
Then
= O.(e).Mo(Q
which
for the
7"o(A1) ~ M ( A I )
e: N~
@:
A tj )
projection.
is an i s o m o r p h i s m
, y = 0
,
T
AI
4.8 PROPOSITION.
tj
I ~ m < n ~ k
Mo(A2)z
A2(A2 )
groups.
k = I , then
be the natural
and
of
~ ~ij(ti 1~i<J~k
with
Yo(A2)
^ tn) = I
If
y =
m,n
of cyclic
~o(A2 )
with
Aft .
the order
i < j .
Otherwise
.
= ~mn(tm
kmn(tm
divides
m A 2 : = ( t m ) × t t n>
below.)
this holds A
kij
pr: A I
(See the figure
l
groups
, a ,direct product
that the order of
for suitable
Since
generated
below.)
Now
Then
to finitely
.
~ F
M(Q)
~0
-. N
,Q
be evaluated.
such that
eo - e ~o(X)
.
be the free p r e s e n t a t i o n
By 2.2 there
e = f.ab(e o)
For any
= p
= q .
~o(y)
is a Q - h o m o m o r p h i s m
; thus we have a m o r p h i s m
of extensions. and
at
p,q e Q , choose Then
46 ¢(p A q) = ~ - 1 [ g ( x ) , g ( y ) ] On the other hand, and
e.(e)
Consider
given
tion a s s o c i a t e s :
a unique
pick f,g E G
[f,g] ~ K e r
w = ~N ; put
A
subgroup
be an a b e l i a n generated
inclusion.
Then
e.(e) M(1) For a proof, e.(elA)
by z
Xo(A)
:
A2(A) - M(A)
to
elA " e
" M(Q)
4.8 to
~Q
construc-
w(g) = y , then
x
and i: A "
y
as follows.
(e.g.
~ Q
the
be the
x ^ y e A2(A) - N
construc-
under
.
elA and use the formula
expresses
as in 1.8.
~ G
w i t h such pairs
of S c h u r m u l t i p l i c a t o r s
y ) and let
; the l a t t e r
N~
This u b i q u i t o u s
containing
e M(Q)
Hence
The f o l l o w i n g
and
is just the image of
apply Proposition
= e.(e)M(1)
with respect
and
e=(~,w):
z = ¢(x,y) ~ N
z = -1[f,g]
subgroup x
extension
w(f) = x
in terms
fc > N .
of ~ .
[]
xy = yx .
element
with
tion can be i n t e r p r e t e d Let
with
= [x,y].[R,F]
, R/[R,F]
= ~(p ^ q)
a central
x,y E Q
from the d e f i n i t i o n
Xo(Q)(pAq)
M(Q)[
= (f.nat)[x,y]
4.9 REMARK.
~x,yl
by 4.7 and 3.6,
is the c o m p o s i t e
e.(e)~o(Q}(p^q)
and assume
= (f.nat)[x,y]
the n a t u r a l l t y
of
8.
47 5. Compatibility with Other Approaches
We kept our treatment of group extensions and Schur multlplicators as elementary and self-contalned as possible; so far we did not treat cohomology and homology groups explicitly.
Most of the pre-
vious results appear in the literature in terms of (co)homology and this section gives the transition.
Since these other formulations
make use of resolutions, cocycles, and derived functors, the customary machinery is temporarily needed.
This section is rarely
used within these notes except for the following section, but will be necessary for most readers who want to do their own research. No originality is claimed.
At first, we are to distinguish three different meanings of the "Schur multlpllcator" of a group group
Q: M(Q)
as defined by the
Schur-Hopf formula 3.2, as the second homology group of a certain chain complex, and as the value of the functor by axioms. group
Likewise, if
Opext(Q,A,~)
(A,v)
Tor~(Z,2)
defined
is a Q-module, we deal with the
of extensions, and the second cohomology group
of some cochain complex, and the functor
Ext~(Z,(A,~))
We will
exhibit systems of natural Isomorphisms which translate our sequences (3.3) and (2.3) into the important (co)homologlcal sequences (5.~)
and
(5.5).
Above all, under these isomorphlsms, our formu-
lation 3.8 of the Universal Coefficient Theorem agrees with the one commonly known (in dimension 2).
Here you must keep in mind that the axiomatic (co)homology theory is not unique.
For example, assume given a sequence of Ext-functors
and connecting homomorphisms
wn
satisfying the axioms.
obtain an equivalent theory if you keep the functors
Ext
Then you but for
48 some
n
replace all
that an axiomatic
~n
map
by
a
- ~n "
Now consider the a s s e r t i o n
agrees with some explicit map
certain identifications.
Such a statement
invariant against the indicated
tion
slgn-changes.
Typically
Because
ment of the translation
of subtleties
llke these,
between the different
We are first headed for an i n t e r p r e t a t i o n of standard free presentations;
5.1 LEMMA.
free p r e s e n t a t i o n
generated by the elements ~ Jp,q
Let
as a basis of
as a system of coset representatives
5.2 and
R
condition.
I P,q e Q , q ~ I I for
RQ
By Theorem 2.7,
Here
Rab
RO
process in a simple
FG
and
mod RQ .
form a basis of
RO
is generated by
satis-
J1,q = I
[]
LANE [2; Thm.
Jp,q
T
elements of Now
Write
determines an exact sequence,
o(e'A),Homo(Rab,A)
T = S U ~X1=11
Clearly
Hence the n o n - u n l t y
p $ 1 $ q .
form due to EILENBERG/MAC Der(F,A,~)
is
p,q ~ O ; and
in the standard free p r e s e n t a t i o n e
RG
We use
Let O be a group and (A,~) a O-module. for
Then
e(O)
I P,q E O , p $ I $ q I
I q ~ O, q ~ I I
Jp,q $ 1
3.3.
Jp,q = Xp.Xq.X -Ipq for
S =
but
be a group and
as in D e f i n i t i o n
cf. ROTMAN [1; Lemma 11.9, p.244S.
Jp,q
is necessary.
of factor sets in terms
W e invoke the R e l d e m e l s t e r - S c h r e l e r
fies the S c h r e l e r
a thorough treat-
concepts
Q
setting, ~ Xq
and this
this has been known for a long time.
(MAC LANE [I; p.747]).
its standard
PROOF.
the identi-
must be made explicit before you can prove the asser-
a = ~ .
is free on
after
can be true only if it is
fications will depend on the connecting homomorphisms, relationship
~
F
for
FQ
e = e(Q)
of
Q .
in this special
13.1S:
e*(e,A)~opext(Q,A,~) := Jp,q.[R,R]
,0
.
and is free abelian
49 on
{ Jp,q
I P,q ~ Q , P ~ 1 ~ q I
satisfies
fl,q = fp,1 = 1
and is Q-llnear exactly,
(5.1)
Pfq,r'fp,q.r
where
fp,q := f(Jp,q)
those
f , for which a set function
= fp,q'fp.q,r .
for all
The image of
(5.2;
fp,q = g(p).pg(q).g(p.q)-I
exists.
If
I fp,q I
Every homomorphlsm
f: Rab " A
if
p,q,r e Q
o(e,A)
consists precisely of
g: Q - A for all
with
g(1) = 1
and
p,q e Q
f e HOmQ(Rab,A ) , then the extension
e*(e,A)f
has
as one of its factor systems.
We notice that conditions resp. principal
(5.1) and (5.2) specify factor sets
factor sets, as defined in 1.1.
tion, we now obtain the same correspondence extensions as in 1.1.
By the last asser-
between factor sets and
The reader may also consult GRUENBERG [1;§5.3]
for the treatment of extensions in terms of arbitrary free presentations.
PROOF.
Since
f: Rab - A
F
is free, we have
is Q-linear exactly,
= f(Xp.Jq,r.X;1.[R,R])
if
for all
Opext(F,A,~)
= 0 .
Now
Pf(Jq,r ) = f(PJq,r ) =
p,q,r
in
Q .
Condition
(5.1) is
introduced by the identity Xp.jq,r.X;1
(5.3) Let =
d
be a derivation
of
d(Xp).Pd(Xq).d(Xp.q)-I
onto
g(p).pg(q).g(p.q)-I
Conversely, f
= Jp,q'Jp.q,r .j-1 p,q.r
let
g: Q - A
be defined by (5.2).
there is a homomorphlsm clearly also
F and
in
d
of
F
in
f = 0(e,A)d
.
Finally,
F = FQ
(A,~)
, then
d(Xl) = 1
Thus
, where
g(r)
Since
F
~(e,A)d
:= d(X r)
is free on
h: F - ( A , ~ ) ~
(A,~) ~q,
F
with
for
I Xq
in
Jp,q
Q . and let
I q e Q , q ~ 1 ~,
h(Xq) = (g(q),Xq)
d(Xq) = g(q)
~Xq-[R,R]I:
r
maps
g(1) = 1
Thus by 1.9, with
"
d(Jp,q) =
be a set function with
h(Xl) = h(1) = (1,1)
derivation
in
h
defines a .
Q - F/JR,R]
Obviously is a set
,
50 section of the projection map in a factor system of has
ab(e)
~ fp,q = f(Jp,q)
5.3
If
}
and exhibits
f e HomQ(Rab,A ) , then
(~,~): N~
short for
(A,~)
resp. the G-module
Hn(Q,(Z,O))
as
8*(e,A)f
be given, let
~G-.----~Q
denote the integers with trivial operators, let
the Q-module
I Jp,q I
as one of its factor systems by 1.5.
Let an extension
(Z,0)
ab(e)
.
A
(A,~w) , while
denote HnQ
is
Under these assumptions, the (co)homology
theory of groups yields the exact sequences (5.4)
H2 G .
(5.5)
~H2Q
B ,N/IN,G]
O--~Der(Q,A)
> Der(G,A)
> Gab
> Qab"
)HomQ(Nab,A)
cf. HILTON/STAMMBACH [I; VI,(8.4) + (8.1)].
~0 ,
8, H2(Q,A)
) H2(G,A)
These sequences enjoy
the naturality properties of 5.4 (b) below.
5.4 REMARK. functors
The following proposition and its proof apply to all
H2G
and
H2(G,A)
actual constructions.
satisfying (a) - (c), regardless of the
(Method: so-called "abstract nonsense", no
chain complexes involved.) (a)
H2G
and
H2(G,A)
are covarlant functors from the category
of groups and the category of copairs (in the sense of 1.15), respectively, to the category of abellan groups. (b)
There are exact sequences (5.4) and (5.5) natural with
respect to morphisms of extensions; in addition, (5.5) is natural in the G-module
A
for fixed
G .
The unlabelled maps in (5.4) and
(5.5) are zero or identity for the degenerate extensions N
N (c)
H2F = 0
~0
and
Whenever and
0~ F
~ Q--
Q .
is a free group and
H2(F,A) = 0 .
A
an F-module, then
51 5.5 PROPOSITION.
For all groups
there are isomorphisms " H2(Q,(A,~)) ~
~
PROOF. e: R"
~
and
~
a) Definition
~F
~Q
Then both
of
e.(e)
of
translated into 45.4)
into 45.5).
~ .
The listed prop-
Choose any free presentation
Then
8(e)
R/JR,F] - Fab
M(F) = O
by 3.4 and also
are monomorphisms
M(Q) ~ Ker(R/[R,F]
a morphlsm
of extensions.
(3.3) and (5.4) are natural and and
~(e 2)
Q1 = Q2
and
a.,~(el)
presentation). = ~(Q2).M(~)
= e(e2).H2(~)
b)
The cohomology case.
moreover
Opext(F,A)
group splits. then define
~(e)
instead.
= 0
When
F
~(e)
~(Q)
both sequences maps, we obtain
as
~(free
H2(~)o~(Q 1) =
H2(F,A) = 0 ;
since any extension with free quotient
as the composite - HomQ(Rab,A))
(~,f):
with
~ .
e
of
8o8*(e,A)-1:
~ H2(Q,(A,~))
does not depend on the choice of Let
we conclude
does not depend on
is free then
Choose a free presentation
Coker(Der(F,A) that
~
Thus the above formula becomes of
and
H2(~).~(e 1) = ~(e2).M(~)
and define
; this is the naturality
i=1,2
Since both sequences
~ = 1 , we find that
the choice of the free presentation
for
is monomorphic,
a.: RII[R1,F 1] " R21[R2,F 2] , and finally Putting
- Fab) ~ H2Q;
is uniquely determined by 5.4 (b). el: RiC----,Fi----*Q i
e I - e2
H2F = 0 .
and we define
Now given free presentations
a..e.(el) = e.(e2).M(~)
-
Under the assumptions
(E,E,I,1,1)
as the composite
(s,.,~):
,
unique.
Q .
and
= ~(e)-l.e.(e) observe that
~: Opext(Q,A,~)
with respect to copalrs.
(2.3) by (1,1,1,~,~)
erties render
and
(A,~)
is natural with respect to group homo-
of 5.3, sequence (3.3) is by and sequence
and all Q-modules
~: M(Q) - H2Q
Here
morphisms and
Q
(Q1,A1) - (Q2,A2)
Q
as in a).
We
Opext(Q,A,~) It follows as above
e , write
~ = ~(Q,A)
be a copair map.
Since
(2.3) and (5.5) are natural with respect to copair H2(~,f).~(Q1,A1 ) = ~(Q2,A2).Opext(~,f)
follows as in the homology case that
~
translates
.
It
sequence
(2.3)
.
52 into sequence
(5.5)
5.6 ADDENDUM. A: Opext(Q,A)
- H2(Q,A)
A
agrees
is defined
, consider
; put
~ .
2.7 gives
- H2(Q,A)
by 5.5, we have
(MAC LANE
abellan
group
erators.
A
Then
sequence
Indeed,
e*(1 A) = e .
The c u s t o m a r y
Theorem
[1; VI.IO],
A(e) = e(1 A) e H2(Q,A)
w i t h our map
a biJectlon Given
(5.5)
for
e
and the
We claim that the map
the d e f i n i t i o n Since
8=~.e*:
of
8" = 8*(e,A)
HomQ(A,A)
of the Universal
[2; III T h e o r e m
for each
[7
-
A(e) = 8(I A) = ~e*(1 A) = ~(e)
formulation
, which
by this property.
as follows.
the exact
in T h e o r e m
5.7
determined
In H I L T O N / S T A M M B A C H
e e Opext(Q,A) G-module
and is u n i q u e l y
4.1]).
is regarded n e N
Given a group
as a G-module
there
Coefficient Q
and an
w i t h trivial
is a natural
short
op-
exact
sequence (5.6)
Ext(Hn_IQ,A):
where
HnQ
denotes
are defined resolution
Represent n-cycle
groups
with
Then
as follows.
= HnQ
is the map
) Kn_ I
i
I v
Then
8(z)
is the c o h o m o l o g y
~.bn:
K n - bnK n - A
.
The maps Let
~
a
and
be a Q-free
Hn(Hom(K,A)) and
~f(c)l
~Hn_lG
is free-abellan,
tl z Hn_IQ of the n - c o c y c l e
by an
Pick in
~E
diagram
of
= Hn(Q,A)
y ~ Hn(Q)
~ Hn_IQ
class
,
is a c h a i n - c o m p l e x
ez: A:
the following
(5.7) ; E
f ~y,
of (n-1)-cycles
bnK n >
A ;
and
extension
maps can be found as to render
ez :
@Q P
by an n - c o c y c l e
a(x)
Cn_ I
as in 5.3.
K = (Z,O)
Hn(K)
an abelian
the group
a ~Hom(HnQ,A)
homology
Then
x ~ Hn(Q,A)
z ~ Ext(Hn_lG,A) Since
integral
(Z,O)
c .
:Hn(Q,A)
via c h a i n - c o m p l e x e s
of
free abellan
B
-
some dotted
commutative:
.
53 5.8 PROPOSITION. ~: H2Q " M(Q) Q-module
There are natural isomorphlsms
P: HIQ " Qab '
~: 0pext(Q,A,~) -
for every
, and
(A,~)
H2(Q,(A,~))
such that, in the case of trivial group action on
A, the following diagram with (3.6) and (5.6) is commutative: Ext(Qab,A))
¥ ~ Cext(Q,A)
~I p*
~* m Hom(M(Q),A)
~!~
Ext(HIQ,A) )
~ )H2(
~I H°m(°'A) )
a
A~ Hom(H2Q,A)
This proposition provides a conceptual description of The interpretation of
a
ECKMANN/HILTON/STAMMBACH
PROOF.
We evaluate
e=e(Q): R" H2(Q,A)
~F
and
resolution part of
~Q
HIQ B(Q)
(notation as in 5.1 and 5.2).
b5
Ki
for
at the standard free presentation
H2Q
at the normalized inhomogeneous bar K(Q) = (Z,0) ®Q B(Q)
b2
• K2 i = 1,2,3
~Z
ments; let
[Xll...Ixl] = 0
xj
run through all non-unity group elewhenever some
boundary operators are given by (5.8.1)
b2[xly]
=
(5.8.2)
b3[xlylz]
[x]
and thus defines homomorphlsm
+ [y]
bI = 0 -
[x.y]
= [ylz]+[xly.z]
a) The definition of
~ 0
is the free abelian group on all i-tuples
, where the
[[x]i
The relevant
b1
• KI
[Xll...Ixl]
morphism
We compute
is
... ----~K 3 Here
8 .
[1; Thm. 2.2].
resp. at
K = K(Q)
and
seems to be "folklore", see also
M(Q)
and
a
xj = I .
and ,
- [x.ylz]
- [xly]
O , cf. MAC LANE [2; p.290].
)x.[Q,Q]I: Kl=Ker(b I) - Qab P: HIQ " Qab "
Qab " HIQ
Then the
sends
The homob2K 2
to
0
It is not difficult to construct a
inverse to
# .
If
f: Q - Q'
homomorphlsm, there is an obvious induced chain map
is a group
54 K(f): K(Q) " K(Q')
.
Inspection gives that
~
is compatible with
!
the pair
~ Kl(f)
' fab: Qab " Qab
b) The definition of those elements (5.9)
of
o .
Note that
~ nx,y[Xly ]
we allow
y = 1
By 5.2 a typical element of b = n (Jx,y) nx'y exactly,
if -I = Xx.Xy.Xxy l[xly]i
Now
Rab
in
F , this condition
abellan groups).
sends
b
to
.
nx, I = 0 = nl,y
.)
for some
I
Since
is equivalent
o(Ker
- Fab )
Jx,y =
to (5.9).
defines an isomorphism
Comparing
.
M(Q) = Ker(R/[R,F]
The above argument gives
= (R N [F,F])/[R,R]
x E Q\[I]
b-[R,R]
lies in
*F
K 2 - Rab
for all
is
R
)Jx,yl:
consists p r e c i s e l y
etc., but put
b.[R,F]
• Fab
Ker b 2
for which
D (n x ,~ , + ny,x - nxy-l,y) yEQ
(For convenience,
=
I
Clearly
(between free-
52) =
(5.8.2) with (5.3), we find
b[xlylz] = [Jy,z,Xx]'[R,R] and conclude morphism argument
~(b3K3)
= [R,F]/[R,R]
for the n a t u r a l l t y of • .
of
e I = 8*(e,A)f
for some
o
induces an iso-
f ~ HOmQ(Rab,A)
;fx,y}=f(Jx,y):
B2(Q) - A
~el!
~fx,yl):
B(f): ~(Q) - B(Q')
The
H2(Q,(A,~))
at
~ . ~(Q)
.
can be written as
Now 5.2 implies that
is a 2-cocycle which is uniquely Let
Opext(Q,A,~)
Since every group h o m o m o r p h l s m
.
is similar to that used for
e 3 e Opext(Q,A,m)
d e t e r m i n e d modulo 2-coboundarles.
map
~
Here we compute
By 2.2 every extension class
morphism
Hence
o: H2Q = (Ker b2)/b3K 3 - M(Q) = (R n [F,F])/[R,F]
c) The definition
I[xly]l
.
~
be the resulting
iso-
- K e r ( b 2 ) / I m ( b I) = H2(Q,(A,~)).
f: Q - Q'
induces an obvious chain
, the proof for the n a t u r a l l t y
of
~
with
respect to copair maps is straightforward. d) Claim: ab*(e 2)
with
~.¥ = 8.p* . ab: Q
) Qab
Given
e 2 ~ Ext(Qab,A)
the natural map.
, then
Y(e 2) =
By Lemma 2.2 we find
55 a Q-11near map
f: Rab - A
such that
ab*(e2) = f.[ab(e)]
, and we
have a commutative diagram ab(e)
Hence
:
Rab)
ab*(e 2) :
A ~
e2 :
A )
ToT(e2)
)F/JR,R]
.'; Q
~ g
~
:, E
~
~ Qab
"
is represented by the 2-cocycle
It is immediate that where
))Q
~, = - 1
.
= hg(Xx.[R,R])
0*(e) Define
Then
can be described as @: K1=Ker(~ I) - E
~'¢[Xx]
gram (5.7) and yields
e
sented by the 2-cocycle
~fx,y=f(Jx,y)}
= x[Q,Q]
by restriction. eb2: K 2 - A
Ar
by
, thus Now
~:E
w' :~HIQ
~[x] = ~
fits into dia-
8.p*(e)
which maps
.
Is repre-
[xly ]
onto
'-lhg(X'Xy'X; ) f(?x,y) =
e) Claim:
~.~ = H o m ( c , A ) . e . .
f ~ HomQ(Rab,A)
such that
wlth the coefficients ~(e I)
Given
e I = f.[ab(e)]
satisfying
= fc.e.(e)
{c,
Let
, flnd
c =x$1~'~'ynx'y[xly]'
(5.9), be a typical 2-cycle.
Is represented by the 2-cocycle
represented by
e I E Cext(Q,A)
• ~ nx,yfx,y }
ifx,yl
and
aoT(el)
On the other hand
by 3.7, while 3.6 exhibits
Hom(o,A)oe.
Then Is
8.(ei) =
as the composite
homomorphlsm f H2 Q ,
°
> M(Q) = (R n [ F , F ] ) / [ F , R ]
which maps the homology class of
c
; H/JR,F] also onto
we employed additive rather than multlpllcatlve
c > AQ = A ,
D nx,yfx,y notation
.
(Here
for the
sake of readability.)
5.9 REMARKS.
(a) From the proof of Proposition
the following description
of the isomorphism
5.8, we recall
56 : Evaluate
H2Q - M(Q) H2Q
by at the bar resolution and
free presentation
e(Q)
M(Q)
at the standard
A typical cycle has the form
c =
D n x v[x~y] , the coefficients nx,y being subject to (5.9) x,yeQ '~ and (without loss of generality) nx, 1 = 0 = nl,y Then ~ maps the homology class of
c
onto
nx'y n Jx,y -[RQ,FQ] x,y~Q (b) Assume given
M(Q)
x,y ~ Q
with
of 4.9 for the central extension
z = in
c(e(Q))
= [Xx,Xy].[RQ,F Q] =
M(Q)
, evaluated at
e(Q)
the image of the Ganea map (4.3).
xy = yx , then the construction
Now let also
yields the element
x,y.J
x.[RQ,FQ]
In particular,
~
elements generating
can be obtained in this fashion, cf.
e ~ Cext(Q,A)
with factor system
f
be given.
Then we conclude from Proposition 5.8, combined with the description of
e
in (a) and that of
~
in 5.7:
e.(e) z = f(x,y)'f(y,x) -I Finally, when
Q
is abelian and
e.(e)Xo(X^y) = f(x,y).f(y,x) -I In a different vein, the function by IWAHORI/MATSUMOTO
Xo = Xo(Q) for all
is as in 4.5, then x,y ~ Q
f(x,y)-f(y,x) -I
has been studied
[I; §1].
We now can infer results from the (co)homology theory of groups. For example, the direct limit argument asserts that homology (not cohomology) variable.
groups of groups respect direct limits also in the group This principle has long been known, a proof is spelled
out in BEYL [3]. position,
Of greatest interest to us is the following pro-
in the special case that a group
G
is regarded as the
directed union of its finitely generated subgroups.
57 5.10
PROPOSITION (Direct Limit Argument).
Let
wS:, G a - G~ I a _< ~ ~ I ~ be a direct system of groups over the directed set
I , let
G
be the direct limit group and
the canonical homomorphisms.
M(~a): M(G o) I M(~): In short,
- M(G)
Then
M(G)
~o: Go " G
together with
is the direct limit of the direct system
M(Q~) - M(G B) I o~
~
I I •
M(dir.lim. Go) = dir.lim. M(Go)
[7
58 6. Corestrlctlon
(Transfer)
This section aims at Propositions Schur multiplicator the multlpllcators these results
6.8 and 6.9, which relate the
of a finite group
G
to the order of
of the Sylow p-subgroups.
is self-contalned
G
and to
The formulation
and the remainder
of
of this section
may be omitted on the first reading.
For the proofs, morphic
to
functor
we use that the functor
H2(-,2 )
by Proposition
Cor2: H2G - H2U
needed properties
the paper by ECKMANN Cor I
the opportunity Eckmann's
Cor2: M(G) - M(U)
of finite groups.
is by dimension
(6.1)
to
U
U in
a
(At present,
is
ideas was that cf. 6.12.
as a "higher transfer".
co/homology
We use
in the same fashion. rather than the Tate
In the latter case,
and NOTATION. into the group
a common approach
Let
i: U c
~G
be the inclusion
G ; choose a right transversal
G , thus
1UXk = m
though usually
shift.)
6.1 ASSUMPTIONS
~Xkl
The
to present an approach which slightly differs from
(Note that we deal with ordinary
of a subgroup
is finite.
transfer homomorphism,
and treats homology and cohomology
cohomology
iso-
source on corestriction
One of his motivating
is dual to the classical
Thus we think of
IG:UI
are well-known,
The fundamental [I].
is naturally
5.8 and invoke the corestrictlon
, defined whenever
of corestriction
stated for cohomology.
M
x[ k=l
may be infinite.)
gives rise to the U-module "A can be considered
V(A,~)
Then every left module = (A,~oi)
as a U-module".
(A,~)
, what we express as:
(Modules are now written
59 addltlvely: G
and
U
O,+,etc.)
The integers
Z
are given trivial
action
by
.
Obviously
V
is an exact
m
m
functor.
As
(6.1)
implies
-I
• ZU.x k = ZG = @ x k -ZU k=l k=l V(ZG)
is U-free
is U-free
for every
We regard (6.2) and
Hn(U,VA)
(6.3)
(6.4) and
for
n ~ 0
A, likewise
Hn(U,VB )
as defined
= def
Ho(G,B )
as functors
from the category
of left
= Tor~(B,Z) def
identify
AG
P .
both
and t e r m i n o l o g y
We usually
V(P)
= Ext~(Z,A) def
Hn(G,B)
tation
free G-module
Hence
both
Hn(G,A)
G-modules
and
as a left as well as a right U-module.
B. -
is that of H I L T O N / S T A M M B A C H
H°(G,A)
~ a E A
with
for right G - m o d u l e s
Unexplained C1; chp.
no-
VII.
with
I ga = a
B G = B/B.IG
for all
g ~ G
I
as in 3.7, w i t h o u t
explicit
mention
of the i s o m o r p h i s m s H°(G,A)
~ HomG(Z,A ) - A G
Ho(G,B)
~ B ~G Z ~ B G
In each case,
the l e f t - h a n d
. isomorphism
tion of any G-free
resolution
~f~
f: 2 - A
for
~f(1)~
for
t ~ Z, b t e B .
of resp.
is induced
Z , while
by the a u g m e n t a -
the other
l ~ ( b t ~ t) J
isomorphism
# ~ ( t b t + B'IG) I
is
60 6.2 REMARK.
There are isomorphlsms
:
Hn(U,VB)
= Tor (VB,Z) - TOrn(B,ZG
:
Hn(U,VA)
n = EXtu(I,VA)
that are natural described
e U Z) ,
- Ext,( ZG e U Z,A)
in the G-modules
A
and
B
and for
n = 0
are
by the formulas
~ ( b + VB-IU) = b ~ (1 ~ 1) ~ B ~G (ZG % Z) resp.
~(a) =
Here
g ~ G
~g ~ t~
acts on
>g(ta)l
ZG ~U Z
(These maps are well-known,
~ HomG(ZG
by
~U Z,A)
g(a ® t) = (gx) ® t
cf. HILTON/STAMMBACH
for
[I; Prop.
x e ZG . IV.12.2;
Lemma VI.6.2] Take a G-free
resolution Z
resolution
of
6.1.
~-I
Now
B %
P
ZG e U ~
and
Z ; then
@U V£,A)
The connecting
homomorphisms
J~B"
complexes.
of
ZG ~
Z
by
isomorphisms
~ HOmu(VP,VA)
This description
Wn(e)
are respected,
is a U-free
% V~ ~ VB % VP ,
~ HOmu(VP,HOmG(ZG,A))
cochaln
V(P)
resolution
is induced by the standard
of chain resp.
~B
of
a G-free
( z a % V~) ~ (B % z a )
H°mG(ZG
e: B ~
~ 2
also implies:
of any exact sequence
i.e. the diagrams
• n (e)
.~ Hn_I(U,VB' )
Hn(U,VB" ) (6.5)
G . Torn(B ,ZG ~ are commutative;
similarly
6.3 DEFINITION. define
restriction
Res n = Hn(i,B) Hn(U'VB)
~n(e) Z)
With
G ) T O r n _ I ( B ' , Z G e U 7)
in cohomology.
¢ =
~ ~3 g ~ tgl
~
tg
I :
as = Torn(B,¢)o
G " T°rn(B'ZG
~ :
G Z @U Z) - Torn(B , ) = Hn(G,B)
,
ZG ~ j Z -', Z
61 Res n = Hn(i,A)
= ~-l.Extn(z,A)
Hn(G,A) = Ext~(Z,A)
- Ext;(ZG @U Z,A) - Hn(U,VA)
One easily verifies that inclusion
AGc
:
~(VA) U .
¢
is G-linear and
Moreover,
map agrees with the "more obvious" and HILTON/STAMMBACH
6.4 LEMMA.
~A =
ones as in MAC LANE [2; p.1161
m = [G:U[ < = .
m -1 ,k~__lXk= ® xk.a~
ral in the left G-module A. g(x ~ a) = (gx @ a)
:
A-
for
for
Here
x E ZG
The first assertion
(uxk)-I
Then the map
ZG @U A
g ~ G and
to
U
in
Note that
G , for
of transversal,
g ~ G .
acts on
2G ~U A
and natuby
a E A .
@ UXk-a = Xk I ® xk.a
u ~ U , the naturallty is obvious. is G-linear.
is G-llnear,
follows from
® (ux k)'a = xklu-1
~A
Just is the
this definition of a restriction
does not depend on the choice of transversal,
PROOF.
Res °
[ 1 ; p. 1901.
Assume
~a|
.
We are left to show that
lxkg ~
is again a (right) transversal
Since
~A
is independent
of the choice
we conclude m (xkg)-
g'~A(a) = g • D k=l
= ~
1
® (xkg)'a
-1
k__lXk 6.5 DEFINITION. define corestrlction
® x k.(g.a) = ~A(ga) Assume
IG:UI < - •
Wlth
~ = ~z:Z - ZG @U Z
as
C°rn--~-l°T°rn (B'~): Hn (G'B)=T°rG(B'Z)
" T°rG(B'ZG
~U Z) - H n ( U , V B )
corn=Ext( ~,A ). ~ : Hn(U,VA) - Ext;( ZG @U Z ,A ) - Ext;( Z,A )=Hn(G,A)
.
,
62 This description puts into evidence that
~Corn}
and
respect connecting homomorphlsms in analogy with (6.5). formulas for
Coro: B G - (VB)u
and
Cor°:
m = D b'Xk I + VB-IU k=l
(6.6)
Coro(b+B'IG)
(6.7)
Cor°(a) =kL__m~lX~l.a for
~cornl Tracing the
(VA) U - A G , one finds
,
a ~ (VA) U
(If, in the context of finite groups, homology groups are identified with Tate cohomology groups of negative degree, then our homology restriction is usually called corestrlction and vlce-versa.)
6.6 THEOREM.
Assume that
the multiplication by and B
m
m = IG:UI
is finite. Let
in any abellan group.
Let A
m.
denote
be a left
a right G-module.
a)
Resn.Cor n = m. :
b)
Cor n . Res n = m. :
PROOF.
Hn(G,B) - Hn(G,B) Hn(G,A) - Hn(G,A)
We first invoke the notation of 6.3 and 6.5 and find that
¢'~=m.: Z - Z .
In the homology case, we have
Resn.Cor n = Torn(B,e).~.~-1.Torn(B,~) = Torn(B,t.9 ) = Torn(B,m.) = m . . In the cohomology case, we likewise conclude corn.Res n = Extn(9,A).~.~-l.Extn(¢,A) = Extn(e.9,A)
6.7 COROLLARY. and
Hn(G,B )
left G-modules
PROOF.
If
G
= Extn(m.,A) = m . .
is finite of o r d e r
have exponent dividing A
IGI , then
JGI , for all
and all right G-modules
Apply Theorem 6.6 for
[]
n > 0
Hn(G,A) and all
B .
U = 0 , thus
m = IGI
Note
63 Hn(U,VA) = 0 = Hn(U,VB)
IG[. =
for
n > 0 .
ResnoCor n = 0 :
6.8 PROPOSITION.
If
G
Consequently
Hn(G,B) - Hn(G,B)
is a finite group, then
finite abelian group of exponent dividing
PROOF.
, etc.
As the bar resolution of
degree two, the homology group
G
M(G)
IG I
is finitely generated in
H2(G,Z )
is finitely generated.
Corollary 6.7, this group is finite of exponent dividing Finally
M(G) ~ H2(G,Z)
V ~
6.9 PROPOSITION.
proof of this proposition and a
is an exponent of
Let
G
p-subgroup for some prime is the Sylow p-subgroup
U := P
M(G)
.
be a finite group and p .
M(G)p
Then the image of of
M(G)
isomorphic to a direct summand of
Put
~G I .
due to SCHUR [I], will be given with Corollary
II.3.10: Roughly,
PROOF.
, then
.
P
be a Sylow
res: M(P) - M(G)
Moreover,
M(G)p
is
M(P)
p
does not divide
m = IG:P~ •
Proposition 5.8 there is a commutative diagram
M(G)
c
-='T°
Cor2
H2G where
(6.8) Now
~ M(P)
~M(Q)
~l ° H2(i,Z) • H2G ~-I° Recall
res = M(i)
We invoke Theorem 6.6 for C.M(i) = m. :
M(P)
M(i)
~ H2P
C := a.Cor2.o -1
3.14 (b).
n = 2
as in Proposition to conclude
M(G) " M(G)
is a finite p-group by Proposition 6.8, thus
Im M(i) S M(G)p
By
by Proposition 5.8.
A representation-theoretlc sharper estimate,
is a
and
By
64
R.C, = m. : M ( a ) p ~ MC' (P) where
C'
and
R
respectively. tion by and of
m
R :M(a)p
are the obvious restrictions of
As
M(G)p
Ker R = Ker M(i)
6.10 PROPOSITION, finite index m.M(G)
m =
Moreover, and
M(U)
,
.
Hence
M(P)
is the internal direct sum
I x ~ M(G)
R
is surjective
.
cf. JONES/¥IEGOLD [I].
If
U
is a subgroup of
G , then I
is isomorphic to a subquotient of Note: If
M(i)
M(G)p
Im C' ~- M(G)p
in the group I xm
and
is a finite abelian p-group, multiplica-
is an automorphlsm of
Im M(i) = M(G)p
C
M(U)
.
is a finite group, then any subquotient of
M(U)
is isomorphic to a subgroup of it.
PROOF.
By Theorem 6.6 (together with Proposition 5.8), there is
a homomorphism
C: M(G) - M(U)
resoC = m. :
M(G) - M(G)
such that .
Hence m.M(G) = Im(res.C) S Im(res) ~ M(U)/Ker(res)
.
~]
For practical purposes we are going to describe the corestriction in terms of cycles and cocycles, with respect to the (normalized) bar resolution
on homogeneous generators
-I -I (go,gl,...,gn) = go[g ° g11g~Ig21...Ign_lgn ]
(6.9) and
B(G) = IBn(G)~
B(U)
.
First, the naturallty assertion of Lemma 6.4 gives that
I ~ n ( G ) : Bn(a) - za % B n ( G ) is a chain transformation lifting
I ~: Z - ZG @U 2 .
presses chains and cochalns at the resolution
B(G)
Next
~
ex-
, considered as
65 a U-free resolution. While
B(1):
opposite
B(U) - B(G)
direction
(6.1) to
U
the coset boundary
Third,
in Ug
B(G)
is a U-llnear
is needed.
G .
Let
, then
~
To this end, denote
g = (g.~-1)~
bn(go,g I .... ,gn) =
B(U)
I Z , a map in the
fix the transversal
the chosen representative with
.
g.(~)-1
e U .
of
Due to the
~ (-1)J(go,...,~i,...,gn) J=1
- where the roof means deletion ~(g) = g(~)-1
of the term under it, the projec-
onto the U-component
transformation
B(,):
steps together,
we obtain the following
6.11
llft of
with
formula
(6.10)
tion
is to be compared
PROPOSITION.
transversal
~Xk}
of the coset module.
B(U)
Assume
that
over
Assume
that
Corn: Hn(G,B)
IZ .
m = IG:UI
as in (6.1) and let
Ug .
Then
B(G)~
induces a U-llnear
A
g
Putting
Fix a
the representative
is a left and
~ Hn(U , VB)
the three
is finite.
denote
chain
B
a right G-
is induced by the chain
transformation
B ~
I ~n :
on generators
Bn(G) - VB @U Bn(U)
~ '
given by m
~n (b ® (go .... 'gn )) = while
corn: Hn(U,VA)
-1
~ b'x-1 k=l k
- Hn(G,A)
® (Xkgo'Xkgo
-1
"
' ""
'Xkgn'Xkgn
is induced by the cochain
)
'
trans-
formation I n
:
HOmu(Bn(U),VA)
, HomG(Bn(G),A )
with -1
m
"n(f)(go' .... gn) = k=1 6.12 COROLLARY. ~.COrl.0-1:
Xk-1"f(Xkgo'Xkg o
(a) W i t h
Gab - Uab
~: HIG ~_ Gab
is given by
-1
..... Xkgn'Xkg n
as in Proposition
)
[]
5.8,
66 m
gIG,G]' (b)
-I
, IZ xkg-xkg k=l
-[U,U]
Cor2: H2(U,VA) - H2(G,A)
f: U x U - VA
I
maps the class of the factor set
onto the class of the factor set _
,2(f)[xly ] =
-I
~ x k l"f[xkX'XkX k=1
-I
IXkX'y'xkxY
] •
The formula of (a) describes the transfer homomorphism (Verlagerung)
~
of Burnside and Schur, cf. 2ASSENHAUS [I; p.167] and
HUPPERT [I; IV.1.4 (b)].
PROOF. 6.11 for tors
This amounts to rewriting the formulas of Proposition n = I
and
[g] = (1,g)
Note that the
[x]
n = 2
and
in terms of the inhomogeneous genera-
[xly ] = (1,x,xy)
of the bar resolution.
in Proposition 5.8 here reads as
CHAPTER
II.
SCHUR'S THEORY OF PROJECTIVE
1. Projective
Representations
Throughout
this section let K be a field, K* its multipllcatlve
group, V a vector
space over K of arbitrary
denotes the group of linear automorphisms
1.1 DEFINITION. the vector
REPRESENTATIONS
A projective
dimension,
while GL(V)
of V.
representation
of a group Q over
space V is a map
P : Q - GL(V) which
satisfies
P(g) P(h) = P(gh) for all
g,h ~ Q.
m(g,h),
The map
w(g,h) ~ K* ~: QxQ - K*
is called the correspond-
ing factor system.
Quite often projective irreducible
linear representations.
is algebraically
closed,
linear representation If N is a central onto dilatations
and let
g ~ G
§49,
be an irreducible
P: G/N - GL(V)
Let T be a transversal
defined by
A more general
between representations
§51], and HUPPERT
of N are mapped
has a unique decomposition
of finite index is contained [I;
with
for the moment that K
3: G - GL(V)
lemma.
representation.
the connection
Assume
in connection
of G, then the elements
by Schur's
t ~ T , and the map
appear
of a group G, where V is finite dimensional.
subgroup
Then each element
a projective
representations
g = nt
[1],
, n e N,
:= ~(t)
is
concept which describes
of groups and normal
in CLIFFORD
[I; V §17].
P(tN)
to N in G.
subgroups
see also CURTIS/REINER
68 Now we return to the projective representation given in Definition 1.1. The assoclatlvlty of the multiplication in GL(V) easily yields the relation I.(1.6) for w, where K* is regarded as a Q-module with trivial action.
The factor system is normalized in the sense of
1.1.1, if and only if P maps the identity element of Q onto the identity of V.
1.2 DEFINITION.
Let V1, V 2 be vector spaces over K, and PI' P2
projective representations of Q over V1, resp. V 2 with corresponding factor systems ~1' w2" (i) P1 and P2 are called (proJectively) exists an isomorphism
equivalent,
~: V 1 - V 2 and a map
if there
c: Q ~ K*
which
satisfy e Pl(g) = c(g) P2(g ) ~ , g e Q . (ll) If the preceding condition can be satisfied with for all
c(g)
= I
g ~ Q , then P1 and P2 are called linearly equivalent.
Assume that PI and P2 are equivalent.
Then an easy calculation
shows (1.1)
~l(g,h) c(gh) = w2(g,h) c(g) c(h)
, g,h ~ O ,
i.e. the corresponding factor systems differ by a principal one, cf. 1.1.1, and we call them equivalent as well.
Linearly equivalent
projective representations have the same factor system.
On the other hand, if w I and ~2 are equivalent factor systems satisfying (1.1), then P~----~cP is a one-to-one correspondence between the projective representations of Q with factor system ~I and those with factor system ~2' respect-
69 ing projective and linear equivalence.
Assume that c'. c -1
c,c': Q - K*
is a homomorphlsm,
ter of Q.
are two maps satisfying (1.1).
in other words a
system
one-dlmenslonal charac-
Now we fix two equivalent factor systems ~1 and ~2 and a
map c, leading to the above correspondence tations.
Then
for projective represen-
Assume that Pi is a projective representation with factor ~i ' i = 1,2 , and let Pl and P2 be equivalent.
there exists a map equivalent.
c': Q - K*
Then by 1.1
such that c'P 1 and P2 are linearly
Furthermore we have
c'P 1 = (c'c-1)(cP1)
and
( c ' c - 1 ) ~ Hom(O,K*)
Hence, t h e c l a s s e s under p r o j e c t i v e
r e p r e s e n t a t i o n s , whose f a c t o r
.
e q u i v a l e n c e o f those p r o j e c t i v e
systems belong t o a f i x e d
equivalence
class represented by a factor system ~, are given by the orbits of the character group Hom(Q,K*) acting by multiplication on the classes with respect to linear equivalence of projective representations with factor system w.
1.3 REMARK.
Beside Definition 1.1 there are two other ways de-
scribing projective representations. (1) One of them uses the notion of the twisted group algebra. For each factor system ~ of Q over K we can consider the set of all finite formal sums plication
~ agg , ag E K , g E Q
endowed with the multi-
o generated by
g.h = ~(g,h)gh
.
This yields an associative K-algebra
(KQ) w of dimension
IQI, called
the twisted group algebra, and it easily follows that there is a one-to-one correspondence between its modules and the projective representations of Q with factor system w.
The isomorphism of modules
corresponds to the linear equivalence of projective representations.
70 (ll) For each K - v e c t o r
space V we consider
the following
central
extension: 8
(1.2)
ov
where
8(k)
:
K* ~
is the dilatation
of 6 is the center is d e f i n e d Let ¥=~P:
P: Q " GL(V)
~: Q - PGL(V) TP = ~
, for all
be a p r o j e c t i v e
is a homomorphlsm,
~2 equivalent.
any map
of Q over V.
lence,
' w2P2 = ~2
in p r o j e c t i v e
we can regard
projective
and h o m o m o r p h i s m s Jective
representations
tlon of Q.
Then
ment a s s o c i a t e d
extension (1,8,~):
Let
with
that
OVa: K*~ CV~ - o V
if satisfying
Let exists
, then we call
~I and
representations equivalent,
if
if we are solely
up to p r o j e c t i v e
equiva-
into projective
do not strictly
(in the sense groups,
a K-
groups.
distinguish
of D e f i n i t i o n
w h i c h will
1.1)
be called pro-
as well.
~*(OV)
Let us recall
we usually
into p r o j e c t i v e
1.4 DEFINITION.
Hence,
them as h o m o m o r p h i s m s
representations
Then
isomorphism
are p r o J e c t l v e l y
representations
In view of the preceding, between
If there
B.~ 1 = ~2~.
and only if ~I and ~2 are equivalent. interested
~.
P: Q - GL(V)
It is easy to see that p r o j e c t i v e ~1PI = ~I
group PGL(V)
projection
such that the induced satisfies
linear
The image
On the other hand,
be homomorphisms.
8: V I " V 2
.
representation.
representation
, i=1,2
8.: PGL(V 1) " PGL(V 2)
k e K*
, and the p r o j e c t i v e
is a homomorphlsm.
~i: Q " PGL(Vi)
PI' P2 w i t h
kv
group with canonical
is a p r o j e c t i v e
isomorphism
v:
of GL(V)
as the factor
Q " PGL(V)
"~ PGL(V)
> GL(V)
~: Q - PGL(V) ~ Cext(Q,K*)
be a p r o j e c t i v e is called
representa-
the c o h o m o l o g y
ele-
~.
~*(OV) ~ G
is r e p r e s e n t e d
~ Q , which
of extensions
by the b a c k w a r d
induced
yields a m o r p h i s m
as in (1.3)
and is d e t e r m i n e d
by
71 cf. Section
this property up to congruence, X K*>-----~ G
p
Ov~ :
ov
K* >
= PGL(V)
~
1.1:
Q
(1.3) :
)GL(V)
1.5 PROPOSITION.
Let
tive representation),
?: Q - PGL(V)
and
let ~ denote the factor represented
PROOF.
P: Q - GL(V)
system of P.
by • corresponds
a map with
(projec-
TP = ~ , and
Then the element of H(Q,K*,O)
to ?*(aV),
Let G denote the Cartesian
be a homomorphlsm
cf. I.I.1.
product
of K* and Q with the
multiplication (kl,ql)(k2,q 2)
=
(klk2~(ql,q2),qlq 2)
Then we have the central e :K*; where
~G
~ = (kl
extension
~Q, ~(k~(1,1),1~
corresponding
~: G - GL(V) a(k,q)
extension
to ~.
• P(q)
, i.e.
e
is an
We also have a homomorphlsm
,
such that the following e : K*;
diagram
~G
II cV : K*.~
io
;GL(v)
such that the map
y,
I, ~PGL(V)
There
1.6 PROPOSITION.
is commutative:
>> Q
which has the form (1.3).
surJective.
> q)
defined by
= 6(k)
(~.4)
, w =((k,q)i
,
~]
exists a vector
space V of dimension
~ y*(aV): Hom(Q,PGL(V))
- Cext(Q,K*)
is
IQ]
72 PROOF.
There are two obvious ways of proving
x E Cext(Q,K*),
and ~ a factor
system representing
form the twisted group algebra module.
1.6.
(KG)w
x.
, and consider
This module gives rise to a projective
follows
Alternatively
Then we can
its regular
representation
with factor system ~ and a vector space of dimension assertion
Let
left of Q
IQI, and the
from 1.5.
let
e : K*: ~ G - ~ - ~ Q be a central
extension with
one-dimenslonal,
[e] = x ~ Cext(Q,K*)
linear representation
of
.
U = ~(K*)
Then - 1
is a
, and we denote
by c the representation
of G which is induced by - 1
sentation
of dimension
IQI, cf. CURTIS/REINER
[1;p.73].
a maps the elements
of U onto dilatations,
which
space
As e is central,
KG@uK
yields a commutative
diagram
In fact, both constructions sentations.
above yield equivalent
A projective
representation
is called irreducible,
modules,
P(g)
, g E Q .
by an irreducible
projective
arbitrary
, resp.
sub-
of Cext(Q,K*)
representation.
3 we shall see that finite dimensional
Cext(Q,K*).
repre-
As the regular
the first proof of 1.6 shows that each element
tations yield only elements
projective
In
represen-
of finite order in the abelian group
This is the reason why we consider
representations
of
dimension.
Let us return to a situation section.
P: Q - GL(V)
of any ring with unit element contains maximal
can be realized Section
projective
.
if 0 and V are the only subspaces
of V that are iavarlant under all left module
x = [e] = ?*(o V)
[]
1.7 REMARK. = mP
(1.~), and proves
and has repre-
If
B : G - GL(V)
considered
at the beginning
is a finite dimensional
of this
irreducible
73 representation e : N~
of a group over an algebraically
~ G
~ Q
a central
Jective representation ~=vP: Q - PGL(V) e
:
>G
Then
diagram
(1.5)
~Q
K*~ 6 > G L ( V )
~I:~L(V)
In the following we say, whenever
.
a commutative
(1.5) is glven, that the projective in e.
we have obtained a pro-
P of Q by Schur's Lemma.
fits in the commutation
N~
ov :
extension,
closed field K and
diagram of the form
representation
(Here e need not be central.)
~ can be lifted
As cV is central
for all V,
? can be lifted in e, if and only if it can be lifted in c(e), centralisation to central
of e.
Thus we restrict
extensions
From 1.5, resp. in Q itself,
in order to llft projective
1.1 .12 we obtain that
i.e.
in the trivial
only if its cohomology sponding projective equivalent
be lifted
element
representations
A projective
in the central
is the map
Recall al
diate from Theorem
Propositions
extension
representations. can be lifted
~ Q
~ Q , if and
i.e.
P: Q - GL(V)
if the corre-
with
~P = 7
are
In general we obtain:
a: N - K*
with
e*(e,K*):
~: Q - PGL(V)
>G
can
~ Q , if and only
~*(aV)
= e*(e,K*)a
Hom(N,K*)
.
~ Cext(Q,K*)
Now the proof of the proposition
is imme-
1.1.10.
1.6 and 1.8 clearly
certain projective
vanishes,
e: N~
from 1.2.7 that ~ [ae]
O>
representation
extension
if there is a homomorphlsm
PROOF.
~*(OV)
in the following
~: Q " PGL(V)
extension
to linear representations.
1.8 PROPOSITION.
our attention
the
representation
e, is not a question
only on the image of 8*(e,K*),
show that the question whether a
can be lifted in a given central
of representation
theory,
hence on the structure
but depends
of e and the
74 abellan group K*.
So we may replace oV in (1.5) by any other central
extension with kernel K*.
It is only important (by I.I.10) that the
extension which is backward induced by the homomorphism on the right side lies in the image of e*(e,K*).
The study of lifting homomor-
phlsms will be continued in Section 2. Extensions e having the property that e*(e,K*) is an eplmorphlsm
(i.e. all projective K-re-
presentations of Q can be lifted in e) will be called generalized K*-representatlon groups, cf. Definition 2.2.
Trivial examples are
given by the centralizations R/[F,R]~
>F/[F,R]
) Q
of free presentations R~
> F
DQ
.
Now we consider a fixed central extension ~ •G
e : N>
~ Q ,
and describe the projective representations of Q which can be lifted in e, in terms of linear representations of G. be a transversal to aN in G, and factor system, cf. 1.1.1.
f: Q×Q - N
For each
Let T = {UqlqEQI be the associated
~ ~ Hom(N,K*)
we define the
factor system ~(a,T) = af : QxQ - K* , and it readily follows that w(~,T) corresponds to 8*(e,K*)s, cf. 1.1.5.
Hence by 1.8 each projective representation
lifted in e, has a factor system
1.9 PROPOSITION. extension,
i.e.
if and only if
Let
can be
equivalent to some w(a,T).
a i ~ Hom(N,K*)
~N S [G,G]
that
,i=1,2.
, w(a1,T) and ~(a2,T)
(1) If e is a stem are equivalent,
51 = ~2 •
(il) If K is algebraically closed, W(~l,T ) and ~(a2,T ) are equivalent, if and only if ~1 ~-1 and s2 a-1 coincide on ~(Im
8.(e)).
~N n [G,G] =
75 PROOF.
Theorem 1.2.7 yields that any two factor systems ~(al,T)
and w(o2,T ) are equivalent, if and only if (~n:
~ ~l(n) ~2(n) -1) : ~N - K*
(01-o2)~ -1 =
can be extended to G.
proofs of (i) and (ll) follow easily.
Now the
For the second statement one
has to keep in mind that K* is a divisible abelian group, if K is algebraically closed. []
Let
~: G - GL(V)
be a linear K-representation of G which maps
the elements of ~N onto dilatations.
(1.6) for
~(~n)
some
=
v:
Hence we have
~a(n)v
o E Hom(N,K*)
~ and
we o b t a i n
a projective
representation
of Q by P(~,T)
= q:
>~(uq)
having factor system w(a,T).
1.10 PROPOSITION.
(1) A projective K-representation can be lifted
in e, if and only if it is proJectively equivalent to some P(p,T). (ll) If ~ runs through all linear representations of G satisfying (1.6) for a fixed a, then P(~,T) runs through all projective representations of Q with factor system w(~,T). (ill) Let ~1 and B2 satisfy (1.6) with the same ~ ~ Hom(N,K*). Then ~I and ~2 are equivalent, if and only if P(~I,T) and P(~2,T) are linearly equivalent. (iv) Let ~1 and B2 be as in (iii).
Then P(~I,T) and P(~2,T) are
proJectively equivalent, if and only if there exists such that
(~)~2
= ~g:
~ (ywg)~1(g)~
~ ~ Hom(Q,K*)
and ~2 are equivalent (as
linear K-representations).
PROOF.
Assertion (i) is Just a slight reformulation of the def-
inition of lifting projective representations,
cf. (1.5), and (iii)
76 is trivial. P(~,T).
As mentioned above,~(a,T)
On the other hand let P be any projective
with factor system w(a,T). morphism
group.
with a map
have the same factor system ~(~,T) This proves
D
P = P((~)~,T)
homo-
Then 1.9 gives rise to a linear
~ of G with commutative
P(q) = ~(q) P(~,T)
representation
Let ~ = ~P be the corresponding
into the projective
representation tain
is the factor system of each
diagram
(1.5).
~: Q - K*
, we have
and completes
.
Thus we ob-
As P and P(~,T)
~ ~ Hom(Q,K*)
.
the proof of (ii) and (iv~
77 2. The Problem
of Lifting Homomorphisms
In this section, e = (~,,): N3
we fix an abellan group A and a central
> G
exist commutative e :
~ Q .
bG
I I
e'
:
>>Q
(.,.,~):
A >
; G'
class @. e - e'
In the previous @ =
~
Q'
~ and all central
In other words,
of extensions
section we considered
space V.
completely
We now say that
(with respect to A),
all extensions According
1.8),
properties"
Im 8*(e,A) of e.
the case
n @ ?
A = K*
representation ~: Q - O'
and
over the fixed
can be lifted in e
if the problem has a solution
2.2 DEFINITION,
for all
for
(combined as in the proof
of Cext(O,A)
cf. YAMAZAKI
0 @ .
which controls
if
and only
Thus it is
"the lifting
we have:
7:0
- O'
to A (now O' varying),
- Cext(O,A)
answer,
e' e Cext(Q',A)
Every homomorphism
with respect
e*(e,A): Hom(N,A)
1.1.10 and 1.2.7
In particular,
2.1 PROPOSITION. completely
e' ~ Cext(O',A)
our problem has a positive
7*(e') e Im e*(e,A)
the subgroup
e' in some
of A, i.e. @ contains Cext(O',A).
to Theorems
of Proposition
extensions
can ~ be lifted to a morphism
for all
~cV } , thus 7 being a projective
K-vector
if
[
1 1
for a given homomorphism specified
We study the problem whether there
diagrams
N)
(2.1~
extension
precisely
is an epimorphism.
[I;§3.2].
e is a generalized A-representatlon
can be lifted when
[]
(1) The central
group resp.
extension
an A - r e p r e s e n t a t l o n
78 group of Q, if e*(e,A)
is an eplmorphlsm
(il) In case A = C*, where £ denotes eralized)
C*-representatlon
representation
We'll
see below that C in
from Proposition
K-representations
and
In Section
1.6,
@ =
group of Q.
~V
resentation
groups of Q do not always
spaces VI.
of extensions
groups of Q for all A. exist.
which are
However,
For example,
A-rep-
let
A = Z ; then
ICext(Q,A)l
= 2 ~ IHom(N,A)I
abelian
A-representatlon
group N is chosen.
group of Q.
Let
The concept
Assume p: Q - Q
y ~
to epimorphisms:
that e is a generalized A-representatlon be an eplmorphlsm,
Then p- induces a generalized A-representatlon •
of generalized
groups behaves nicely with respect
2.4 PROPOSITION.
Ker(p~)'
zero.
This answers our initial problem
examples
A-representatlon
whatever
could be replaced
if, and only if, e is a gener-
I all K-vector
I we gave (trivial)
and
2.2(ii)
for any field K, that all projective
generalized
Q = Z/2
a (gen-
closed field of characteristic
of Q can be lifted
alized K*-representation A = K*
the complex numbers,
group is simply called a (generalized)
by an arbitrary algebraically
for
isomorphism.
group of Q.
2.3 REMARK.
It follows
resp.
and
K = --1(Ker
p) .
group
(P~)'-~..
[a,z] o f ~. PROOF. definition. central
Let ~ denote the extension Let
extension
e'=(-,o): and
A)
~: Q - Q'
is a generalized A-representatlon
above,
• G'
mQ'
which is central by be an arbitrary
a homomorphlsm. group,
The extension
e
and by 2.1 and 2.2 we have
79 a morphism
(-,~',~p):
~'(K) S K e r a .
e - e'
As K e r a
Thus o~' = ~p~, which implies
is central in G', we obtain [G,K] 2 Ker ~',
and ~' induces a morphism
(-,~,7): ~ - e'.
of Q can be lifted in ~ completely;
Hence,
each homomorphism
i.e. ~ is a generalized A-repre-
sentation group.
Our fixed data e,A give rise to the following diagram Hom(N,A) ~ e . ( e ) *
(2.2)
le*(e,A) Ext(Qab,A)3
¥ Y Cext(Q,A)
~
e. ~ Hom(M(Q),A)
,
the exact row of which is I.(3.7) of the Universal Coefficient Theorem, while e.(e)* is defined as the indicated composite map. By Lemma 1.3.7, we have
e.(e*(e,A)c)
= ae.(e)
for all
m ~ Hom(N,A).
Thus e.(e)* = Hom(e.(e),A)
= (c~
;ae.(e)):
Hom(N,A)
" Hom(M(Q),A)
,
the dependence on A being suppressed from the notation.
2.5 PROPOSITION.
(i) If e is a generalized A-representatlon
group, then e,(e)* is an epimorphism. (ii) Let e I and e 2 be central extensions with factor group Q. If in e I and e 2 the same homomorphisms
of Q are liftable with respect
to the same central extensions of A, then
PROOF.
The homomorphism
generalized representation
Im e.(el)* = Im e.(e2)*
e*(e,A) is an epimorphism,
.
if e is a
group, and (i) follows from diagram (2.2).
The arguments leading to 1.8 and 2.1 show that e I and e 2 have the same lifting properties Hence,
if, and only if
(ii) follows from (2.2).
Im e*(el,A) = Im e*(e2,A)
[]
In the following we quite often study the important special case
.
80 where Ext(Qab,A)
= O.
From the c o n s i d e r a t i o n s
above we easily
obtain 2.6 PROPOSITION. 2.5(I)
and 2.5(ii)
(i) If
Ext(Qab,A)
= 0 , the converse
in
holds.
(ii) The following
statements
(a) e is a g e n e r a l i z e d
are equivalent:
A-representatlon
group and
e,(e)*
an
isomorphism. (b) e is an A - r e p r e s e n t a t i o n
group and Ext(Qab,A)
(c) e.(e)* is en isomorphism
and Ext(Qab,A)
2.7 LEMMA.
Let L(Q,A)
denote
, where
~ runs through Hom(M(Q),A).
epimorphlsm,
if, and only if the following
(a)
L(Q,A)
_o Ker
(b)
Each homomorphlsm
Furthermore,
= O.
the intersection
Ker a 2 M(Q)
= 0
of the subgroups Then
e.(e)*
two c o n d i t i o n s
is an
hold:
e.(e)
e.(e)*
c':
Im e.(e)
" A
is an isomorphism,
can be extended
to N.
if, and only if we have
in addition: (c)
Horn(
~
,A)
= 0
.
~NO[G,G] Condition
(b) is satisfied
if
Ext(
(c) hold if e is a stem extension,
PROOF. each
The h o m o m o r p h i s m
a e Hom(M(Q),A)
o = ee.(e) induces
.
Assume
a':
~ e Hom(N,A)
a = ee.(e)
.
(b).
By (a),
extending
On the other hand,
a = ee.(e)
~ e Hom(N,A)
Im e.(e)
for all
- A
each with
a', and which
let
(b) and
1.3.11.
is epimorphlc,
exists
(a) and
we have a
relation
e.(e)
there
a homomorphism
cf.
,A) = 0 , and aN ~Nn[G,G]
e.(e)*
if
and only if for
, such that ~ e Hom(M(Q),A) a'e.(e)
= ~ .
By (b)
therefore
satisfies
be eplmorphlc.
Then the
a e Hom(M(Q),A)
yields
81 L(Q,A) 2 Ker 8.(e) morphlsm
.
~: N - A
N, proving
Let
e.,A)
if and only if
Im e.(e) = ~-I(~No[G,G]) e.(e) ~
Then we have a homo-
a'e.(e) = Se.(e) . Thus B extends ~' to
with
the first part of the Lemma.
is inJective,
Coker
a' ~ H o m ( I m
The homomorphlsm
Hom(Coker
e.(e),A)
= 0 .
8.(e)*
As
, we have
~N
~NO[G,G]
which proves the second assertion.
The remaining
part of the proof
is trivial.
2.8 COROLLARY.
If A is a divisible
abelian group,
(i) e is a generalized A-representation Hom(Ker
e.(e),A)
e.(e),A)
PROOF.
= 0
Assume
and
group,
Hom(Coker
resp.
reformulation
= 0 .
Hence
group,
an isomorphism.
tion (b) in 2.7 is satisfied, e.(e),A)
e.(e),A)
= 0 .
Ext(Qab,A)
= 0 ,
e is a generalized A-represen-
resp. A-representation
is an eplmorphlsm,
Hom(Ker
if and only if
that A is divisible.
and we can apply 2.5 and 2.6; i.e. tation group,
if and only if
= 0 .
(ll) e is an A-representation Hom(Ker
group,
then
whereas
The condition
e.(e)*
if and only if
As A is divisible,
(a) is equivalent Hom(Coker
condi-
to
e.(e),A)
= 0
is a
of (c) in 2.7, and we are done.
From 2 . ~ 2.6 and 2.7 we obtain that a generalized A-representation group, which is a stem extension, group.
This can only hold if
1.3.8 guaranties with
N = M(Q)
the existence and
is even an A-representatlon
Ext(Qab,A)
= 0 , but in this case
of such extensions,
e.g.
choose
e.(e) = IM(Q)
Let us return to the general
case, where A is an arbitrary
e
82 abelian group, and assume that e is a stem extension.
Then
e.(e)*
is a monomorphism and we obtain from (2.2)
e*(e,A)
Im
n ~ Ext(Qab,A ) = 0 .
This relation and Lemma 2.7 yield
2.9 PROPOSITION.
Let e be a stem extension.
Then the following
properties are equivalent: (1)
L(Q,A) ~ Ker e.(e)
(li)
e.(e)* is an isomorphism
(Ill) Cext(O,A) Im e*(e,A)
and
2.10 EXAMPLE.
is the internal direct sum of
¥ Ext(Oab,A ).
[]
The preceding proposition can be applied as follows.
For every group Q and every abellan group A, we "construct" generalized A-representatlon groups of Q in a different manner than before. The abelian groups Hom(N,A) and Cext(Q,A) a:
~ ~.a
and [e'];
[e'] m Cext(Q,A), implies that
~.[e']
~ ~ End(A)
¥ Ext(Qab,A)
The calculus of induced extensions
Let [ei]i~ I be a system of End(A)-gen-
¥ Ext(Qab,A ) Gi
Let
.
, ~ ~ Hom(N,A),
is a submodule of Cext(Q,A) and e*(e,A)
is an End(A)-homomorphlsm. erators of
= [@e']
are End(A)-modules by
H Gi i~I
~ Q .
be the (unrestricted)
direct product of the groups Gi,
and let
GI =
Ix I x=(xl)i~ I
be their fibre product. eI
:
n
As
~
G1
H G i , ~i(xl) = -~(x~) for all i,J E I I Hence we obtain a central extension ~ Q
.
l~I
For two extensions el, e 2 we obtain
(elxe2) ~
, which is part of the
83 sum defined in Cext(Q,A), J-th projection. Hom(HA,A)
onto
cf. I.(2.2).
Let
pie 1 = ej .
Then we have ¥ Ext(Qab,A)
Let
H A ~ A be the ieI Hence, e*(el,A) maps
e2: N2~
pj:
t G2
~ Q
be a stem
extension satisfying one, and hence all the conditions of 2.9, and consider e0 = (elxe2)AQ
:
E A x N 2~. iel
~ GI~G 2
~Q
.
Then we have
Hom(IIAxN2,A)
=
Hom(NA,A)
x Hom(N2,A)
and e*(eo,A)
=
e*(el,A) x 8*(e2,A) .
As e*(el,A) maps onto be a complement of
¥ Ext(Qab,A)
¥ Ext(Qab,A)
and
Im e*(e2,A)
in Cext(Q,A),
is assumed to
e*(eo,A) is an epl-
morphlsm, and thus e o is a generalized A-representatlon group.
2.11 EXAMPLE. n 2 3 and
Assume
Q = S n , the symmetric group on n letters,
A = Z/2 , which can be regarded as the multlpllcatlve
group of GF(3).
We have
(Sn)ab ~ M(Sn) ~ Z/2 , and we can identify
A = M(Q) = Z/2 , cf. 3.8.
Thus we obtain
Ext(Qab,A ) ~ Z/2 , Hom(M(Q),A) ~ Z/2 , Cext(Q,A) A generator of eI :
Z/2 ~
¥ Ext(Qab,A)
is given by
~ Sn~Z/4
~S n ,
where (Sn)ab and (Z/4)/2(Z/4) are identified. e2: Z/2~
r D
~ Sn
such that
~
Z/2 x Z/2 .
Choose
e.(e2) = 1Z/2 .
From the consid-
eration in 2.10 we obtain a central extension e O : ZI2 x ZI2;
~ DE
(Sn~
Zl4)----~S n ,
which is a generalized A-representation group for S n. and the domain of e*(eo,A) have the same order, isomorphism,
As the image
e*(eo,A) is an
i.e. e o is even an A-representatlon group, and it is
a minimal extension in which all projective GF(3)-representations
of
84 Sn can be lifted.
It is easy to see that the representation which
maps a generator of Z/4 onto the matrix
(~ -~)
over GF(3) induces
an irreducible projective representation of S n, whose cohomology element is a generator of
Y Ext(Gab,A)
, and which therefore cannot
be lifted in any stem extension of S n.
In the following we restrict our attention to the case where Ext(Qab,A)
= O.
Then the (generalized) A-representation groups are
characterized by 2.5, 2.6 and 2.7.
As mentioned above, A-represen-
tation groups for all A with Ext(Qab , A) = 0 are given by the special stem extensions e satisfying N = M(Q) and e.(e) = 1M(Q).
Let e be
any A-representation group for Q, and B an abelian group with Hom(B,A) = O. e' :
Then
N x B~
~G
x B
~Q
is also an A-representation group.
Hence, an A-representatlon group
(in the sense of Definition 2.2) need not be a stem extension, not even if
Ext(Gab,A)
2.12 REMARK.
= O.
Assume
Ext(Qab,A) = O
all proper subgroups U of N.
and
Hom(N/U,A)
$ 0
for
Then e is an A-representation group
for G, if and only if e is a stem extension satisfying L(Q,A) ~ Ker
PROOF.
e.(e)
It remains to show that each A-representatlon group e
has to be a stem extension.
If e.(e) were not surJective,
at least two elements of Hom(N,A) vanishing on Im
e.(e)*
e.(e).
there are Hence,
cannot be inJective, a contradiction.
The group theoretical significance of the condition was given in 1.3.13.
As mentioned above,
Ext(Qab,A)= 0
it is satisfied for all Q,
if A is a divisible abellan group, and in this case the A-represen-
85
tation groups were described in 2.8.
Examples for divisible groups
are the multiplicative groups of algebraically closed fields.
For
a particular O, where Oab is either finitely generated or a torsion group, it is sufficient for having
Ext(Oab,A) = 0
that the ele-
ments of A can be divided by all natural numbers which appear as orders of elements of Oab.
(For finite groups Gab, such groups
sometimes are called lOabl-divisible , cf. YAMA2AKI [I].)
Both
conditions in 2.12 hold, if A is divisible and contains elements of finite order n, for all n which appear as orders of elements in the quotients of N.
This is satisfied if A is divisible and contains
elements of arbitrary finite order.
In the category of abelian
groups, these groups are exactly the inJective objects which are cogenerators.
Examples are given by Q/Z, the torus group B/Z, the
multlplicative groups of algebraically closed fields of characteristic O, i.e. C*.
Summarizing part of the results above, we obtain:
2.13 PROPOSITION. (i)
The following conditions are equivalent:
e is a generalized A-representation group for some divisible
abelian group A containing elements of arbitrary finite order. (ii)
e is a generalized A-representation group for all divisible
abelian groups A containing elements of arbitrary finite order. (iii)
(iv) (v)
(vi) in e.
e is a generalized representation group (case
A
= C*).
e.Ce) is a monomorphism. The inflation
M(~): M(G) - M(O)
vanishes.
All complex projective representations of 0 can be lifted
86 2.14
PROPOSITION.
(i)
The following conditions are equivalent:
e is an A-representatlon group for some divisible abelian
group A with elements of arbitrary finite order. (ii)
e is an A-representation group for all divisible abelian
groups A with elements of arbitrary finite order. (iii)
e is a representation group.
(iv) e.(e) is (v)
an isomorphism.
e is a stem extension and
(vi)
M(,) = 0 .
e is a stem extension and all complex projective represen-
tations of Q can be lifted in e.
PROOF of 2.13 and 2.14.
The equivalence of (i), (li), (lii),
(iv) is immediate by 2.8, the equivalence of (iv) and (v) follows from 1.3.5, and the equivalence of (ill) and (vi) from 2.3.
2.15 COROLLARY. (i)
The following conditions are equivalent:
Each central extension of Q is a generalized representation
group. (ii)
The trivial extension
0 ~
> Q
~ Q
is (up to isomorphism)
the only representation group of G. (iii)
M(Q)
(iv)
=
0
Each complex projective representation of
Q
is equivalent
to a linear representation.
2.16 COROLLARY. central)
such that
this holds whenever
Let e be any group extension (not necessarily M(,): M(G) " M(Q) M(G) = 0 .
vanishes.
In particular,
Then the centralization c(e) is a
generalized representation group of Q.
PROOF. M(.')
=
Let
0 .
c(e) = (.,.')
Apply
2.13.
[]
.
Then
Im M(.') = Im M(~)
yields
87 Now we return to more general abellan groups A.
2.17 DEFINITION.
Assume
Ext(Qab
group e of Q is called minimal, Ker 9.(e) = L(Q,A)
,
A) = 0 .
A n A-representatlon
if it is a stem extension satisfying
.
In this terminology,
all A-representatlon groups are minimal,
A is divisible with elements of arbitrary finite order.
if
Now let e
be a minimal A-representatlon group (for an arbitrary A), and e': N'>
~ G'
DQ
a generalized A-representatlon group.
Then we
have
N ~ M(Q)/L(Q,A)
and
Ker e.(e') ~ L(Q,A)
.
Hence there exists a monomorphlsm from N into a quotient of N', which Justifies Definition 2,17.
If Q is finite, we have by 3.10 that
M(Q) is finite, and we obtain:
2.18
PROPOSITION.
Let Q be finite and
Ext(Qab,A)
=
0 .
Then
the minimal A-representatlon groups of Q are exactly the finite generalized A-representatlon groups of Q whose middle groups have minimal order.
Let us summarize some of the results above in the case where A is the multiplicatlve group of an algebraically closed field K of characteristic
p ~ 0 .
L(Q,K*) = M(Q)p
, the group of all p-elements in M(Q), resp.
M(Q)p = 0
for p=O.
Then K* is divisible, and we have
Hence the minimal K*-representation groups
of Q are given by the stem extensions e satisfying Ker e.(e) = M(Q)p
If p=O, we obtain the representation groups,
and no other K*-representatlon groups occur, cf. 2.14.
2.19 LEMMA. abellan group,
Let M, N1, N 2 be abelian groups, A a divisible el: M - N i
homorphisms,
and L, resp. L i the inter-
88 section By
of the k e r n e l s
of all h o m o m o r p h i s m s
el* we denote the h o m o m o r p h i s m
by e i.
from M, resp. N i to A.
from H o m ( N i , A )
to Hom(M,A)
induced
T h e n we have:
(i)
Ker
81 + L = Ker
(ii) The c o n v e r s e
e2 + L
implies
in (i) h o l d s
Im el* = Im e2*
if one of the f o l l o w i n g
. conditions
is satisfied: (a)
L i 0 Im e i S el(L)
(b)
A is d i v i s i b l e
with elements
(c)
M is a t o r s i o n
group.
PROOF. a
(i) A s s u m e
e Hom(NI,A)
.
Ker
I=1,2
e I + L = Ker
: Im 82 - A
: 82(x):
A s A is divisible, definition
a n d let
all
.
81 + L . From
8 ~ Hom(N2,A)
a n d we o b t a i n
L = Li = 0 .
elements primes p
~ae1(x)
of M, resp.
. 8: N 2 " A
and
Thus we have
with elements In g e n e r a l Ni, w h o s e
for w h i c h no e l e m e n t s g r o u p we t h e r e f o r e
(b) a n d
(c) are c o n t a i n e d
of 8'.
L i O Im e i S el(L), COl(X)
e2(x) E I m case
of a r b i t r a r y
L, resp.
= 0
we o b t a i n
e2 + L , f i n i s h i n g
torsion
and
F r o m the
= 882 = 82-(a)
This i m p l i e s
x E Ker
If A is d i v i s i b l e have
I ~ Hom(M,A)
Im 81" = Im e2*
.
order.
is w e l l - d e f l n e d
Im 61" = Im e2*
x ~ Ker
o ~ Hom(N1,A)
82*(8)
finite
e2 + L , and let
we have an e x t e n s i o n
follows
(ii) A s s u m e
of a r b i t r a r y
e2 + L .
T h e n the f o l l o w i n g h o m o m o r p h i s m 8'
.
e I + L = Ker
el*(a)=ae
Thus,
K e r 5e I ~ Ker
,
~e2(x)
for ,
finite order we
L i consists
Im e i n L i = e l ( L ) []
= 0
(a).
of all t o r s i o n only by those
of order p exist in A.
in (a).
,
for all
e 2 n L 2 ~ e2(L)
orders are d i v i s i b l e
have
I=1,2
If M is a Thus,
cases
8g For later purposes we need:
2.20 PROPOSITION. characteristic
p,
Let K be an algebraically
el: Ni~--->G i
~ O, i=1,2
closed field of ,
central extensions,
and assume that either M(Q) is a torsion group or same projective K-representatlons
p = 0 .
Then the
of Q can be lifted in e I and e 2 ,
if and only if Ker e.(e 1) + M(Q)p = Ker e.(e 2) + M(Q)p
PROOF.
.
By 2.19 the equation above is equivalent to
Im e.(el)* = Im e.(e2)*
, and from 2.5 (li), 2.6 and 1.6 this holds,
if and only if the same projective
representations
can be lifted.
The connection between various situations occurring above and some of the results are shown by the following diagrams:
K alg. closed field
~
A
=
K*
K alg. closed field,l A = K * char K = 0
In the two following diagrams, refer to the properties
~
A divisible
~
A divisible with elem. of arb. finite order
the numbers beside the arrows
of A numbered in the first diagram.
boxes contain the numbers of the corresponding propositions.
• ~Ext(Qab,A
The
definitions and
[]
90 O*(e,A)2.2eplm. 1
[
'e.(e)*epim.'
5
e gen. A-rep. grp.]_. 2 > e.(e) monom. 2.13
2.5,2.6,2.7
I Hom(Ker8.(e),A)= 0 ] 2.8
e*(e,A) isom. I 2.2
l
e.(e)* isom2.6,2.7
e A-rep. grp.
Hom(Ker e.(e) ,A) = 0 Hom(Coker e.(e'),A) = 0 2.8
l e minimal A rep gr~]
2.17
e.(e) isom. 2.14
gl
3. Representation
Groups
In this section we restrict tions over algebraically tlon groups
e.(e)
for
Q
isomorphic.
groups
ourselves
to projective
closed fields K.
Certain K*-representa-
are given by the central These extensions
of Q in Section
representa-
extensions
e by Q with
were called the representation
2, and they proved to be the only ones if
char K = 0 .
3.1 DEFINITION. A morphlsm
(i) Let e I and e 2 be group extensions
of the form
(a,8,1Q):
over Q, and if B is isomorphic, morphisms
are the morphlsms
GRUENBF~G
[I; chp.
(li) Let
e: N)
eI - e2
an isomorphism
in the extension
K
)G---~Q
over Q.
category
~U
be an arbitrary
is normal
KI
extension
is called a homomorphism These homo(Q-) in
9].
group of N, such that induced
by Q.
e/U
Then
denotes the
~t
N/U ~
G/~U ~
Now we are going to describe terms of isomorphism
in G.
extension and U a sub-
over Q.
Q .
the representation
Let
e: N ~
groups of Q in
~G---~Q
be a represen-
tation group of Q, and X e'
: M(Q)~
Then
e.(e,)
other
hand,
~ G--~>Q
= 1M(Q) let
ei:
e.(el) = e,(e2)
1.3.8,
X =
~e.(e)
, and e and e' N;
¢ G1
are isomorphic
¢ Q , 1=1,2
, then the naturallty
and 6 is isomorphic, over Q, i.e.
,
whenever
[eli = [e2]
we obtain:
.
On t h e
be stem e x t e n s i o n s
of I.(3.4)
(a,B,IQ):
o v e r Q.
eI - e2
yields
with
e = IN ,
is a homomorphlsm
If we combine these observations
with
92
3.2 PROPOSITION.
(1) Each representation group of Q is isomorphic
Q to a central
over
e.(e')
extension
e': M(Q))
• G
~ Q
with
= IM(O)
(ll) The classes of representation groups of Q - with respect to isomorphism over Q - are uniquely parametrized by their members in
e.-I(1M(Q)), which
is a coset of
¥ Ext(Qab,M(G))
in
Cext(Q,M(Q))
. []
3.3 REMARK.
The classes of representation groups under arbitrary
isomorphism of extensions will be considered in Chapter III.
One
should keep in mind that isomorphism of extensions is a much weaker condition than isomorphism over Q as in 3.2.
In group theory it is
often important to distinguish only between non-lsomorphlc middle groups of representation groups, isomorphism of extensions.
a condition even weaker than
Hence, the size of
Ext(Qab,M(Q))
is
an upper bound for the number of these groups.
If Q is finite, M(Q)
is also finite by 3.10,(i).
Let
and
the invarlants of
M(Q)
common divisors of
Gab
and
(ai,bj)
Ext(Qab,M(Q))
al,a2,...,a n respectively.
, I ~ i ~ n, 1 ~ J ~ m
, cf. III.4.5.
bl,...,b m
be
Then the greatest are the invariants
This bound for the number of rep-
resentation groups of finite groups is due to SCHUR [2; Sstz I, p.95].
As already mentioned in Section 2, the Universal Coefficient Theorem yields the existence of representation groups. proposition,
The following
essentially due to Schur, gives a more explicit
description.
3.4 PROPOSITION, (i) Let
e(F,R): R "
such that
S/JR,F]
R/[R,F]
Then
.
cf. SCHUR [2; §3], GRUENBERG [1, chp. 9.9]. ~F
~Q
be a free presentation of Q,
is a complement of
e(F,R)/S
(Rn[F,F])/[R,F]
in
is a representation group of Q.
S S R
93 (ii) For each representation subgroup S of F, contained
group e of Q there exists a normal
in R, such that
over Q to e; for each such S, the group
e(R,F)/S
S/JR,F]
is isomorphic
has the property
of (i). (iLl) Let e' be a stem extension resentation that
e/U
Then there exists a rep-
group e of Q and a subgroup U in the kernel of e, such and e' are isomorphic
PROOF.
of Q.
Let
over Q.
e': N ~ - - - ~ G ~ Q
be a stem extension.
The iden-
tity of Q gives rise to homomorphisms
a: R ~ N , ~: F " G , such
that diagram
For sake of simplicity we
evaluate
1.(3.6)
M(Q)
is commutative.
at
e(F,R)
induces the epimorphism and 8 are eplmorphisms, satisfies
and that
and we put
S/[R,F]
and
~ RICRn[F,F])
T/(Tn[F,F])
M(Q)
= (Rn[F,F])/[R,F]
8.(e'): M ( Q ) ~ N
JR,F] ~ T H R
TICTn[F,F])
, l.e.
T := Ker c = Ker 8 , which = R .
~ R[F,F]I[F,F]
is a complement
Then c
, which implies that
T.(RN[F,F])
is free abellan.
.
This yields
,
Let S be a subgroup of T such
of
(Tn[F,F]/[R,F])
in
T/[R,F]
,
which implies S n IF,F] = S n (Rn[F,F]) l.e. S satisfies obvious
= S n (Tn[F,F])
the conditions
isomorphlsms
e' ~ eCF,R)/T
in (i).
= [R,F]
,
We also have the following
over Q:
~ (eCF,R)/S)/(T/S)
.
We have
As
8.(e(F,R)/S)
= (x[R,F]!
e.(e(F,R)/T)
= (x[R,F],
8.(e(F,R)/S)
resentation
group,
if and only if
xS)
~)
is an obvious and it follows
T = S .
, x e R0[F,F] , x ~ Rn[F,F] isomorphism,
e(F,R)/S
is a rep-
that e' is a representation
These observations
easily prove
(i),
group, (ii)
g4
and (ill). D 3.5 REMARK.
Let us consider
The isomorphisms
e.(e(F,R)/S):
allow us to identify tation groups
e.(e)
which
M(Q)
e(F,R)/S
and
R/S
.
, cf. proof of 3.4,
extensions
e by Q for
and from 3.4 (il),(ill)
we obtain
of Q is forward induced from such an
description
3.6 PROPOSITION.
Assume
tors and r relations, the torsion
subgroup
k+r-n elements.
of this situation will follow in of isoclinic
and let k be the rank of of
Qab
IV.I as P. Hall's
Let
RC
"
Then
M(G)
with n genera-
Qab/T
, where T is
can be generated
will be discussed
in more detail
~ F
~ Q
be a free presentation closure
R/(RQ[F,FJ
On
is free abelian of rank n-k.
The following proposition 2.14 and sometimes
of
of r elements.
is an abelian group with of most r generators.
the other hand,
by
Inequality.)
Q, where F is free of rank n and R the normal R/JR,F]
groups are studied.
that Q has a presentation
(This proposition
SKETCH OF PROOF.
Then
from 3.4.
Thus we regard the represen-
as those central
III.4, where presentations
in Section
D R/S
e(F,R)/S
e.
A more detailed Section
M(Q);
is the identity,
that each stem extension extension
the extensions
is an easy consequence
quite useful
of 2.16,
resp.
in order to exhibit representation
groups.
3.7 PROPOSITION. assume
M(G) = 0
3.8 EXAMPLES multlplicator
Let
e: N)
> G---a~Q
be a stem extension,
Then e is a representation
OF REPRESENTATION
GROUPS.
are their own representation
and
group of Q.
(i) Groups with trivial groups.
In particular,
95 by Proposition 3.6, this covers finite groups which can be presented with the same number of generators and relations.
W e mention in
passing that cyclic and free groups have trivial multlplicator by Lemma 1.3.4. (ii) The generalized quaternion group entation
R-
~ F ~ • 2 n-1
closure of
~x
of order 2 n+l has a pres-
Q , where F is free on 2
y , x2[y-l,x-1]l.
Ix,yl, and R the normal
Hence we can apply 3.6 in order
to see that the multiplicator vanishes. (iii) Let O be abelian on two generators. (ii)
an extension e by O is given with
it is a representation group. O = Z/p
Z/p
x
follows
In Example 1.4.6(i),
e,(e)
where F is free on
Hence
Let us consider the group
more closely, where p is a prime.
M(Q) = Z/p
isomorphic.
From 1.4.6 it
Now we have the presentation
R~
~F
~Q
x,y , and
R = <x p, yP,[ x, y]>~ , the upper index F denotes normal closure. Rn[F,F] = [F,F]
, and
and the complements of Cn,m/[R,F] Cn, m
=
Thus we have
[R,F] = ([x,y]P,[F,F,F]) F , [F,F]/[R,F]
in
R/[R,F]
are given by
, where (xP[x, Y]n, YP[x, Y]m, [R, F] >F , o ~ n, m ~ p-1
Hence we have p2 complements.
On the other hand
Ext(Qab,M(Q) ) ~ Ext(Z/pxZ/p,Z/p)
has also the order p2.
By 3.2
and 3.5, the central extensions
~/Cn,m~-"* F/Cn, m
~Q
are a system of representatives of representation groups of Q with respect to isomorphism over Q, see also III.4.7. (iv) The dihedral group D of order 2n+l has the following presentation
Re
~F----~D
, F free on x,y and
96 R = (x 2 [ and let
Xp
y]y2p[ X ~ y]2n-l>F
G = F/(
x2
Ix,y]
to the generalized
-2
,[x,y]y
2
,Ix,y]
2n F > .
Then G is isomorphic
quaternlon group of order
2 n+2
and gives rise
to a stem extension
Z/2 ;
~G
,~ D .
By (ll), G has trivial multlplicator,
and by 3.7 the extension above
is a representation group of D, in particular we obtain that 2n-1 [x,yJ JR,F] is a generator of M(D) = (Rn[F,F])/[R,F] -= Z/2. The elements
x2[R,F]
R/[R,F].
Hence we have exactly
Clj[R,F],
, [x,y]y2[R,F]
o _< 1,J _< 1 2
clj = <x Ix,y]
2n+i-1
4 = IExt(Dab,M(D)) I
M(D) in
complements
given by
,Ix,
y]2n+J-l+ly2
The same argument as in example R/Ci j r
generate a complement of
,[R,F] >F
(lii) shows that the extensions
~ F/Ci j ~ D
are representatives
of the representation
over D.
F/Cij
The group
groups under isomorphism
is a dihedral group if
alized quaternion group if
i=O, J=1
i=J=1
, a gener-
and a quasl-dlhedral
group if
J=0, i=0 or 1 (v) The representation
groups of the symmetric and alternating
groups were studied by de SEGUIER [I] and SCHUR [3], and SCHUR [2] obtained the representation
groups of
SL(2,q)
and
PSL(2,q)
, see
also HUPPERT [I; V.25]. (vl) Representation
groups of perfect and of metacycllc
groups
will be discussed in II.5 and IV.2, respectively.
3.9 PROPOSITION ?: Q " PGL(V)
(Schur).
Let K be an arbitrary field and
a projective K-representatlon
Then the associated cohomology finite order dividing n.
element
of finite dimension n.
?*(aV) ~ Cext(Q,K*)
has
97 PROOF.
The extension
gram (1.3).
Let
~*(o V)
is defined by the top row of dia-
~: K* " K* : x:
~ x n , and
Then the definition of the addition in yields
n.y*(Ov) = a,~*(OV)
and 1.I.3 implies
Cext(Q,K*),
cf. 1.2.4,
On the other hand we have
a.~*(OV) = 0 .
3.10 COROLLARIES
d=det.¢: G ~ K* .
[]
Let Q be a finite g r o u p .
(Schur).
a = d.k ,
Then:
(i) M(Q) is a finite abellan group. (il) If K is an algebraically closed field of characteristic O, then
Cext(Q,K*) ~ H2(Q,K *) ~ M(Q)
(ill) If
R"
~F
~Q
is the torsion subgroup of
is a free presentation of Q, then M(Q)
R/[R,F]
(iv) Let e be the exponent of M(Q).
Then e 2 divides the order
of Q.
REMARK.
3.10 (ll) usually does not hold if Q is infinite.
example is given by while 1.3.8 yields
Q = Z × Z.
Here
we have
M(Q) ~ Z
an algebraically closed field of characteristic O. Cext(Q,K*) ~ Hom(M(Q),K*) Hom(M(Q),K*)
by I.A.6 ,
Cext(Q,K*) ~ H2(Q,K *) ~ K* .
PROOF of 3.10. (i) By 3.6, M(Q) is finitely generated.
that
An
Let K be
Then
by 1.3.8, and from 1.6 and 3.9 it follows
is a torsion group.
Thus M(G) is a torsion
group as well, and M(Q) is finite. (il) The assertion follows from the proof of (i). (ill) As mentioned above, R/(RD[F,F]) and by (1),
M(Q) = (RN[F,F])/[R,F]
(iv) Let K be as above, and
is a free abelian group,
is finite.
x ~ Cext(Q,K*)
Thls proves (Ill). .
As Q is finite,
we have by (1) a finite representation group e of Q: R
e
: M(Q):
t G
~
Q ,
g8 and there is a (unique) 8*(e,K*)a = x .
Let
a e Hom(M(Q),K*)
61,62,...,6 n
of the induced representation
satisfying
be the irreducible constituents
(a~-1) G .
Then each 61 gives rise
to a projective representation of Q having cohomology element x. The Frobenius Reciprocity Theorem yields that 61 is contained in (a~-l) G
exactly (dim61)-tlmes and we obtain
IQI = dlm(a~-l) G = ~(dim61) 2 This equation and 3.9 show that the square of the order of x divides IO[, and (iv) follows from (ii).
[]
The theory of representation groups can be used to study projective representations of finite groups by applying the theory of linear representations.
A nice example is the following theorem
of FRUCHT [I]:
3.11 THEOREM.
Let Q be a finite abellan group, K an algebraically
closed field and
x e Cext(G,K*)
Then any two irreducible projec-
tive representations of Q with cohomology element x are proJectlvely equivalent.
PROOF. for Q and M(Q).
Let
N~
>G
p = char(K)
D Q .
be a minimal K*-representatlon group
Then N is isomorphic to the p'-part of
Without loss of generality we can assume
N ~ G , and we have
G = G' × Qp , where Qp is isomorphic to the p-part of Q, and G' is a nilpotent p'-group of class at most 2, containing N.
By 1.10 we
have to show that any two irreducible linear representations
61 , 62
of G', whose restrictions to N contain the same (one-dimensional) constituent ~ are equivalent up to a one-dlmensional
factor.
As G'
is an M-group, we have subgroups At, A 2 satisfying AlGA 2 ~ Z(G') ~ N , and one-dlmenslonal representations a i of A i G' with a i = 61 , which thus satisfy all N = a Hence,
gg
o I := ~Ia2 a 2 resp.
-1
can be regarded as a character
e 3 be extensions
c3B 2 = (a2a2)G' by Mackey's
.
As
induction
common constituent.
Frucht's see MANGOLD
a2a 2
and
and
G'/N on
.
, and let
Now we have
AINA 2 , we obtain
c382
have a non-trlvlal
they are equivalent.
to arbitrary
finite groups,
[I], TAPPE [2].
Let Q be an elementary
nary irreducible
representations
have dimension
pn.
abellan group of order p
a stem extension
In HUPPERT
theorem
81
(AINA2)/N
resp.
coincide
can be generalized
is extra-special.
irreducible
elements
characters
cannot exist projective having the cohomology
representations
elements
order of the cohomology
~ Q , where E
greater
I all
projective
repre-
Using Frucht's
representations
(beside
pn, we obtain that there of lower dimension
from above.
elements
,
of order p, because they of Z/p.
ones) have dimension
2n
it is shown that the ordi-
or Just the fact that the irreducible
the one-dlmenslonal
r E
of E of dimension
The corresponding
from irreducible
Z/p:
[I;V,16.14]
of Q have cohomology
are derived
aI
theorem that
and consider
sentations
A2/N
As they are irreducible,
theorem
3.12 EXAMPLE. p a prime,
of a I to
of
n than p ,
Hence the bound for the
given by Proposition
3.9,
is some-
times very bad.
3.13REMARK.
Another
application
tations has been given by FAHLINGS with faithful terized.
irreducible
[I;4.1].
projective
irreducible
projective
has no normal p-subgroup
p.
represen-
Here the finite groups
representations
The result reads as follows:
closed field of characteristic faithful
of lifting projective
are charac-
Let K be an algebraically
Then the finite group G has a
K-representatlon,
and is the central
group H, whose socle is the normal closure
if and only if G
factor group of a finite of a single element.
100
We close this section with a few remarks on unitary projective representations. By
U(H)
Let H be a Hilbert space with inner product
we denote the subgroup of
satisfying
(ux,uy) = (x,y)
GL(H)
for all
that
( , ).
consists of all u
x,y ~ H , and let T denote the
multlpllcative group of all complex numbers of absolute value 1. Then we have the central extension ~H : where
T~ ~(t)
?: Q - PU(H) and
t U(H) = (xl
~ PU(H)
> tx)
and
,
PU(H) = Coker ~
.
is called a unitary pro~ectlve representation of Q,
~*(~H) E Cext(Q,T)
the associated cohomolcgy element.
lar argument as in 1.6 shows that each element of induced by a unitary projective representation o v e r space
121Q I
Any homomorphism
As
T ~ ~/Z
Cext(Q,T)
A simiis
the Hilbert
is divisible with elements of arbitrary
finite order, the question of lifting unitary projective representatlons also leads to the representation groups of Q by the results of Section 2.
101
4. Representation Groups of Free and Direct Products
In this section we construct (generalized) representation groups for the free product
Q1.Q2
and the direct product
QlXQ2
of two
groups from central extensions (4.1)
ek = (~k,~k) : Nk~----->Gk-----~Qk ,
k=1,2 ,
which are (generalized) representation groups of Qk according to the context. M(QI*G2)
and
In homological terms, the resulting formulas for M(QI×Q2)
are well-known; a different proof of these
formulas, also non-homologlcal, was given by C.MILLER [1].
Concern-
ing the direct product, the basic idea is taken from SCHUR [2] and WIEGOLD [2].
Since we here allow infinite groups, the proofs need
to look different. M(QlXQ 2) 4.1.
Actually, SCHUR [2;Satz VI,p.109J described
for finite groups Q1 and Q2" Recall that the category of groups has s ~ s ,
ditionally called free products. G 1 and G2, one has a group for k=1,2
G1.G 2
these are tra-
In other words, given any groups and homomorphisms
Jk: Gk " G 1 ~ 2
with the following ~ i v e r s a l property: For every pair of
homomo~hisms
~:
Gk - H
J~-/"G 1 \ f l (4.2)
GI*G2A / f
~ H G2
with common range, there is a ~ i q u e f = ~fl,f2}: G1.G 2 - H the projections
with
homomorphlsm
f'Jk = ~
ql = 11,0}: GI.G 2 - G 1
which b y the very definition ~ t i s f y
for k=1,2 . and
We also need
q2 = ~0,1}: G1.G 2 - G 2
102 qk.Jk = 1 = Identity yon G k
for k=1,2
q2.Jl
= 0 :
;
(4.3) = 0 :
G 1 " G2 , q l . J 2
G2 ~ G 1
Free products exist for arbitrary (possibly infinite) families of groups, but our main concern is with two free factors.
4.2
Ve infer the following explicit construction of the free
product.
The group
GI.G 2
is defined as F/R where F is the free
group on the generators ~ and b, one a for each for each
a ~ G1
and one
b ~ G 2 , and R is the normal subgroup of F generated by
~1.~2-(a-'~2)-1
and
are defined by
bl.b2-(b-"~2)-1
it(a) = ~
and
The homomorphisms J l
J2(b) = b
It turns out that every element in
GI.G 2
for
with
a i e GI\O
morphlsms Jl and J2 are monomorphlc.
and
for example
, b i e G2\O .
The homo-
If the context permits,
often regards G 1 and G 2 as subgroups of
b e G2 .
has a unique reduced or
normal form as 1 or an "alternating product", bl~2b2...a2n_Ib2n_la2n
a e G1
and J2
one
GI.G 2 , suppressing Jl and
J2 and the bars from the notation.
4.3 LEMMA.
(a) The free product is a functor from pairs of groups
to groups; given
fk: Gk " Hk
for k=1,2
, then
fl*f2 = lJl.fl,J2-f21: GI*G 2 - H1.H 2 (b) If fl and f2 are monomorphlc resp. eplmorphic,
then so is
fl*f2 • (c) If fl and f2 are arbitrary homomorphlsms, is generated by of
J1(Ker fl)
and
J2(Ker f2 )
then
Ker(fl,f2)
as a normal subgroup
GI*G 2 . (d) If
G k = Fk/R k
are free presentations of G 1 and G2, where
F k is the free group on the set Xk, then F ~ F1.F 2
the free group on the disjoint
G1.G 2 ~ F/R XIUX 2
with
and R the normal
103
subgroup generated by
PROOF.
Jl(RI)
and
J2(R2)
Part (b) is immediate from 4.2 and (d) from (c).
cerning (c), this is the salient point: lies in
Ker(fl.f2)
, one must have
Con-
if an alternating product z
fl(ak) = 1
or
f2(bk) = 1
for at least one k, by appeal to the reduced form in
HI.H 2 .
One
proceeds by induction on the "length" of z.
4.4 THEOREM.
Given generalized representation
and e2 of Q2 as in (4.1). e = el.e
2
:
K ~
Let
K = Ker(.1.w 2)
~ G1.G 2
and consider
~1"~2.~
.. Q 1 . Q 2
.
(a) Then c(e) is a generalized representation (b) If moreover e I and e 2 are representation Q2' respectively, (c) M(Q1.Q2)
then c(e) is a representation is the internal direct sum of
Im M(J2 ) , where
M(Jk): M(Qk) - M(QI.Q2)
M(QI*O 2) ~ M(O1)xM(Q 2)
JlZlNq
By Lemma 4.3, .1.~2
and
J2~2N 2
group of
QI.Q2
.
groups of Q1 and group of
Q1.Q2
Im M(J 1)
.
and
are monomorphic;
This direct sum decomposition
with respect to pairs of homomorphlsms
PROOF.
groups e I of Q1
thus
is natural
Q1 " Q1 ' Q2 " Q2 "
is epimorphic and K is generated by
as a normal subgroup of
G1.G 2 .
In general,
K is not central. (a) Given any complex vector space V and a homomorphism ~: QI*G2 - PGL(V) groups,
. Since the e k are generalized representation
there are homomorphisms
ak
oV : Let
C*~
~=181,82}:
6 ~ GL(V)
8k: G k - GL(V)
with
~Sk = (~Jk)"k '
~k .~ Ok
~
PGL(V)
G1.G 2 - GL(V)
, then clearly
~8 = ~.(.I.~2)
.
104 Thus we obtain a morphism centralizing,
(',8',~):
in this fashion, QI*Q2
(',B,7): c(e)
-
e - CV
C(av)=a V .
of extensions and, by Since all ~ can be lifted
c(e) is a generalized representation
group of
by Remark 2.3.
(b) We show that c(e) is stem provided e I and e 2 are stem. the kernel group of c(e) is to prove
K ~ [G,G]
by assumption;
Now
K/[K,G]
with
G = G1.G 2 , it suffices
JkKkNk E Jk[Gk,Gk ] S [G,G] for k=1,2
use the initial remark.
(c) We analyze the situation of (b) in greater detail. as K is generated by K/[K,G] where of
J1~1N1
is generated by ak: N k - K/[K,G)
,I.~2
and
cIN I
e.(e k)
e.(e)
as an abelian group, By the definition
ek - e
of extensions.
diagrams
~ Nk t~k [K,a]
The horizontal maps are isomorphisms
combined with Proposition 2.14. with
M k := Im M(Jk)
thus
M(J2)
M 1H
as a normal subgroup,
are the obvious maps. (.,Jk,Jk):
Again,
K
M(Q1.Q2)> .
a2N 2
of 8. yields commutative
M(Qk) ~ ~M(Jk)
kzl,2
J2~2N2
and
, we have morphisms
The naturality
for
Since
.
Now
p2.J2 = I
is monomorphic.
Ker M(P2 ).
Therefore
Consequently
Next,
implies
p2.Jl = 0
by assertion
(b)
M(QI.Q 2) = MI"M 2 M(P2).M(J2) and
M(O) = 0
= 1 , imply
MIDM 2 ~ (Ker M(P2 ~ Q M 2 = 0 . The natu-
rality of the direct sum decomposition
is immediate from the func-
torlality of the free product.
4.5 PROPOSITION. with
M(G i) = 0 .
PROOF.
Let G be the free product of groups Then
Gi, i e I,
M(G) = 0
By 3.8(i), every complex projective
representation
of G i
105
can be lifted to a linear representation.
This property carries
over to G by the universal property of the free product. M(G) = 0
by Remark 2.3.
Alternatively,
Thus
this is immediate from
Theorem 4.4 (c) for finite I and in the general case follows by a direct limit argument; consider the directed system of the finite subsets of I and apply Proposition 1.5.10.
4.6 EXAMPLES.(a) The modular group known to be isomorphic to of order 2 and
Z/2.Z/3
b = ± (~ -~)
the multlpllcator of
[]
PSL(2,Z) = SL(2,Z)/Center with generators
of order 3.
PSL(2,Z)
As
is trivial.
for a different approach to this result.) complex projective representation of linear representation.
projective representation,
a = ± (~ -~)
M(Z/2) = M(Z/3) = 0 , (See Example IV.1.7 (d)
Consequently,
PSL(2,Z)
In particular,
is
every
can be lifted to a
this applies to the defining
treated as a complex one.
Indeed, the
assignment )(0
~)
with
~ = exp(~)
determines a faithful complex representation of degree 2, i.e. PSL(2,X)
appears as a subgroup of
GL(2,C)
(b) The infinite dihedral group is defined as
D® = z ~ ( z / 2 ) It is known that D groups
( a M > and
= group ( a,y
: y2
= I
, yay-!
= a -1)
.
is isomorphic to the free product of the sub(y>
of order 2, therefore
M(D ) = 0 .
(Continued in Example IV.I.7 (e).)
We now treat the direct product of two groups. injection and projection maps, as depicted in (4.4)
iI GI~---~--~G I x G 2 ~ _ L ~ G 2 , Pl P2
The canonical
106 satisfy identities analogous to (4.3). can: G1.G 2 - G I × G 2 for
is given by
k ~ i, 1 ~ k, 1 ~ 2 . 4.7 LEMMA.
Pk.can.Jk = 1
and
can: GI.G 2 " G 1 × G 2
is
the subgroup generated by all The conventions
PROOF.
Clearly
is normal.
G1.G 2
Ker(can) S [GI,G2]
4.8 DEFINITION. A.B/D
.
[A,B] denotes
with
of 4.2 explain the notation
[GI,G2] ~ Ker(can)
b e J2G2 , c ~ [G1,G2]
and B is
[a,b] = aba-lb -I
has the form Then
a ~ A
G1.G2/[GI,G 2j
z = a-b-c
can(z) = (a,b)
and
[G1,G 2] .
By 1.4.2 (a)
An appeal to the factor group
every element of
[JlGI,J2G2 ] ,
[GI,G 2] .
Recall that, for subgroups A and B of some group,
b e B .
Pk.can.Jl = 0
It is epimorphic.
The kernel of
usually abbreviated to
The canonical homomorphism
with
[GI,G2] shows that a ~ JIG1
,
and
[7
The metabelian product
where
group of A.B such that
D = [[A,B],A.B] [A,B]/D
A o B
of the groups A
is the smallest normal sub-
is central in
A.B/D
.
The name "metabellan product" was coined by GOLOVIN [2]. This construction
(in the case of two factors as above) agrees with the
earlier "S-product" potent product";
of LEVI [1]. Another common name is "second nil-
this is also due to GOLOVIN [I], exept that he
used a different convention for counting the nilpotency class ("first" instead of "second"). to construct a representation
Following WIEGOLD [2], we are going group of a direct product as a met-
abellan product of representation
4.9 PROPOSITION.
groups.
Let A and B be arbitrary groups. The subgroup
[A,B] of A*B is a free group with basis
i [a,b]
I aeA, b~B, a~1~b
}.
107
T h i s r e s u l t and the f o l l o w i n g
PROOF.
Every
"elementary" [b,a]
commutators,
= In,b] -q
generality, inverse.
e l e m e n t x of
with
commutator
W e p r o v e by i n d u c t i o n survive u n c h a n g e d
plained
in 4.2.
that,
n = 1
Wn+ q = W n [ b n + 1 , a n + q ]
fication.
Thus
last two l e t t e r s
Wn+ I
reduces
to
Consequently,
the set of e l e m e n t a r y
commutators
(MacHENRY
[I],
Here
A @ B
is d e f i n e d as
PROOF. [A,B]/D A
x
B
factors
is central -
A o B over
~A
A x B
-
by Lemma 4.7.
extension
and the
n ~ I
is
and
For a r b i t r a r y of groups,
x B ,
and
, NCbO)
= (1,b)
a = a[A,A]
A o B , the set map
yields
r e m a i n un-
by our speci-
for all
can: A . B ~ A x B
A a b x Bab
to
w n . . . . [bn,an]
[I]).
,CAD) = (a,1)
is a b l h o m o m o r p h i s m
the t e n s o r p r o d u c t = Im ~
in
reduces
an+ I + a n
cf. W I E G O L D
Aab@Bab
It is clear that
Wn+ q
If
is a basis.
> A o B and
by its
w n = . . . b * a n - q b n -q
is possible.
wn ~ 1
(4.5)
= [a,b].D
loss of
form of x, as ex-
to
The case
exists a central
x(~eb)
or
Consider
, then
groups A and B, there
with
Without
...b*(anlan+l)bnlan+1
survive unchanged.
e : A ® B~
reduced
, then
h a n d l e d by symmetry.
4.10 T H E O R E M
[a,bJ
; the last two l e t t e r s
Wn+ 1 = W n - [ b n , a n + 1 ]
n ~ 0
n ~ q , the last two
reduces
bn+ q + b n
...b*an-1(bn-lbn+l)an+Ibn~lan~1 If
for
, no c a n c e l l a t i o n
with
of some
of the form
is clear.
w h i c h by i n d u c t i o n
[1;chp.IJ.
in x is not f o l l o w e d
in the u n i q u e
The case
Wn+ q = W n [ a n + q , b n + q ]
touched.
is a p r o d u c t
a e A, b e B, a ~ I ~ b .
letters
If
[A,B]
the l a t t e r b e i n g
an e l e m e n t a r y
w n = ...[an,bn]
p r o o f are due to G O L O V I N
by Lemma
1.4.1.
, and the u n i v e r s a l
~ as specified.
etc..
induces (a,b)I
Now
W e are g o i n g to c o n s t r u c t
.
,.
Since
~ [a,b]D
:
This map property
of
Ker , = [A,B]/D = 4: Im ~ - A ® B
108
with
*.~'
To this
= 1
end,
~: [A,B]
where
we invoke
- AeB
formula
by
extends
identities
~, = ~ I A e B , I m ~ ; c o n s e q u e n t l y Proposition
~a,b]
to the
1.4.1
= a~b
cases
(a),(b)
4.9 and define for
a = 1
in the
a ~ A and
and
Ker
~ = 0 .
a homomorphlsm b e B
b = 1 .)
.
Recall
(The the
form
X[a,b] = [ ~ a , b ] [ x , b ] -1 , Y[a,b] = [a,y]-1 [a,yb] where
now
a,x ~ A
and
~(X[a,b])
= (~.~)~
~(Y[a,b])
= -
Consequently a n d all
and thus
@'~'
~
where
c(~)
Thus
: N •
t([a,b].D)
Given
central
(b)
If b o t h
(c)
If e I a n d
:
group
Im M ( i l )
map
= 0
D and
= a@b
.
for all
u e [A,B]
induces Clearly
of
extensions
of 4 . 1 0
Iz(aeb)
-
the
= aeb
stem,
extension
-
QI x Q2
representation group then
e 2 are representation
e 2 by Q2 a s
'
GI x G2
representation e2 are
e I by QI a n d
consider
P ~ QI x Q2
GI o G2
of
and
groups,
Q1 x Q2
N = K e r p. then
"
so is ~. groups,
then
~ is a r e p r e -
Q1 x Q2 " multipllcator
M ( Q I × Q 2)
and
and
Im M ( i 2 )
?: G I ~ 2 - M ( Q I X Q 2 )
composite
~[u,z]
e 2 are g e n e r a l i z e d
e I and
(d) T h e S c h u r
where
,
= ,[a,b]
resp.
with
is a g e n e r a l i z e d
sum of
= ,[a,b]
= ~
~ G1 o G2
If e I a n d
sentation
= ~
= annihilates
In the n o t a t i o n
p=(wIX,2).,
(a)
= ~(u)
Thus
= 1
THEOREM.
in (4.1). (4.6)
.
- AeB
- ~ + ~(~.~)
~(Zu)
z e A.B
t: [ A , B ] / D
4.11
~
b,y E B
in t e r m s
is the
Ker M(Pl) of (c)
internal
R Ker M(P2)
is d e s c r i b e d
direct = Im ~ ,
as the
109 "1 @"2
Q1 @ 02 4~
with ~' restricting Schur-K~nneth
(4.7)
~'
¢ G1 ~ G2
3: G I e G 2
e.(~)
- M(%xQ 2)
> Ker p @ -
G I
o
%
G 2
There results the
.
formula
M(Q 1 x Q2 ) ~ M(Q1) x M(Q2) x (QI ® Q2 ) "
PROOF.
(a) We reduce the problem to Theorem 4.4 (a).
According
to Proposition 2.4, we obtain a generalized representation
group of
the form
G/[K.G] P' --""-~Q1 [L,G]/[K,G]
G [L,G]
where
x Q2 '
G = G1.G 2 , K = K e r ( . 1 . ~ 2 )
, L = ( ~ 1 . . 2 ) - 1 [ Q 1 , Q 2 ] = K'[G1,G 2]
and p' is induced by
o a n Q . ( . l * . 2) = ( . l X . 2 ) . c a n G : Since
[G1,G2]
i s a normal subgroup o f G, we have
D = [G,[G1,G2] ] ~ [L,G] group of
c(~)
G/D [L,G]/D
=
GI*G 2 - QlXQ 2 .
and
N = L/D .
Consequently,
is
G [L,G]
(b) We r e c a l l ~1N1 ~ [G1,G1]
from above t h a t and
N = K'[G1,G2]/D .
~2N2 ~ [G2,G2]
of 4.4 (b), we conclude
K ~ [G,G]
by assumption. and finally
are left to show that e is a central extension. in G, the Three-Subgroups [~1N1,G2] ~ [[GI,GI],G2] have
the middle
[~INI,GI] = 0
[~2N2,G] ~ D .
Now As i n t h e p r o o f
N ~ [G,G]/D
.
We
Since D is normal
Lemma 1.4.3 gives ~ D .
Since
and consequently
~INI
is central in GI, we
[~INI,G] ~ D .
By Lemma 4.3 (c), K is generated by
as a normal subgroup of G.
Together,
K'[GI,G2]/D
Similarly ~INI
and
is central in
u/D
(c) This assertion is the combination
~2N2
of the previous
steps;
110
note that ~ itself is central by (b). (d) We keep the assumptions
of (c). The identities M(Pl).M(i I) = 1,
M(Pl).M(i 2) = 0 , etc. exhibit
M(ik): M(Qk) - M(QlXQ2)
as mono-
morphisms and imply M(QlXQ 2) = Im M(il) x Im M(i 2) x IKer M(p 1) n Ker M(P2)} an internal direct sum.
Note that
(G1)ab@(G2)ab " (Q1)ab®(Q2)ab assumptions,
(Wl)ab
and
In view of Thm. 4.10,
~1@~2 = (~l)ab@(~2)ab
by definition.
(W2)ab
and thus
combined with Proposition 2.14, all of
:
Under the present
wle, 2
~' is a well-defined
,
are isomorphisms.
monomorphism.
e.(e k)
and
By (c)
e.(E)
are
isomorphisms. We finally derive the formula for (Bk,Pk.,,pk):
e " ek
Im ~.
Consider the morphisms
of central extensions,
denote the restrictions
of
where
8k: N - N k
pk.~: GlOG 2 - GlXG 2 - G k .
The natu-
rallty of e, yields the formulas (4.8) Hence
~k-e.(e) = e.(ek).M(Pk)
for
k=1,2
.
%.(e) Im ~ ~ N 0 Im ~ ~ Ker 61 0 Ker 82
Ker M(Pl) N Ker M(P2) together with
implies
Im 7
The reverse inclusion follows from (4.8)
Ker 81 n Ker 82 ~ Ker(p1,)
0 Ker(P2, ) = Ker , = Im ~ .
D 4.12 EXAMPLES.
Assume that A and B have trivial multipllcator,
thus are their own representation
groups.
Then Theorem 4.11 exhibits
the metabelian product extension (4.5) as a representation A x B .
Specifically,
allowed: m=O (4.9)
or
G = group
m=n=O
consider .
( x,y,z
A = Z/m
B = Z/n
with
nlm ;
Then we obtain : z=[x,y], xm=yn=l,
together with the obvious epimorphlsm sentation group.
and
group of
[x,z]=[y,z]=l
G - Z/m x Z/n
)
as a repre-
In the finite case, these groups were already
found by FRUCHT [1; p.19].
We assert that the groups (4.9) are
isomorphic to the representation
groups described in 1.4.6.
Indeed,
111 the assignment morphism
over
x~--~T(I,0,O) 2/m x Z/n
4.13 PROPOSITION.
, hence
Given
If e I and e 2 as in (4.1) representation
groups,
eI x e2 :
Q1 x Q2
, yl
~ T(0,1,0).
(Q2)ab
groups
Q1 and
Q2
with
resp.
QI @ Q2 = 0 . generalized
then e G 1 x G2----~Q
resp.
generalized
QI @ Q2 = 0
are torsion
perfect.
groups
The p r o p o s i t i o n
products
and in this
the direct p r o d u c t
PROOF. ever both
Clearly
I x Q2
representation
(pk,Pk,Pk):
group
e I x e2
of
It suffices
groups;
group.
Hence
to prove
The obvious yield
we obtain
)"N I
~
e.(el)xe.(e2 )
M(Q1) x M(Q2)
KerlM(pl),M(P2)l Consequently morphisms.
direct
these are
it is stem whenthat
eI x e2
morphisms
PkOe.(elxe2 ) = a commutative
diagram
xN 2
11 'pI'p2' NI x N2
e . ( e l ) x e.(e2) is monomo~phic by assumption. (d) and the present
and
or when QI is
to finite
extension,
e.(elxe 2)
M(QIXQ2)
(Q1)ab
subgroups.
of extensions
k=1,2.
extension
finite n i l p o t e n t
is a central
stem.
whenever
sets of primes,
has an immediate
representation
for
IM(pl)'M(P2)I
on disjoint
of their Sylow
elxe 2 - e k
= e.(ek).M(p k)
is satisfied
form handles
e I and e 2 are
is a g e n e r a l i z e d
4.11
is a homo-
"
The a s s u m p t i o n
where
K
an isomorphism.
are r e p r e s e n t a t i o n
N I x N 2t
is a r e p r e s e n t a t i o n
•K
Moreover,
assumption,
= Ker M(pl)
the composite
0 Ker M(P2)
map and finally
~ QI @ G2 = 0 . e.(elxe2)
are mono-
by
112
The free and direct products are t~e extreme cases of regular products in the sense of GOLOVIN [I]. obtained the Schur multiplicators
We remark that HAEBICH
[I]
of arbitrary regular products G
and, when G is finite, representation groups of G. (In the case of two factors A and B, such a regular product is of the form with
T ~ [A,B]
.
Representation groups of
A.B/T
A.B/T
can always be
obtained in the spirit of our proof of Theorem 4.11.) mention that WIEGOLD [3] and ECKMANN/HILTON/STAMMBACH
We finally [2] have
treated the multipllcator of central products by arguments related to those given above.
113
5. The Covering Theory of Perfect Groups
STEINBERG [I] presented a new approach to the problem of how to find free presentations of certain matrix groups.
The matrix groups
in question were perfect, and KERVAIRE [I] found that many of Steinberg's results follow from this property alone.
We here pre-
sent the covering theory of perfect groups, which is analogous to the well-known covering theory of nice topological spaces and sheds a new light on Schur's representation groups.
Although the covering theory of perfect groups is quite pleasing, most applications of it require the computation of the Schur multiplicator of interesting groups and this may be quite hard.
Such
applications are the study of finite simple groups and of algebraic groups and algebraic K-theory.
In the first case, most of the inter-
esting information seems to be known, though much of it is still to be published,
cf. GRIESS ~2~.
In the other areas, we can barely
mention some of the existing work and research is going on.
5.fl DEFINITION. equivalently,
A group G is called perfect if
G = [G,G]
perfect cover of Q, if
An eplmorphism G
perfect group Q precisely when sion.
~: G " Q
(hence Q) is perfect and
Due to the exact sequence I.(3.3'), Ker ~ r
Gab = 0
or,
is called a Ker(~) E Z(G)
w is a perfect cover of the ~ G
~ Q
is a stem exten-
In this section, we feel free to identify ~ with the latter
extension.
5.2 EXAMPLES of perfect groups.
(a) The most familiar perfect
groups are various families of matrix groups, e.g.
SL(n,K)
and
114 PSL(n,K)
for n ~ 3 , or for
[1;II§6].
n = 2
IKI
when
> 3 , cf. HUPPERT
Here we allow finite as well as infinite
fields K, of
course. (b) Every simple non-abelian smallest non-trlvlal the alternating
group
(c) Another
integral
homology
n ~ 2 .
Indeed
spheres,
i.e.
isomorphic
(nice)
(KERVAIRE
e = e/O : M(Q)~
e.(e/O)
[1;§1]). ~ Q0
= I
is a perfect
Sn
for some
Actually,
the result
indicates
cf. also BAUMSLAG/DYER/
Let Q be a perfect
U = Ker
e.(e I) 2 M(Q)
e.(el)
~Q
Moreover,
is isomorphic
classes
of perfect
e/O
: M(Q) ~ U
covers
> QU
e/U
e.(e/U)
= U
cover
with and
IV,p.38].
'~Q
of
M(Q)
of the representaRecall
by 3.1, nU
between the iso-
of Q and the subgroups
Q, the uniqueness
extension ~U
every perfect
correspondence
is due to SCHUR [1;Satz
defined as an induced
, the extension
to e I over Q.
In the case of finite perfect tion groups
M(Q)
over Q to
Ker
This theorem gives a one-to-one morphism
U of
Conversely,
of Q is isomorphic .
Then
~ Q
cover of Q.
~G
group.
group
For all subgroups
el: N~
(5.1)
spaces X with
[I] on the "reverse plus-construction"
there is a unique representation
e/Ker
topological
to that of a sphere
groups abound in topology,
5.3 THEOREM
e/U
of the fundamental
[ 1].
HELLER
with
hence is
A 5 m PSL(2,5)
~1(X)ab ~ HI(X) ~ HI(S n) = 0 .
of KAN/THURSTON
The
finite perfect group must be simple,
family of perfect groups consists
groups of homology
that perfect
group is clearly perfect.
QO
QU : -0- "
that
e/U
is
.
115
PROOF
(ECKMANN/HILTON/STAMMBACH
the r e p r e s e n t a t i o n Proposition
3.2.
group As
e.(e/O)
e.(e/U)=nat:
M(Q)
cover
Conversely,
of Q.
By R e m a r k there
e,(el)
In view of
= U .
The reader
perfect
group
Let Q.
e
:
o Indeed, Let
RD[F.F]
~
denote
5.10
5.5 T H E O R E M e = (a,w): group.
N)
by
with
M(Q)
.
then
8 is also
PROOF.
e o , to be r e g a r d e d
to
e/0
,
.
yields
= Q
and clearly
central.
The obvious
morphism
e.c(e)
1.3.5
, we
= i.e.(eo)
perfect
cover
.
Since
conclude
the r e p r e s e n t a t i o n
(ECKMANN/HILTON/STAMMBACH >G---a~Q
be a central ~: X " Q
w°8 = ~
The lifting
of the
group
of Q in view
e/0
resp.
of T h e o r e m
5.6
below.
A homomorphism
~: X - G
group,
the last assertion.
extension
p[F,F]
Proposition
the u n i v e r s a l (iii)
cover.
implies
the inclusion.
of extensions
In this context,
and Prop.
be a perfect
be a free p r e s e n t a t i o n
~Q
due to
e.(eo)
is called
is a perfect
[R,F]
= i
eo
e/U
such that
, is c o n g r u e n t
e.c(e) = e.(e) = I .
~ N
= I , this
IF,F]
" R/[R,F]
e O ~ c(e)
~ Q
easily v e r i f i e s
c(e)
is an e x t e n s i o n
i: M(Q)~
(i,.,I):
of
by
of the r e p r e s e n t a t i o n
~: M ( Q ) / U 8.(e)
hence
> G
~ F--~Q
[R,F]
eo
el: N~
Then the following
as a "subextenslon" (5.2)
e: R "
= 0
of
= I , we have
let
and
The u n i q u e n e s s
Ext(Qab,M(Q))
an epimorphism;
.
5.4 REMARK.
from
and the uniqueness
U ~ M(Q)
[el] = ~.[e/U] Ker
~ M(Q)/U
3.4 (iii)
exists
follows
[1;Thm.5.3]).
[1;Thm.5.7]). extension
can be lifted
if, and only if,
8 is unique.
Let
and X a perfect to a h o m o m o r p h i s m
Im M(~) ~ Im M(~)
If ~ is surJective
and e is stem,
surJective.
By P r o p o s i t i o n
I,I.12 a lifting
exists
in
exactly
if
116
~*[e] E Cext(X,N)
is zero.
By P r o p o s i t i o n
1.3.5 and Theorem
1.3.8
we obtain: ~*[e] = 0 <---~e.(e~) < here
8.
is an isomorphism
Finally,
G = ~N-(Im 8) hence
due to
by Theorem
Xab = 0 .
I.I.10,
as
Im ~ = G .
aN E Z(G)n[G,G]
This condition also
Der(X,N,O)
(KERVAIRE[I ;§1]).
a homomorphlsm
we conclude
with
"V "~ = "U
M ( . u ) : M ( Q U) - M(Q)
M(Q U) ~ U .
,: G - Q
satisfies perfect
p r e c i s e l y when
Conversely
QU "
QO
The key point is
of finite perfect groups Q, the vanishing shown by I W A H O R I / M A T S U M O T O
PROOF.
Exact sequence
(1.3.3')
perfect cover of
Q¥ .
We obtain perfect covers
gives
Thus 8 with
T h e o r e m 5.5, p r e c i s e l y when Since
if the group G in a
in the universal covering group"
M(Q O) = 0 . of
perfect
M(Q O)
(Qu)o
In the case
was already
[1;§2].
and coker M(~U) ~ Eer ~U "
and surJectlve.
cover of QV "
cover.
may also serve as "the universal
of all the groups
U ~ V .
M(G) = 0 , then , is isomorphic
This theorem implies that the group e/O
, there exists
is monlc with image U and
, thus
over Q to the universal
cover
From
aN ~ [Im 8,Im 8] ,
M(Q)
If so, 8 is uniquely determined and is a perfect
Ker "U ~ M ( Q ) / U
=
Let Q be a perfect group, assume
For subgroups U and V of
8: QU " ~
perfect cover
= Hom(Xab,N)
[3
the n o t a t i o n of 5.3.
Moreover,
;
assume that ~ is surJectlve and e is stem. and
5.6 THEOREM
= 0
> Im M(~) ~ Ker e.(e) = Im M(~)
gives the uniqueness = 0 .
= e.(e)oM(~)
Im M(WU) = Ker e.(e/U) = U "V °~ = ~U
by
U ~ Im M(~V) = V ; such 8 is unique
Eer 8 ~ E e r ( ~ U) ~ Z(Q U) Next,
exists,
M(~U)
8U: Q0 " QU
, 8 is indeed a
is monlc by Thm. since
0 ~ U
1.4.4 (ii).
for all
117 subgroups cover
U S M(Q)
Conversely,
~: G ~ G .
Theorem
5.3; and
Then , is isomorphic U ~ M(Q U) ~ M(G)
5.7 PROPOSITION. group O, then
Let
-Z(G) = Z(Q)
= O
Trivially
,(t) = z .
t is central.
Consequently, G/Z(G)
wZ(G) ~ Z(Q)
We claim that
Certainly
the composition
the map
morphism
by Lemma
Now
[t,[g,h]]
Concerning implies
1.4.1.
= [t,ghg-lh-1]=
This suffices
Let
z e Z(Q)
Ker ~
The preceding
g~
for all
indicate
topological
members.
covers,
spaces,
the first assertion
the role of the fundamental
is characterized
by
Z(O) = 0 .
of a "smallest
here.
group".
A n addi-
In detail,
a family of perfect groups with two
or, equivalently, covering
we have
by commutators.
a strong analogy with the covering
The family consists
is the universal
extreme,
is a homo-
[t,gS.[t,h].[t,g]-[t,h] -I = 1
is the existence
Q = G/Z(G)
so that
is again central.
a perfect group G determines distinguished
g ~ G
G - G
group there being played by the Schur multlpllcator feature
t e G
As it was first
~[t,g]:
~ of perfect
results
theory of connected
by
of per-
for every
and
since the perfect group G is generated
a composition
that
.
[t,g] ~ Ker - ~ Z(G)
by GR[~N [I;p.3],
G 1 of
by
cover of the perfect
is centerless
It,g] = 1
observed
tional
~U
by the above.
be a perfect
cover and
for the perfect
group G.
PROOF. with
M(G)
over Q to some
= 0
w: G - Q
fect covers is a perfect perfect
assume
group and
M(G O) = O
Q ~ G1/Z(GI)
of the perfect
the groups U S Z(G O)
and covers all for all
Go/U
covers
where
On one side, GS .
GO GO
On the other
G I , and Q is characterized
118
5.8 EXAMPLE. well-known
We d e t e r m i n e
that
A5
the p e r f e c t
is isomorphic
(5.3) There
icosahedral
G = group is an obvious
~(t) = y .
= Z/2
(x,y
w ~ Z/2
Thus
A5
as a n o r m a l
Ker ~ =
Z/3
, and Z/5
0
subgroup.
cover. .
~M(As)
We are going to show and
1(1-i,-1+i~ ~(s) = ~ 1 + i , 1+i j with
o := ( ~
- 1)/2
(5.3),
shows that
(Our a p p e a l
$L(2,5)
per-
and by
A5
con-
~ G ,
z := s 3
we easily
subgroups
commutes generates
find
of
A5
and
G a b = 0 , thus are
Z/2 x Z/2
M(cycllc)
M(A 5) = 0
or
,
= 0 ,
M(A5)
~ Z/2
sequence
~ Ker ~
~ O
This i m p l i e s
Ker ~ ~ M(A 5) ~ Z/2
To this end, we define a 2 - d l m e n -
representation
~ of G by
and
1 ~-~i, -1 ) ~(t) = ~( 1 ,m+ci
and
~ := ( / 5 +
~(s)3 = ~ ( t ) 5 = [ ~ ( s ) - ~ C t ) ] 2 This
= 0
in G; it o b v i o u s l y
1.6.9 yields
z ~ 1
(unitary)
and
I •
M(G) = 0 , h e n c e the claim.
sional c o m p l e x
.
Thus
5.6, we have an exact
~ M(G)
M(G)
M ( Z / 2 x Z/2) ~ Z/2
argument
w(s) = x
groups determined
is central
The S y l o w
Since
the c o r e s t r i c t l o n By T h e o r e m
and hence
the p r e s e n t a t i o n
w is a p e r f e c t
and
The element
I 1,z,z-l,z2,z-2,...
Abellanlzing
with
One may d e d u c e
W e p r o c e e d to prove the claim.
Ker ~
the
and that ~ is the u n i v e r s a l
IGI = 120
alone.
w i t h the g e n e r a t o r s
group,
: x 3 = y5 = (xy)2 = I)
~: G - A 5
; the family of p e r f e c t
sists of G and
It is
: s3=tS=(st) 2 )
Ker
A5 .
A5 .
g r o u p be d e f i n e d by the p r e s e n t a t i o n
eplmorphlsm
We claim
fect cover of M(A5)
( s,t
of
to the i c o s a h e d r a l
l a t t e r h a v i n g a free p r e s e n t a t i o n Let the b i n a r y
covers
1)/2 = ~
-1
.
One may check
= (-~ _~) •
~ is indeed a r e p r e s e n t a t i o n
to this r e p r e s e n t a t i o n
of G and gives
is not at all a c c i d e n t a l ,
z ~ 1 cf.
.
119
Example IV.1.8).
It is sometimes useful to know that ~ is faithful
and z is the only element of order two in G. (Proof: Use the simplicity of
A5
5.9 REMARK.
and the order of
~(t) .)
The Schur multlpllcators of all finite simple groups
have been found, often by exhibiting a universal perfect cover (representation group).
For the results see the summary by
GRIESS [2] and follow the references given there.
Note that the
Schur multlpllcator of the Mathieu group
Z/12
Z/6 , according to MAZET [I].
M22
is
rather than
Relative to the existence question,
LEMPKEN [1] computed the multlplicator of Janko's simple group as trivial; later the existence of S. NORTON [I]~
J4
J4
has been asserted by
Let us not forget that already SCHUR [2] determined
the representation groups of all
PSL(2,q)
, de S~GUIER [1] and
SCHUR [3] those of the alternating groups.
The following proposition is often handy when you have a candidate ,: G - Q resp.
for the universal perfect cover, but do not yet know
M(G)
M(Q)
.
5.10 PROPOSITION
(Steinberg and Kervalre).
perfect cover of the perfect group Q.
Let
~: G - Q
be a
Then the following are equiva-
lent. (1) , is isomorphic over Q to the universal perfect cover of Q; (li) every representation group of G splits (as an extension); (ill) for every central extension
e: A:
; X
exists a (necessarily unique) lifting homomorphlsm ~.~
=
.
by Q, there
~: G - X
.
PROOF. M(G) = 0
~ Q
Note the assumption by Theorem 5.6.
Gab = 0 .
If
G ~ Qo ' then
Thus (il) holds due to
with
120
Cext(G,A) ~ Hom(M(G),A) 5.5.
Conversely,
of G.
= 0
while (lii) is immediate from Theorem
assume (li) and let e be a representation group
Then e is direct split and stem at the same time, thus
M(G) ~ kernel = 0 ; then (1) follows by Theorem 5.6.
Finally,
assume (ill) and let e be a representation group of Q; then M(G) m Im M(~) ~ Im M(~) = Ker e.(e) = 0
5.11 EXAMPLE:
the functor
an arbitrary ring with 1. GL(A) = ii b GL(n,A) Here the typical
K2
Let
by 5.5 and 5.6.
of algebraic K-theory. E(A)
[]
Let A be
be the subgroup of
which is generated by the elementary matrices.
nxn matrix X of
GL(n,A)
As identified with the
matrix
in
GL(n+I,A)
A matrix is called elementary,
if it agrees with
the identity matrix except for at most one off-dlagonal entry k ~ A ; call this matrix e~j
if k appears as the (i,J)-th entry
(Think of elementary row and column operations.) calculus, E(A) tator.
is centerless and each
Let the Steinberg group
free presentation:
e~,j , i $ j , is
St(A)
a
COmmu
-
be defined by the following
generators are the symbols
and natural numbers
By standard matrix
x iJ ~
for all
k m A
i $ j , the defining relations are =
(5.4b)
X U [xij,xjl] =
14j4l,i
(5.&c)
k u [xij,Xkl] = I
i~J~k+l~i
.
The point is that the corresponding relations hold in the asslgment
x kijw---~e~j
defines an eplmorphlsm
the kernel of which is Hilnor's relations (5.&b), St(A)
K2(A )
is perfect.
E(A)
, hence
¢: St(A) " E(A)
by definition.
Due to the
KERVAIRE [1] showed that
,
121
K2CA )~
; S t C A ) - - - ~ ECA)
is the universal perfect cover of
E(A)
and
For the proof,
K2(^) = Z(St(^)) ~ M(E(^))
5.10(ii)
is most appropriate;
finite field, then presentation
of
M(St(A)) = 0
the appeal to
see also MILNOR [I;§5].
K2(A) = 0
E(^)
.
, consequently
When A is a
and the above actually is a free
, cf. MILNOR [1;Cor.9.13].
On the other hand,
consult MILNOR's book [1] and the survey by DENNIS/STEIN K2(Z ) m Z/2
and other examples of known
5.12 EXAMPLES:
SL(n,Z).
[I] for
K2(A)
The concept of the Steinberg group can
also be defined for the Chevalley groups over commutative including
SL(n,Z)
KALLEN/STEIN
and
Sp(2n,Z)
rings,
Here the paper of van der
[I] is the last in a series, leading to complete
results whenever the Chevalley groups in question are perfect.
In
particular,
are
the Schur multlplicators
isomorphic to
Z/2 x Z/2
results of J.R. Silvester, KERVAIRE
:
SL(4,Z)
Combining the
as presented by MILNOR [I;§10], and of ~ Z/2
for
n~5 .
Finally
For every perfect group G, let
M(G)~
~G O
~ G
be a fixed universal perfect cover with (a) Every homomorphlsm a lifting
and
this will be shown in IV.1.7(d).
5.13 PROPOSITION. e(G) o
SL(3,Z)
by van der KALLEN [I].
[1], we find M ~ L ( n , Z ~
M(SL(2,Z)) =0,
of
~: G O - H o
map on the kernels is
~: G " H
e(G) o : M ( G ) ~ - - - - ~ G o ~ G
e(H) o : M(H)~
~Ho---@~H
.
.
between perfect groups yields
which is unique by a = M(?)
e.(e(G) O) = IM(G)
~o ~ = Y~o "
The induced
122 (b) If groups,
e=(~,w):
N~
~G
then the following
(5.5a)
NO
(5.5b)
M(N)
M(~)
assertion
is an extension
sequences
~ Go
This p r o p o s i t i o n suggestion
~Q
are exact:
~ 0o
~ O
M(~) ~ M(Q)
M(G)
is due to KERVAIRE
of E C K M A N N / H I L T O N / S T A M M B A C H implies
functoriality.
(a) As
M(G o) = 0
of p e r f e c t
~ 0 .
[I;§2],
our proof
[I;p.119].
Thus,
whenever
follows
a
The u n i q u e n e s s c is isomorphic,
so is ~.
PROOF. diate
from 5.5.
The n a t u r a l i t y
(b) As in I.(3.&), with
Ker
p = S
presentation etc.
and
of N.
,
let
of
.
,
we have normal
IF,R]
R = S-[R,R]
Then
• N
of G
is a free
we choose
e(G) o
; way.
The a s s e r t i o n
= [R,R].[F,S] F = S-[F,F]
Clearly Ker ~ = Im
Since N and G are perfect, As the groups
R etc.
are
in F, we conclude
[F,R] ~ [ R , S ] . [ [ F , F ] , S ] - [ [ F , F ] , [ R , R ] ] The other
inclusion NO~
~N
gives
~R
in the obvious
[F,R]/[F,S].
and
SC
No = [R,R] JR,S]
the Lunique maps ~ and ~ are induced
to
be a free p r e s e n t a t i o n
Then
Go = IF,F] [F,R]
is equivalent
is imme-
a = M(~)
loss of generality,
in (5.2).
with kernel
gives
~G
R = Ker(~p)
is surjective
the first a s s e r t i o n
8.
0: F
Without
as specified
Go = IF,F] [F,S]
by 5.6,
is trivial.
GO
~
>G
the exactness
• Go ~
~
~Q
of (5.5b).
Finally, 0
~0 []
~ [R,R].[F,S] the K e r - C o k e r
lemma
for
CHAPTER
III.
1. Isoclinic
ISOCLINISM
Groups
Throughout e :
ei central
this s e c t i o n we denote by
N~
:
and C e n t r a l E x t e n s i o n s
~ ~ G
Ni ~
~ ~ Q
~ Gi
extensions.
~ i ~ Qi
, i=1,2
Let us denote
by c the c o m m u t a t o r
function
of e, i.e. c
:
and by then
Q x Q
-
[G,G]
c i ,i=1,2
;
(~g,~h):
the c o m m u t a t o r
[G,G] 2 ~N , and the map
rig,hi
function
~-Ic
,
of
e i ; if O is abelian,
coincides
w i t h the c o m m u t a t o r
form of 1.3.10.
1.1 D E F I N I T I O N .
The central
isoclinic, p r e c i s e l y and
~: [GI,G2]~
when
there
~ [G2,G2]
extensions
eI
and
exist i s o m o r p h i s m s
e2
are called
~: QI:
, such that the f o l l o w i n g
~ Q2
diagram
is
commutative:
c1 Q1 x Q1
~ [GI'G1]
~x~ c2 ~ Q2 x Q2 i.e.
~[gl,hl]
,2g 2 = ~ 1 h 2 to
~ [G2'G2]
.
= [gR,h2] The p a i r
e 2 , d e n o t e d by
1.2 REMARK. morphism.
'
for all (~,~)
(~,~):
gl,hl e G I , ~2g 2 = ~ l g I , is called an i s o c l i n i s m
eI ~ e2 .
In 1.1 it s u f f i c e s
Furthermore,
from
to a s s u m e
~ determines
that
~ uniquely.
~ is a m o n o -
eI
124
Let G be a group.
Then we obtain
the f o l l o w i n g
central
extension:
nat (1.1)
eG :
As m e n t i o n e d
Z(C) ~
~ C
above,
In terms of 1.1,
Isocllnlsm
cllnlsm
Let G and H be groups.
precisely
(~,~)
when
from
G to H, notation:
(~,~):
~I(N1)
eH
G ~ H
Let
eH
PROOF.
Let
with
e I ~ e2
Isoclinisms
tion of 8 to e2 .
G2 . ~2(N2)
= Z(G 2)
if and only if
[g2,h]
= 1
for all
h e G2
.
This p r o v e s
Thus
follow
[G1,GI]
.
and that
(~,~):
~IZ(GI )
(~,~):
e ~ e , resp.
to
relation.
(~,~):
to
implies
G ~ G
are
a group e2
induces
A c l ( e 2)
be an i s o m o r p h i s m
Then
eI
onto
of iso-
e I ~ e2
G, and they c o n s t i t u t e from
and
[]
is an e q u i v a l e n c e
isoclinlsm
Acl(el) 2
immediately.
isocllnlsm
Each
e l s e
that ~ maps
sees that the c o m p o s i t i o n
of e resp.
from
(a,~,~):
to
g ~ G
, resp. Acl(G).
Let
G1
, and
for all
of the form
called autoclinlsms
an i s o m o r p h i s m
n,: G 1 / Z ( G 1 ) ~ - - ~ G 2 / Z ( G 2 )
= 1
(il)
from
be an isoclinism.
[gl,g]
One r e a d i l y
e2 ~ el
A n iso-
.
is a g a i n an isoclinlsm,
( -1,~-I):
T h e n G and H are c a l l e d
, then
(i) and
1.5 REMARKS. cllnlsms
from
gl E Z(G1)
Now,
[1].
is also c a l l e d an i s o c l l n i s m
(~,~):
w2g 2 = ~wlg I .
,2Z(G2 ) .
in H A L L
are isoclinic.
= Z(G 1) , if and only if
By 1.1 this holds,
to
to
is an I s o c l i n i s m
(ll)
Acl(e)
and
~ i n d u c e s an i s o m o r p h i s m
(~',~)
g2
eG
eG
1.4 P R O P O S I T I O N . (1)
of g r o u p s was d e f i n e d
it reads as follows:
1.3 D E F I N I T I O N . isoclinlc,
~G/Z(G)
(?,8')
and
8'
is an i s o c l l n l s m
the r e s t r i c from
e1
125 The e q u i v a l e n c e
classes
under
classes",
the classes
P. Hall.
By 1.4 the families
classes
The groups the a b e l i a n
to
extensions
"isocllnism
were c a l l e d
"families"
as those
e with
by
isoclinism
~(N) = Z(G)
, i.e.
eG .
groups,
isoclinlc
w h i c h thus
isocllnic
and the q u a t e r n i o n (~,~):
groups
are c a l l e d
can be r e g a r d e d
of central
w h i c h are
two n o n - a b e l i a n
Let
of isoclinlc
that consist
e is isomorphic
isoclinism
group
to the trivial
form a family.
groups
group are exactly
An easy example
is given by the dihedral
for
group D 8
Q8"
• I ~ e2
be an isocllnism.
~2
~-1~2
Then the b a c k w a r d
induced
extension
~*e2
: N2~
is isomorphic Thus
~ G2 to
~Q1
e 2 , and
(1Q1,E):
in many cases we can restrict
in w h i c h
~ is an i d e n t i t y
1.6 PROPOSITION.
(i)
Let
"2 ~g' = ~ I g '
(ii)
PROOF.
elements of
I = ~.lg'
tators
(~,E):
e I ~ e2
, g' " [Gl,Cl]
(n,~)
be an isoclinlsm.
Then
,
g' ~
= .2~g, follows
, x ~ [G1,G1]
~: [G1,GI]~
(i) By 1.1,
(il) Let
(iii)
to i s o c l i n i s m s
map.
~(gxg -1) = h E ( x ) h -1
~ 1 g = ~2 h ; i.e.
converse
ourselves
is an isoclinism.
~(~1N1 n [G1,G1] ) = ~2N2 n [G2,G2] .
(iii)
for all
e I ~ ~*e 2
~[G2,G2]
the a s s e r t i o n
holds
, y ~ G
, h ~ H
with
is an operator
isomorphism
for commutators,
and hence
[G1,G 1]
(~1N1 0 [G1,G1]) , which
implies
Then
(i) yields
~g' e (~2N2 N [G2,G2] ) .
The
analogously.
It is sufficient x = [gl,g2]
.
Let
to show that the a s s e r t i o n hi ~ G2
with
holds
.2hi = ~.lg i .
for commuThen
126
w2(hhi h-l)
-1
= ~,2(ggi g
)
and t h e r e f o r e
~(gxg -1) = g([gglg-l,gg2g-1])
= [b_hlh-l,nh2 n-l]
= h [ h l , h 2 ] h -1 = h x h - 1
As we have m e n t i o n e d induce
isoclinlsms.
This
1.7 DEFINITION. clinic, with
if there
(?,~'):
morphlsm,
e I ~ e2 .
(a,B,~)
8 0 [GI,G1] (ii)
~,
=
of central
gives
(a,B,?):
e I " e2 6':
Let
an isoclinic
(a,B,~):
is isoclinlc,
is called
iso-
[G1,G1]~---~[G2,G2 ] an epl- resp.
epi- resp.
e I - e2
precisely
extensions
rise to the following:
If 8 is in a d d i t i o n
is called
mono-
monomorphism.
be a morphism.
when
~ is isomorphic
and
= 0
If (~,~,~)
is isocllnic
and
6'
as in 1.7,
then
~[[GI,GI]
PROOF.
As
(~,8,?)
8[gl,g 2] = [hl,h 2] proves
(ii).
Hence
is a m o r p h i s m
, where
morphism,
cf.
1.2.
Ker 8 n [G1,GI]
Then
if and only if we call
and
(~,8,~)
S[[G1,GI]:
is isocllnlc,
[GI,GI]
- [G2,G2]
is e q u i v a l e n t
Let G and H be groups
B induces Ker
an isocllnlc
8 0 [G,G]
.
This
if and only is a mono-
to
= 0
and
= 0
and
morphism
and
6: G - H
a homo-
from
to
Im(8)Z(H)
eG
is obvious
Im(0)Z(H)
= H
by 1.8. .
Let
eH
,
= Hi; in this case
homomorphism.
The o n l y - i f - p a r t
8 N [G,G]
that
The last c o n d i t i o n
8 an isocllnlc
PROOF.
we have
= 0 .
1.9 PROPOSITION. morphism.
of extensions,
gi e G I , h i e G 2 , ~ I g i = ,2hl
we obtain
if ? is an i s o m o r p h i s m
Ker
observation
exists an i s o m o r p h i s m
1.8 PROPOSITION.
Ker
[]
isomorphlsms
A morphism
(a,B,?)
(1)
in 1.5,
=
Assume g e G
with
127
8(g) e Z(H) Ker
Hence we have
8 0 [G,G]
= 0
implies
g e Z(G)
, we have
Z(Im(~))
= Z(H)
that
8 induces
Im(8)Z(H)
[g,g']
8g e Z(Im
0 Im(B),
8)
implies
= 1
for all
= I , proving .
As
and thus
a monomorphism
= H
by 1.8.
8[g,g,]
Im(8)Z(H)
G/Z(G)
, and
g e Z(G)
8g e Z(H)
from
g, e G
= H
.
, we obtain
Now we have into
H/Z(H)
that it is an isomorphism,
If
shown
, while
and we are done
[]
1.10 REMARKS.
(i) A m o r p h l s m
(e,8,Y):
if and only if ? is an i s o m o r p h i s m homomorphlsm
and
e I " e2
is isocllnlc,
8: G 1 " G 2
is an isoclinlc
of groups.
(li) The c o m p o s i t i o n
of isocllnlc
morphlsms
is an isoclinic
morphlsm. (lii) E a c h epimorphlsm
isocllnic
and an isocllnlc
In the f o l l o w i n g Q = Q1
"
morphlsm
we consider
an i s o m o r p h i s m
T h e n we have the following
w i t h the fibre p r o d u c t sometimes
denoted
~ (gl,g2) ~
by ~IgI
~i: N I X N 2 " - - ~ N i
and
71 = 1Q : Q ~ - - ~ Q 1 we can readily
1.11
I (gl,g2)
GI& G 2
, and
N 1 x N2~
~ = ~IX,2
~i: ~
$ Gi
e _~ Q
(1.2)
ei :
Ni'~
~ Gi-
"
Having
~ Qi
} ,
,
Furthermore
proposition,
(1) The diagrams > ~.,
n~Ig I = ~2g 2
we denote
the i-th p r o j e c t i o n s
# Q2
and put
,
I gi e Gi,
I.(2.2)o
)'
~: QI " Q2
extension
~ ~Q
I , cf.
the following
PROPOSITION.
central
~ ~ G
G =
' ~2 = ~: Q~
prove
of an Isocllnlc
monomorphlsm.
= (e I x ~*e2)A Q : N 1 x N 2 ~
=
is a c o m p o s i t i o n
by
and put
fixed the notation, cf. J O N E S / W I E G O L D
[2].
128
are c o m m u t a t i v e . (li) The i s o m o r p h i s m e I to e2,
~: Q I ~ - - ~ Q 2
if and only if
(oi,~i,~i)
i n d u c e s an i s o c l i n i s m , i=1,2
are i s o c l i n i c
from
epi-
morphisms.
PROOF.
The p r o o f of (1) is trivial.
isoclinlsm
from e I to e 2.
~: [G1,G1]
- [G2,G2]
~ l g i = ~2hl
.
an
~[gl'g2 ~ = [hi'h2]
' gi E G I , h i ~ G ,
(gl,hl) E G , and we obtain =
[(gl,hl),(g2,h2)]
that n induces
Then we have an i s o m o r p h i s m
with
Thus,
Assume
•
([gl,g2],~[gl,g2])
This i m p l i e s
= I (g',~g')
[G,G] As
Ker T 1 = kN 2
By 1.8,
Let
follows
e: N)
group.
and
(oi,Ti,~i)
converse
from D e f i n i t i o n
~> G
W~Q
1.7.
~': N;
xxl>
> (g,1)}:
~ N x A
e x A
:
e
:
N x Ay
are c o m m u t a t i v e , (U',U,IQ):
As
~ = ~ 2 ~ 1 , the
F~
be a central
G x A
G - GxA
similarly.
N
Ker T i n [G,G] = 0 .
extension of
G x A
and A an a b e l l a n onto G and we
extension
: N x A~
u=~g,
Let
epimorphlsms.
We denote by m the p r o j e c t i o n
e x A
and
Ker T 2 = ~N 1 , we obtain
are isoclinlc
o b t a i n the c e n t r a l
Let
I g' ~ [CI,G 1 ] I
p
(~',m, IQ): e - exA
el,e2,e,oi,~i,~i
, define
. ~': N × A
N
and
T h e n the d i a g r a m s
~xl ~ G
'¢
~L~ ~ Q
x A
~
ffQ
G
;~-Q
exA - e
is an i s o c l l n l c
is an i s o c l l n l c be as above,
eplmorphlsm,
monomorphlsm. ~: O 1 ~ - - ~ O 2
an isomorphism,
129 and
A = ~ab =
, and
I n,
) (~In,ab(~n))
~ :
= I g~--~(~lg,ab(g)) 1.12 LEMMA. Then
(a,B,~l):
PROOF.
Let
from
[G1,GI]
Then
Tl(g7
e - elXA
and it follows that
Let
(n,~7:
tative
diagram
of isocllnlsm. sN I =
ab(g)
g2 = 1
=
=
[G,G]
=
,
Ker
is a
by ~ the i s o m o r p h i s m (n,~7:
(g1,g27
~ = O
e I ~ e2 • E [~,~]
.
I gl ~ [GI'G1 ] ~'
, and by 1.8 we
monomorphism.
[]
Now we consider
w h i c h will
the commu-
yield two c h a r a c t e r i z a t i o n s
holds:
I nI E N1 I ,
I (Xlnl,ab(~lnl,1))
I g ~
(~,8,~1)
i (gl,Egl)
be an isocllnlsm.
The f o l l o w i n g
I (g,~g)
, i.e.
is an isoclinlc
below
that
by n, i.e.
= 1
from e I to e 2.
monomorphlsm.
implies
Thus we have
eI ~ e2 (1.37
an i s o c l i n l s m
B , and denote
of 1.11 we obtain
I (nl,ab(,lnl,1)
Ker(nat)
of (1.27
induced
and
(a,~,Y17
~G 1 ×A
~ induces
e Ker
[G2,G2]
= gl = 1
C
pN 1 × A
is an isocllnic
g = (gl,g27 to
F r o m the p r o o f
[G,~]
that
The c o m m u t a t l v i t y
morphism.
obtain
Assume
I :
NI × N2
I nI ~ N1 I ,
[G1,G 1]
t •
This yields Ker(nat)
n [GlXA,GlXA]
= O ,
aN 1 O ~'N 1 = 0 . Furthermore
we have
1.13 L~.MMA. (U',U,IQ1)
Ker(nat'a)
= N 1 = Ker
(1) The c o m p o s i t i o n
is an isoclinic
(ll) The c o m p o s i t i o n
of
of
monomorphlsm
~2 ' and we obtain:
(nat',nat,1Q17 from
(nat',nat,1Q17
eI and
into
and nat'(elXA7
(~,~,~17
induces
.
130 Diagram
(1.3)
e2
:
~
:
N2~
) G2
'~ Q2
"I' //I
~Q
N1 x N 2 ;
1
:['¢ 1
!
!
I eI xA
rr2
~2
:INlxA~
;I
I \ \
at'
IG 1 x A I I
at I
nat'(elXA)
__t1 -
aN 1
:
N1 x A
G1 x A
aN 1
IB~.N1
/ I
at'
I I
e I xA
/ I !
\
\
U'
"T
'T
I
: I N I x A; I
nat
~IG 1 xA
~r
"- Q1
I
\
\
rT1 eI
:
N1
G1
131
an isoclinic
monomorphism
The following
from
theorems
e2
summarize
can be easily read from diagram
1.14 THEOREM.
(ll) There epimorphisms
extensions
the proofs
are equivalent:
e I and e 2 are isocllnlc. extension
e' together with isocllnlc
from e' onto e I and e 2. exists a central
clinic monomorphlsms
1.15 REMARK.
extension
If e I and e 2 are finite,
sketched by
In particular,
e" together with iso-
from e I and e 2 into e".
1.14 can also be chosen finite.
generated
[]
the results above,
properties
exists a central
(ill) There
roughly
nat'(e IxA)
(1.3).
The following
(1) The central
into
eI ~
by isoclinic
extensions
The situations
e'
the equivalence
[]
~ e2
and
relations
epimorphism,
e' and e" in
in 1.14 can be el~
~ e"4
(e 2 .
for central
extensions
Isocllnic
monomorphism
resp.
coincide with isoclinism.
1.16 THEOREM. (i)
The following
properties
are equivalent:
e I and e 2 are isoclinlc.
(il)
There
an isoclinic epimorphlsm (lii)
exists an abelian group A, a central
monomorphism
from e' into
e',
, and an isoclinlc
from e' onto e 2. There
exists an abelian group B, a central
an isocllnic
epimorphlsm
monomorphism
from e 2 into e".
The statements el.~"--~
eI x A
extension
eI
from
eI x B
onto e", and an isocllnic
of 1.16 can be sketched as follows:
x A ~,
< e~
~ e2
extension
e",
132
e ~ e For
I x B----~e"%
isoclinic
groups we obtain:
1.17 PROPOSITION (i)
4e 2 .
(Hall).
Let G be a group.
Let A be an abelian group.
Then G and
G × A
are iso-
clinic. (ii) morphism
Let N be a normal nat: G - - - @ G / N
subgroup
of G.
is an isoclinic
The natural homo-
eplmorphism,
precisely
if
N n [o,o] = o (iii)
If
equivalent
to
(iv)
is finite,
monomorphism,
If
equivalent
then isoclinism
G/Z(G)
to U.Z(G)
of O.
is finite,
The proof of (i),(ii),(iv)
(v)
[G,G]
Assume
that
As obvious.
= I[G,G]N/NI
=
to
G/N
I[G,G]I/I[G,G]
.
Then
N N I ,
and we
G/Z(G)
is finite and G isoclinic
Iulz(G)nul
= Iuz(G)Iz(a)l
to U.
lalz(G)l
.
Then This
= G .
If we restrict theorem,
to WEICHSEL
of G and U is
N N = 0 .
lalz(o)l = lulz(u)l implies U.Z(G)
of U into G is
(iii) and (v) do not hold in general.
be finite and G isoclinic
I[G/N,G/N]I
[G,G]
following
is
= G .
PROOF.
obtain
G/N
= G .
then isocllnism
The assertions
(iii) let
The embedding
if and only if U.Z(G)
Remark:
I[G,G]I =
of G and
N N [G,G] = 0 .
Let U be a subgroup
an isocllnic (v)
[G,G]
[I].
1.16 to isoclinism the equivalence
of groups,
we obtain the
of (i) and (ii) of which is due
133 1.18 THEOREM. properties
G and H are Isocllnlc.
(ll)
There
UZ(GxA)
exist an abellan group A, a subgroup U of
= G x A
, and a normal
N N [U,U] = 0 , such that (ill) G x B
U/N
M 0 [GxB,GxB]
= GxB/M
1.19 REMARK.
8: G
~H
Hence we have the following [H,H].
~-1
A ~ Gab
abelian group
.
G
(~,~): G ~ H , where
; H
,~ Hab
eplmorphlsm. ~ = 81[G,G ]
diagram with exact rows
,
On the other hand,
[G,G] = Im ~-1
if we fix H, while A is any
and
Ker ~ 0 [G,G] = O.
eplmorphlsm.
all groups which map epl-lsocllnlc
Hence,
are called quotient
out subgroups U such that
U.Z(G)
This
(1.4) determines
onto H.
In KING [I], groups G without a normal
subgroup N satisfying
irreducible,
and groups G with-
= G , are called
subgroup
irre-
King proves:
1.20 THEOREM.
(1)
The group G is quotient
only if the socle of G is contained Z(G)/(Z(G)
with
~ A
shows that B is an isocllnlc
ducible.
GxB/M
and G a group which fits into (1.4) for some epi-
8, then
N O [G,G] = 0
subgroup M of
to H.
be an isocllnlc
commutative
I
II
[H,H] c
morphlsm
to H.
= 0 , and a subgroup V of
Then 8 induces an Isocllnlsm
(1.4)
is isomorphic
, such that V is isomorphic
Let
G x A
subgroup N of U with
There exist an abellan group B, a normal
with
VZ(GxB/M)
where
Then the following
are equivalent:
(1)
with
Let G and H be groups.
R [G,G])
in
is a torsion group.
[G,G]
irreducible, and
if and
134 (ll)
If G is subgroup irreducible,
the Frattlni subgroup of G. Z(G)/(Z(G) (lii) ducible,
O [G,G]) If
then
Z(G)
is contained in
The converse holds, if
is finitely generated.
Z(G) ~ [G,G]
, then G is quotient and subgroup irre-
r]
For finite groups,
irreducibility of G means that G does not have
a subgroup or factor groups being isocllnic to G. that a finite group
One might expect
which is quotient and subgroup irreducible, is a
group of minimal order in its family.
But the metacyclic groups in
the following example show that this conjecture does not hold, of. KING [1].
1.21 EXAMPLE.
Let p be a prime and
n ~ 2
an integer.
Let us
define : G n = group
(a,b
n n n-1 : a p =b p =I , [a,b]=a p
We can easily verify n-1 Z(G n) = (aP)~ Z/p n-1
, [Gn,G n] =
)-~ Z/p
Gn/Z(G n) _~ Z/p × 2/p , M(Gn/Z(Gn)) Im 8w(eGn) = [Gn,Gn] Hence,
@.(eGn )
_~ 7/p n-1 n Z(Gn) =
is a monomorphism,
which proves that
generalized representation group of
eGn
is a
2/p x Z/p , cf. II.2.15.
In
2.4 it will be shown that all generalized representation groups of a given group are mutually Isoclinlc, can be verified that for all
Gn
cf. IV.2.15.
Using 1.20, it
is subgroup and quotient irreducible
n > 2 . m
Let us consider a family which contains finite representatives. From 1.6 we obtain that a group G in this family has minimal order, whenever
Z(G) ~ [G,G]
holds, i.e. if
eG
is a stem extension.
135 P. Hall called these groups that each family contains
"stem groups".
stem groups.
Let G be a group and U a subgroup we can choose a transversal
(gi)i~ I
We denote by
Ku(U)
resp.
KG(g)
of u in U.
KG(gIU)
, resp.
For all
= giKu(u)
In 2.6 it will be shown
u e U
of G with U-Z(G)
= G .
to U in G with the conJugacy
and all
gl
gl E Z(G)
class of g in G,
' we have
Hence each class of U corresponds
of G having the same length.
Now we assume
Hence
to
{I{ classes
that G is finite,
and let
w be a set of primes.
It is easy to see that one can choose the
system
gi
(gl) such that
is a ,-element
of
G/U.
(Decompose
,-element and a w'-element, ,-element, resp.
is a w-element,
CO)
where
=
denotes
1.22 PROPOSITION. integer,
k CG)'IHI. = k PROOF. with
gi u
is a
are ,-elements.
Let
k~(G)
,
classes of ,-elements
Then we have
the ,-part of
}G:U}
Let G and H be finite
and , a set of primes.
isocllnlc
n
CHHal..
= E
k~CG).IE:GI.
groups,
Then
By 1.14 we can regard G and H as subgroups
G.Z(E)
Using
of a
,
{G:U{,
positive
gi := gi U
Hence,
denote the number of conJugacy
in G, resp. U having length n. k
gl
when
in a unique product
which commute.)
if and only if u and
k~(U)
gi
precisely
and
H-Z(E)
= E
.
= k~(E) = k~CH)'IE:HI~
IE:GI. = IEI./IGI.
and
of a group E
Our considerations
above show
.
IE:HI. =
IEIJIHI.
, the assertion
follows.
Let
kn(G)
denote the number o f all classes
From 1.22 we obtain the following
o f G having
result of P. Hall:
length n.
136 1.23 COROLLARY.
kn(G)-IHI
Let G and H be finite isoclinic groups.
= kn(H)-IGI
Then
[-1
We close this section with a few remarks o n the correspondence given by the isomorphism of the central factor groups of two isoclinic groups.
Let
groups U of G with
U 2 Z(G)
which is defined by (~IU/Z(G),~) Z(H) c nU .
(~,~): G ~ H
we denote by
~U/Z(H) = ~(U/Z(G))
is an isocllnlsm
;~U~U/Z(H)
from
.
Z(G) e
~U
For all sub-
the subgroup of H,
It is easy to see that >U
~U/Z(G)
, and it induces an isocllnlsm
If G and H are finite,
V := PZ(G)
be an isoclinlsm.
P is a Sylow p-subgroup
, and Q a Sylow p-subgroup
Sylow p-subgroup of
~U .
of
to
from U to
of U,
nV , then Q is also a
It can also be shown that ~ induces an
isoclinism from P to O.
Isocllnlsm has many Invariants, nilpotency.
in particular
solvability and
Let
0 5 G I = Z(G) H G 2 E G3 S be the upper central
...
series of G, and
G I = G R G2 = [G,G] ~ G 3 ~ G 4 2 the lower central
series of G, and
0 _~ , 6 2 _ ~ 2 _ ~ 3 ~ _
.-. (~,~): G ~ H
Then
. . .
iS the upper central series of H, and 1.6 yields that H a
~[G,G]
= [H,H] a
gG3 a
gG4 ~
is the lower central series of H.
-..
137
2.
Isoclinlsm and the Schur Multiplicator
In this section we study the connection between isoclinism and the subgroups of the Schur multiplicators.
Assume as in Section 1
that e
:
Ny
ei :
~
> G
Ni~
~
~0
r Gi
are central extensions. xe.(e): M(Q) - G .
~ O i , i=1,2 Let us consider the homomorphism
We have
Im(~e.(e)) = xN O [G,G]
xe.(e)
Thus
.
induces a homomorphism
8': M(Q) - [G,G]
, such that
the sequence (2.1)
Ker e.(e) c-----,M(Q)
e, >[G,G]
is exact, w' being induced by e.(e)
~: G
~ Q .
~'~ [Q,Q] The naturality of
, cf. 1.3.5, yields the naturality of (2.1) with respect to
morphisms of central extensions.
This is quite useful for studying
isoclinic morphlsms.
2.1LEMMA. sions
Let
(~,~,~): e I - e 2
and assume that
~: Q1 " Q2
be a morphlsm of central extenis isomorphic.
Then the fol-
lowing conditions are equivalent: (i) (ll)
PROOF. of
(~,~,~)
is isocllnlc
.
M(~) Ker 6.(e I) = Ker e.(e 2)
For any morphism
(2.1) yields
(a,8,7)
:
e I - e 2 , the n a t u r a l i t y
the commutatlvlty of the following diagram
138 !
Ker e . ( e 1) ~
e1
~ M(Q1 )
) [G1,G1]-----~[Q1,Q1]
t
Ker e . ( e 2) ~ where
> M(Q 2)
e~ , i=1,2
are defined as above
~, : vl[a1,¢1 ] M(~)
As ~ is isomorphic,
, the restriction of
e epimorphic. holds,
e2 ~ [G2,G2] "
By
M(~)
Proposition
if and only if
B'
to
~[Q2,G2]
,
, B' = ~I[G1,GI ] ,
the same holds for Ker e.(el)
1.8 , condition
is isomorphic,
~'
and
is monomorphic,
and
(i) of this lemma
being equivalent to (li)
by the diagram above.
2.2 LEMMA. tion I, let
Let
be an isomorphism and, as in Sec-
e = (elx~*e2)A Q .
Ker e.(e)
PROOF.
~: Qq " Q2
=
Then
e.(e 2)
M(~) - 1 K e r
n
From (1.2) and the naturality
e.(~) = e.(e 1) × M(n)e.(e 2) : proving 2.2.
Ker
e.(e I)
of
e.
we obtain
M(Q I) - N I x N 2 ,
~]
Now we are in the position to prove the main result of this chapter:
2.3 THEOREM.
Let
~: Q1 " Q2
be an isomorphism.
Then the fol-
lowing statements are equivalent: (i)
~ induces an isoclinism from
(li) ~'~le.(el) (ill)
PROOF.
There exists
eI
to
e': [GI,GI]P----~[G2,G 2]
e2 . with
= ~2e.(e2)M(~) M(~) Ker e.(el)
Condition
=
Ker e.(e 2) .
(il) makes sense, because
Im(~ie.(el) ) is
139
contained in ~1,~2
[Gi,Gi]
, cf. the definition of
are monomorphlc,
and
8',M(~)
8'
in (2.1).
are isomorphic,
As
(ii) implies
(lli). Let e be as above and assume (iii). Ker 8.(e) = Ker morphisms
e.(e 1)
This equation reads in terms of the epl-
(~i,~i,~i): e
M(~i) Ker 8.(e)
=
Then 2.2 shows
~e
from (1.2) as follows:
Ker e.(el)
, I=1,2 .
From 2.1, we obtain that these epimorphisms are isocllnic, and (i) follows from 1.11. Let
(~,8'): e I ~ e 2 • = ~il[~,~ ]
Then 1.11 and 1.8 imply
where
~
yield
8'~le.(e1) = ~28.(e2)M(~)
2.4 COROLLARY. defined in 1.5. Acl(e) ~
Diagrams
(i) Let
8' = T~T~ -I ,
(1.2) and the naturality of (2.1)
[3
Acl(e)
be the group of autocllnlsms
Then
I ~ I ~ E Aut(Q)
, M(~) Ker 8.(e) = Ker e.(e)
~ .
(ll) If e is a generalized representation group of Q, i.e. Ker
e.(e)
=
0
, then
Acl(e) ~ Aut(Q)
.
(ill) Any two generalized representation groups of a given group are isocllnic.
REMARK.
Corollary 2.4 (ill) is implicitly
contained in
HALL [I], for other proofs see GRUENBERG [1] and JONES/WIEGOLD [2].
In terms of projective representations,
2.5 COROLLARY.
Let
duces an isocllnism from
~: Q1 " Q2 eI
to
necessarily finite dimensional)
2.3 reads as follows:
be an isomorphism.
Then ~ in-
e 2 , if and only if for all (not projective representations P of G 2
over some algebraically closed field of characteristic
zero, the
140
following holds: P can be lifted in lifted in
PROOF.
e 2 , precisely if
P?
can be
eI
The projective representation
P~
can be lifted in
eI ,
if and only if P can be lifted in e' = (~-l)*e I :
N1)
~Q2
e.(el) = e.(e I ')M(~)
Furthermore we have
follows from II.2.20 and 2.3.
In the sequel,
~1
) G1
" .
Hence,
the corollary
[]
let us fix the group Q and consider an isoclinlsm
class @, which contains central extensions whose factor groups are isomorphic to Q. e: N~ of
) G---~Q
M(Q)
Then * has a representative , and by 2.3
, namely
Ker e.(e)
@ determines at least one subgroup In fact each extension in ~ is iso-
morphic to an extension of the form above. be any subgroup of extension
e'
M(Q)
with
In particular we have
.
Then 1.3.8
8.(e')=~: M(Q) Ker
of the form
On the other hand, let U
implies ~ M(G)/U
e.(e') = U , and
e'
the existence
of a
(natural projection). is a stem e x t e n s i o n
If we combine these observations with 2.3 and the naturality
of
e. ,
we easily obtain the following results of P. Hall:
2.6 PROPOSITION. stem extension.
(1) E a c h central extension is isoclinic
In particular,
each group is isoclinic
to a
to a stem
group. (ii) The isoclinism classes of central extensions with factor groups isomorphic to Q correspond to the orbits of set of subgroups of U S M(Q),
M(Q)
Aut(Q)
with respect to the action
U ~
on the ~ M(~)U
~ ~ Aut(Q)
2.7 PROPOSITION.
Let
~: Q1 ~ Q2
following conditions are equivalent:
be an isomorphism.
Then the
,
141
(1)
There exists a subgroup U of
isocllnlsm from (ii)
Nq
el/U: N 1 / U r - - ~ G 1 / " I U
M(~) Ker e.(e I )
~
such that ~ induces an ~Q1
NI
are given by the
Im(e.(el)) N U = 9.(el)M(~-l)Ker(e.(e2 )) .
PROOF.
By 2.3, condition (i) holds, if and only if
Ker(8.(el/U)) = M(~-1)Ker(8.(e2)). NI
e2 .
Ker 9.(e2)
If (ll) holds, the possible subgroups U of condition
to
onto
N1/U , then
Let ~ be the projection from
e.(el/U) = ~e.(e 1) , as
e.
is natural.
Hence, the equality above is equivalent to M(? -1) Ker e.(e2)
=
~ x I x ~ M(QI)
, e.(el)x ~ U I ,
which holds precisely when Im e.(e 1) n U
=
e.(el)M(~ -1) Ker A.(e2)
and M(~ -I) Ker 8.(e2) and we are done.
2.8REMARKS. of Q.
~
Ker
e.(el)
,
[]
(1)
Let e be a generalized representation group
Then the extensions
e/U
for
U ~ N
represent (in general
not uniquely) all isoclinism classes with factor groups isomorphic to Q. (ll)
Assume that e is a gen. representation group of Q.
xe.(e): M(Q) ~ .NO[G,G]
is isomorphic.
resp. 2.4 it induces an autocllnlsm of Isocllnlsm, where
~'
the action
M(~) , and
e.(e)
(~,~)
of
Acl(e)
from 2.6, and again
of e.
~'~e.(e) = ~e.(e)M(~),
~N 0 [G,G]
Aut(Q)
By 2.3
The definitions
Hence, we have
on the subgroups of
which is equivalent to the action of M(Q)
~ e Aut(O)
yield that
is the restriction of ~ to U~----~'U
Let
Then
KN n [G,G] ,
on the subgroups of
the orbits correspond uniquely to the
142
isocllnlsm tatives
classes
of these
with
orbits,
(e/Ui)i~ I , U i = ~-Iu~ (ill) tation
Assume
of
Q1
that
of
eI
e2
M(Q 1)
into
~ Q .
Chapter used
IV.
by
.
, and
T h e n the situation
eI - e2
with
U = Ker c •
el/U;
~ e2 , which
capable,
(a',8',~)
groups
Nevertheless
if there
e,(e I)
is
of 2.7 is
be any isomorphism.
Let
then
of a free presen-
Ker Then
8 = ~1Ker (a,~,~)
is isoclinic
a , induces
by 2.1.
is even isomorphic.
of 2.8 is left as an easy exercise.
These
are represen-
are r e p r e s e n t e d
NI 2 M(QI)
~: QI " Q2
(a,8,~):
(a',8',~):
A group G is called G/Z(G)
N1 .
Let
is a stem extension,
The p r o o f
classes
is the c e n t r a l i z a t i o n
aiM(Q1 ) = 8.(e2)M(~)
If
(Ui)i~ I '
.
more comfortable.
a monomorphism
If
the isocllnism
Then we have a m o r p h i s m and
Q.
In this case we have
the embedding slightly
factor group
[]
exists a group G with
will be studied
in more detail
we can show here,
how i s o c l i n i s m
in can be
for a first characterization.
2.9 PROPOSITION. sentation (i) (ii)
PROOF.
group
Let
of Q.
e: N; ~ ) G
Then the following
~N
=
Z(G)
conditions
repre-
are equivalent:
Obviously
.
(li)
implies
e': N'~
there
exists a subgroup
clinic
e'
Proposition
of
G/~U
(i).
x' ) G'
By 2.8,(i)
the center
be a g e n e r a l i z e d
Q is capable.
exists an e x t e n s i o n
to
~ ~Q
Assume ~Q
such that
U of N such that
1.4 implies
, and we obtain
(i).
that
~N = Z(G)
N/U
Then there ~'N' e/U
= Z(G') is iso-
is m a p p e d []
onto
143
2.10 REMARK.
Let Q be capable.
Then families of groups exist,
having Q as central factor group.
By 2.9 (the middle groups of)
the representation groups of Q also represent a unique family with Q as central quotient, and all other families with this property are represented by certain factor groups of a representation group. P. Hall called the family, which contains
the representation groups
of Q, the maximal family.
We close this section with a characterization of the groups i s o clinic to finite groups.
2.11 PROPOSITION
(KING [I]).
Let G be a group.
Then the fol-
lowing properties are equivalent: (1)
G is isoclinic to a finite group.
(ii)
G/Z(G)
(iii) G
PROOF. Assume that
is finite.
is isoclinlc to a finite subquotlent of itself.
The implications Q = G/Z(G)
generated subgroup
(iii) = (i) ~ (ii)
is finite.
G 1 of G with
clinic to G by 1.7,(iv).
Look at
are trivial.
Then there exists a finitely
G = Z(G)G 1 el: Z(GI)r
Thus
G 1 is iso-
~ GI
~Q
.
As a
central subgroup of finite index in a finitely generated group, Z(G1)
is finitely generated abelian.
Decompose
Z(GI) = T x A ,
where T is the (finite) torsion subgroup and A is torsionfree. M(Q)
is finite by II.3.10,
Z(G I) N [GI,GI] = Im e.(e I)
finite, hence lies in the torsion part, and follows from 1.17,(ii) that phism.
GI---~GI/A
Z(GI)/A ~ T
is also
A n [GI,GI] = 0 .
It
is an isoclinic epimor-
Thus G is isoclinic to its subquotlent
is finite as an extension of
As
by
GI/A Q .
, and
GI/A
144
3. The Isomorphism Classes of Isoclinic Central Extensions and the Hall Formulae
Throughout this
ei :
Ni ~
section
~ Gi
we consider central extensions
~ Qi
From the results in Section I we know that the following subquotients are invariants of Isocllnlsm: Qi
resp.
Gi/~i(Ni)
,
ci/([ai,ai]~i(Ni))
,
[Gi,Gi]
~i(Nl)
,
n [al,a i]
,
whereas the subquotlents [Gi,Gi]~i(Ni)/[Gi,Gi]
~
~i(Ni)/(~i(Ni) Coker e.(ei)
may differ for isoclinic extensions.
n [Gi,Gi] ) =:
Bi
The situation is illustrated
by the following diagram:
~~(N)[a,a]
~(N)~ N [GG ,] ) n [a,a]
wO
The abellan group
Bi
is called the "branch factor group" of
It vanishes if and only if (in case of finite groups)
ei
factor of a group G.
is a stem extension and its order
is called the "branch factor".
of 1.4 it makes sense to call
ei .
Z(G)/(Z(G)
n [G,G])
Because
the branch
It is the aim of this section to solve the
145
following
problem:
a stem e x t e n s i o n sentatives those
We assume
e o , and that B is an abellan
of the i s o m o r p h i s m
central
extensions
clinic
to
to B.
If we restrict
eo
branch
by either
the groups
if eI
Bi
i.e.
(up to iso-
family w i t h a given out by
in certain
subclasses,
w h i c h are
equivalence
[~,~,~]
8~I
e2
of groups,
can be w o r k e d
We call
and
are iso-
classes
e I ~ e2
and
of
are isomorphic
, we obtain
(~,~):
~2 a
of I.(1.4))
relations:
be an i s o c l i n l s m a strong
coincide
and
isocllnlsm
on
are then called
strongly
iso-
(of the first kind).
(il)
Let
~,~,a'~
(~,~):
a strong
an i s o m o r p h i s m
strongly
e I ~ e 2 , and
isocllnlsm
from
coincides
called
with
clinism
the one induced
by O;
isoclinic
[~,b]
coincide.
have the p r o p e r t y first kind). correspond
eI
to
, where
whose
Similarily
to the pairs
e2
[~,~,~]
{¢,~I
eI
and
a'
of i s o c l i n i s m
e2
,
are then
and strong
the strong
are in one-to-one
a to
~IN1
is a strong
correspondence
from
and
iso-
Isocllnlsms
~INI[GI,GI]
~ to
isocllnlsm
isoclinlsms
, where
induces
kind.
6 is an i s o m o r p h i s m
the strong
if
We call
G2/(~2N2.[G2,G2])
In general
restrictions
that
to
of the second
the n o t i o n s
of the first kind from
~2N2[G2,G2]
of the second kind, )
of both kinds
w i t h the pairs
a': (G1)ab~-~(G2)ab.
GI/(~INI.[G1,G1]
For stem extensions
to
class
Let
an isomorphism.
~ I ( K I N 1 R [GI,G1] ) ;
which
= Z(Gi)
one of the f o l l o w i n g
(1)
groups
Then repre-
which
to isocllnism
The i s o m o r p h i s m
(of the first kind),
clinic
(in the sense
factor
is given by
group.
in an i s o c l l n l s m
isocllnism
3.1 DEFINITION. a: N 1 - N 2
~i(Ni)
class
be determined,
our a t t e n t i o n
group.
the given
will
branch
with
of groups)
factor
dividing given
ei
classes
ei
and whose
to extensions morphism
that an i s o c l l n l s m
[G1,GI] (of the
of the second
z is an i s o m o r p h i s m
kind
from
146 GI/(~IN I n [GI,G1] ) ~,a'
such that
to
G2/(~2N 2 n [G2,G2]
~,~,a'l
It is clear from 3.1 These
equivalence
situation
inducing
is a strong isoclinism
how to define
relations
of the centers",
resp.
"the situation
second kind,
if and only if the corresponding
are isoclinlc. trary central
GI
O [Gi,Gi]
and
c
G2
~G i
The only-lf-part
of the
stem extensions
R [Gi,Gi]
of this statement
) , i=1,2
holds
,
for arbi-
can be used in order to classify
and it is more or less a matter
which notion one prefers.
briefly
Isocllnic
extensions.
or extensions,
ways means
"the
of the commutator
are strongly
~ Gi/(Z(Gi)
Both kinds of strong isocllnism groups
of groups.
for groups were called by P. Hall
The groups
Z(Gi)
of the second k i n ~
strong isoclinism
quotients".
(3.1)
isomorphlsms
In the following,
strong isoclinlsm
of taste
strong isocllnism
of the first kind.
Later on
al-
we
outline how to deal with the second kind in the case of iso-
clinic groups.
It is obvious is uniquely morphism
that a strong isocllnlsm
determined
by ~ and a.
~: [GI,G I] ;
3.2 PROPOSITION.
~ [G2,G2]
Let
following properties
a: NI~
[~,~,a]
The existence
~ N2
and
e2 iso-
Then the
are equivalent:
(ll)
e.(e2)M(~)
from
eI
to
e2 ,
= ae.(el)
~ induces an isocllnlsm
only if there exists
to
in the following
~: QI~---~Q 2 .
~ and a induce a strong isoclinlsm
By 2.3,
eI
of a suitable
will be described
(1)
PROOF.
from
~: [GI,GI]~--@
~ l e . ( e l ) = ~2e.(e2)M(~)
.
If
from
[G2,G2]
[~,~,a]
eI
to
e 2 , if and
, such that
is a strong isocllnism,
we
147
have in addition
~1
= ~2 ~
on
Both con-
~ 1 ( ~ 1 N 1 n [G1,G1])
dltions above are equivalent to (il), and we are done.
3.3 COROLLARY. e2
(i)
Let
[el],[e2] e Cext(Q,N)
[]
Then
eI
and
are strongly isoclinlc by a strong Isocllnlsm of the form
[IQ,~,I N]
, if and only if
[eli - [e2]
lles in
¥(Ext(Qab,N))
,
cf. 1.3.8. (ll)
Let
isocllnlc,
[el] e Cext(Q,N)
if and only if
e 3 e Cext(Q,N) [e 3] e
PROOF.
e2
Then
eI
are strongly
with
¥(Ext(Qab,N))
+
[e I] •
Property (i) follows from 3.2 and 1.3.8, and (li) is a
e e Cext(Q,N)
and denote by
isocllnlsms from e to e A(e)
e2
is isomorphic to an extension
consequence of (1) and the naturallty of
Let
and
e, .
A(e)
[]
the set of all strong
(called strong autocllnlsms).
Obviously,
constitutes a group, and each strong isocllnlsm from e to an
extension
eI
induces an isomorphism from
Hence, we also denote this group by
A(¢)
A(e)
A(e 1) .
where @ is the corre-
sponding class of strongly isocllnlc extensions. extensions in • are represented
to
By 3.3 (il) the
(up to isomorphism)
by a single coset
Q := [e] + ¥ Ext(Qab,N ) . From 3.2 and 3.3 (1) it follows that a pair
(~,a) e Aut(G) x Aut(N)
induces a strong autecllnlsm of e, if and only if it induces one of each
eI
with
[el] ~ Q .
So, having fixed, the coset G (which is
not uniquely determined by ~), we can identify of all pairs
(~,~)
as above.
with the group
Thus we have an action of
the representing set ~ by
(3.2~
A(~)
(~,a~[eI] := [ae1~ -1] ,
A(@)
on
148 which on
is the r e s t r i c t i o n
action
of Aut(Q)
x Aut(N)
Cext(Q,N)
3.4 PROPOSITION. lowing
properties
(b)
(~,a)[e I] e ~
(c)
(~,a)~
Let
classes
=
~
(c). e.(e)
If
for some
is a system
represents
¥(Ext(Qab,N))
Cext(Q,N)
(~,a) ~ A(~)
implies
set I.
Then the
of the i s o m o r p h i s m
, which
the orbits
of
A(~)
on ~.
is fixed by the a c t i o n proves
the equivalence
of
of (b)
, we have by 3.2(ii)
(~,a)[e]
By I.I.10,
~ ~ .
No~
and an a b e l i a n
~
of
classes Aut(Q)
the i s o m o r p h i s m
ourselves
from
we c o n s i d e r
~ G
group B.
~
(1).
a fixed
follows
of central
x Aut(N)
classes
to the elements
(b) follows
In the following
The converse
the i s o m o r p h i s m
to determine
of (a) and
eo :
some
of r e p r e s e n t a t i v e s
([el])i~i
on
we can restrict lence
i runs through
in ~.
N by Q are given by the orbits In order
[eli e ~ ,
= a e . ( e ) M ( ~ -I) = e.(ae~ -11
1.3.8 (li)
The fol-
,
The group
x Aut(N)
.
are equivalent:
The system
(1)
x Aut(N)
.
of extensions
PROOF.
(~,a) E Aut(Q)
[el] E Q , where
(el)iE I
(b)
Aut(Q)
~ A(~)
properties
(a)
Let
are equivalent:
C~,~)
following
Now
(i)
Ca)
(ii)
and
of the w e l l - k n o w n
in ~.
Hence,
[]
stem e x t e n s i o n
we assume
extensions
of
Cext(G,N)
.
of extensions
~ Q , Furthermore
on
analogouslD
that
in ~,
the equiva-
149
di :
Nod
~i
°i
) Ni
~ B ,
i ~ I
runs through a system of representatives of
Ext(B,No)
, and we
obtain the following extensions (3.3)
e(dl) = Xie o :
Ni~
; Gi
.~. 0 ,
which will play an important role in the determination of the isomorphism classes in the isoclinism class of extension,
each autoclinism
a' -- - I ~ of
of
N O , cf. 1.6.
e O , and by
AcI(F)
consists of pairs ously , are
AcI(F)
(~, ~)
is isomorphic to eo
Acl(F) × Aut(B)
(3.4)
on
3.5 PROPOSITION.
is a stem
Aut(O) × Aut(N O) , which a'
Acl(e)
are as above. Obvifor all extensions e that (6,a') ~ Aut(B) x Aut(Nol
Ext(B,N o) , which maps
[dl]
onto
This gives rise to an operation of
Ext(B,No)
((~,~'),~)[d i]
eo
yields an automorphism
For each element
:= [a'di6 -I]
As
By F we denote the Isoclinism class
, where ~ and
we have the obvious action on (5,a')[di]
eo
the subgroup of
(~,a')
Isoclinic to
of
eo .
by
:-- (6,~')[d i]
(i)
.
All extensions
e(di)
from (3.3) are iso-
clinic to e o. (li)
The branch factor group of each
(iii)
e(di)
is isomorphic to B.
Each central extension which is isoclinic to
eO
and
has a branch factor group isomorphic to B, is strongly isocllnlc to some
e(dl) (iv)
The extensions
if and only if
[di]
AcI(F) × Aut(B)
.
PROOF.
We have
Thus it follows
and
e(dl) [dj]
and
e(dj)
are strongly isocllnlc,
are conjugate under the action of
e(di) = kieo , where
e.(e(dl) ) = ~18.(eo)
~i: No " Ni
, and
is inJective.
150
Ker
e.(e(dl)) As
eo
follows
= Ker
e.(e(dl))
el: NI~
factor
, proving
is a stem extension, Coker
Let
e.(eo)
group
e.(eo)
-- Coker
;GI----~Q I
isomorphic
(1) by 2.3. is an epimorphism,
Xi ~ B
, proving
be a central
to B, and
(~,~):
and it
(ll).
extension
eI ~ eo .
with a b r a n c h Hence
we have
(cf. 2.3): ~Xle.(el ) = ~e.(eo)M(~) Let
Xl := ~ 1 ~ - 1
and
, which
Coker
e.(e I) -~ B
is a m o n o m o r p h l s m
from
No
to
N 1 , and
we obtain e.(e I) = Xle.(eo)M(~) As
e.(e o)
is eplmorphic,
Im ~1 = Im e.(e 1)
we have
and
Coker
kI = C o k e r
e.(e 1) ~ B .
X1 Thus we obtain an e x t e n s i o n
d: N o F
Ed] = ~dl]
index
morphism
,
for a suitable
a: N I~---~N i
Xi = aX 1 : Summarizing
the results
3.2(ii)
proving
.~B , and we have
i ~ I .
satisfying
above,
we obtain
= ~ie.(eo)M(~)
, ~ and a induce
= a~le.(eo)M(~)
a strong
Isoclinlsm
= ~e.(e 1) from
eI
to
e(di),
(ill).
It is easy to see that the a u t o m o r p h l s m s isoclinisms Acl(r)
Thus we have an iso-
No~---*N i .
e.(e(di))M(~) By
~ N1
.
from
Hence,
e(di)
to
assertion
e(dj) (iv)
of Q, w h i c h
coincide
w i t h those
is an easy c o n s e q u e n c e
induce
the
appearing
in
of 3.1 and
1.1.10.
3.6 REMARK. resp.
groups
3.5 yields
Propositions
3.4 and 3.5 d e t e r m i n e
in an i s o c l l n i s m
representatives
class resp.
of the classes
the extensions
family up to i s o m o r p h i s m of strongly
isoclinlc
;
151 extensions,
whereas
3.4 yields representatives
of the isomorphism
classes.
The classification using
of isoclinic
strong isoclinlsm
groups
can also be worked out by
of the second kind:
Let S be a stem group
,
and consider s :
[S,S]"
As above, d~ :
~ S
~Sab
-
let B be an abelian group, B ~
~ Ni
be representatives
and let
~ Sab of
group of the backward
Ext(Sab,B) induced
By
Ti
we denote the middle
extension
~s
.
are exactly those, which map epl-isocllnlc group B. tives
factor group. suitable
sponding
Acl(r)
isoclinlc
again use 3.4 to determine of isoclinism clinism. isoclinlsm
the isomorphism
extensions
above,
groups
if and only if the correFor stem extensions,
means the same.
might be useful to classify
For central
of the second kind,
As mentioned
(3.1) are isoclinlc.
and strong isocllnism
yield representa-
once, by again using a
of the second kind,
stem extensions
isocllnlsm
x Aut(B)
factor
to S and have B as branch
Each class will be obtained
action of
are strongly
Hence,
classes.
one can
So both kinds
groups in terms of iso-
with a fixed factor group,
strong
of the second kind does not seem to be as comfortable
the first kind, the kernels
because
the use of backward
of extensions,
[e] E ~ , we denote by C(e)
induced
extensions
St[e]
of Proposition
the stabilizer
the group of all automorphisms
of
3.4. [e]
For each in
as
fixes
but not the factor groups.
Now we return to the situation
and by
Ti
of the classes under strong isoclinism which are isocllnic
these groups
onto S with branch
Then one can show that the groups
among those groups,
By 1.9
A(@)
of e having the form
152
(1N,5,1Q)
; this is a normal subgroup of
3.7 LEMMA.
St[e] ~ Aut(e)/C(e)
(ii)
C(e) ~ Hom(Qab,N)
(iii) C(e) ~ Ext(Qab,N)
By 3.4,
Aut(Q) x Aut(N)
St[e]
Let
that the automorphisms >g.(~fw(g))~
, if Q and N are finite.
is also the stabilizer of
, acting on
lows from 1.1.10.
= {g!
.
We have
(i)
PROOF.
Aut(e)
Cext(Q,N)
e: N; ~ ~ G
"~Q
[el
Hence, assertion (i) fol.
Then one easily sees
(1N,5,1Q) of e are given by , where f is a homomorphism from Q to N, and
different homomorphisms yield different automorphisms, Assertion
in
(iii) is trivial by (ii), see also 4.5.
proving (ii).
[]
A n interesting situation is given, if • contains an extension e whose automorphisms induce all elements of St[e] = A(~)
A(@)
, i.e.
In this case the isomorphism classes of extensions
in @ are in one-to-one correspondence with the orbits of ~(Ext(Qab,N))
.
A(@)
A more detailed consideration of this situation will
be given in Section 4.
For finite extensions we now can prove the
following "Hall-Formula":
3.8 PROPOSITION.
Let Q and N be finite, and
el,e2,...,e n
representatives of the isomorphism classes in @. n 1 = D ~A(@)~ i=I PROOF.
I IAut(ei) I
By 3.4 and orbit decomposition,
iExt(Oab,N) l = I~I = NOW
3.7 implies
on
n D i=I
we have
]A(@) :St[e i] I
Then we have
153
IA(¢):St[e i]l = IA( @)l'IC(e i) l'IAut(e i)1-1 and IC(el) I = IExt(Qab,N) I , and the proposition easily follows.
If we restrict ourselves
in 3.8 to the situation where ¢ coin-
cides with the class of stem groups of a family F, we obtain Hall's well-known
formula
(3.5)
1
=
IAcl(r) l where
G1,G2,...,G
groups in F.
n D
1
9
1=1 IAut(%)l n
form
a complete
system
of
non-isomorphic
Similar formulae can be obtained,
stem
if one restricts
3.8 to Hall's situation of the centers resp. commutator quotients, i.e. strong isocllnism of groups of the first resp. Another Hall-Formula
3.9 PROPOSITION.
turns out to be a consequence
(i)
The s t a b i l i z e r
under the action (3.4) of A(¢I)/U i , where
@i
,
(il)
x Aut(B)
e(dl)
, and
which is isomorphic to
Ui
of
is isomorphic
[di] to
is a normal subgroup of
Hom(B,No)
If Q, B, and N o are finite, and
ferent) classes under strong isoclinism, tives
of 3.5.
is the class of central extensions which are
strongly isocllnlc to A(¢i)
Acl(r)
St([di])
second kind.
¢1,...,¢m and
are the (dif-
el,...,e k
of the isomorphism classes of extensions
representa-
in F with branch
factor group B, then 1 IAcl(F) x AutCB) l PROOF.
m
=
Let
D
1
=
i=I IA(%)I
(~,a) ~ A(¢i) ~ Aut(Q)
a' E Aut(N o) , ~ ~ Aut(B)
, and we have
k
~
1
3=I
IAut(e 3) I
× Aut(Ni)
Then a induces
((~,a'),6) ~ Acl(F)
x Aut(B).
154
Hence we have a homomorphlsm :
A(¢i)
• Acl(P) x Aut(B)
and 3.4 and I.I.10 imply Im ~ = St([di] ) Denote by
Ui
the kernel of ~.
such that a induces INo and 1B. U i ~ Hom(B,No) , proving
It consists of all pairs As in
3.7(ii)
(IQ,a)
we obtain
(1).
The first equation In (li) follows by the same argument as 3.8. Replacing
IA(~i)l -I
by the sum of 3.8 (first Hall-Formula),
obtain the second equation.
[]
we
155
4. On P r e s e n t a t i o n s
of I s o c l i n l c
The c o n s i d e r a t i o n s
Groups
of this s e c t i o n are b a s e d
on those
t i o n 3, some of w h i c h will a p p e a r a g a i n in a d i f f e r e n t start w i t h a short r e v i e w
of the g r o u p
Ext(Qab,A)
of Sec-
guise.
We
and of the h o m o -
morphism :
Ext(Qab,A) ~
from the U n i v e r s a l (4.1)
e :
> Cext(Q,A)
Coefficient
X c
)Y
be a free p r e s e n t a t i o n abelian). induced
if and only if was d e f i n e d T[e']
1.3.8.
of
Qab
(as an a b e l i a n group, Ext(Qab,A)
of
, where
~e
to Y.
; and
by the
[ae] = [Be]
The h o m o m o r p h l s m
, T
by ,
ab
:
Q~Qab
In this s e c t i o n we c o n s i d e r
the f o l l o w i n g
(4.2)
~Q
e
R •
:
be a c e n t r a l
F
;
extension
"
,
B ~_ F
clR
[FIR,F/R]
,
situation.
by the g r o u p Q, and A,B,C
the f o l l o w i n g
A _c R =
i.e. Y is free
are r e p r e s e n t e d
~ E Hom(X,A)
can be e x t e n d e d
a - 8
:= ab*[e']
satisfying
Let
~ Qab
T h e n the e l e m e n t s
extensions
Theorem
Let
subgroups
of F
condltlons:
B 0 R = A
,
C = RB
,
,
and CIB t is a free
BIA
=
~ FIB (abelian)
[FIA,FIA]
diagram:
"'
~Qab
presentation This
situation
of
Qab
"
In a d d i t i o n we have
is i l l u s t r a t e d
by the f o l l o w i n g
II F
156
Qab Q C
C4.3)
B
R
A
0
As
F/B
plies
is free abellan,
the same holds
that A has c o m p l e m e n t s
corresponding ai: R
R/A ~ C/B
,
ki: R
~K i
For any two c o m p l e m e n t s f = f(Ko,K1)
=
For each c o m p l e m e n t
via the i s o m o r p h i s m
with
Ko,K 1
~Ar!
we can regard
,al(r)-ao(r)-l~
:
R/A
f also as a h o m o m o r p h l s m
~Art
)Br~
from
R/A
to
plements
Let
and
the free a b e l l a n C/B
where denote
Hom(C/B,A) group
~i
~ cj
)A from C/B
C/B .
between
I J ~ J
} bj = Bcjl
.
~
J e J
to A
If we fix the com-
be a basis ~
of
is a basis
of A in R are given by
I J e J ) ,
f runs through by
.
K o , then
, and the c o m p l e m e n t s
cj-f(bj)
we h a v e
we obtain
correspondence
of
Ki
im-
r = ai(r).ki(r)
K o , we thus obtain a one-to-one Ki
, which
projections
~A
Obviously
in R.
for
the maps
the i s o m o r p h i s m
from from
~bj} R/K i
to A.
For each
to A defined
by
K i , we
157
Xi(Kir)
:= ai(r)
, and we obtain the extension
e I i e(K i) :
A~
e(Ki)
:= Xi(e/Ki)
,
i
~E(Ki)~Q
Now we assume that in (4.1) we have
X = C/B
and
Y = F/B
.
Then
we can show
4.1 PROPOSITION. f = f(Ko,K1) [e(K1)]
i.e.
Let
Ko,K 1
[e(Ko)]
the extensions
=
~ [f;]
e(Ki)
; in p a r t i c u l a r
f(Ko,K1)
can be extended from
A; with o =
U
=
represent
Cext(O,A)
is r e p r e s e n t e d ; ( A x F/A)/M
C/B
IM(a,Ax)~
Let
with
obtain a commutative
a coset of
M =
to
F/B
~(Ext(Qab,A))
holds, .
e i := e(Ki)
] , u =
la!
G = (A x F/A)/M
~ (f(Ar)-1,rB)
, one can verify that
>M(a,1)}
,
, and
I r ~ R
H e n c e we
diagram: :~ Qab
i. t A."
el-e o :
with
~ =
duced by
V
; G
U T
A~
U
,,lG
II ~" ~ Qab
~
T
~ Q
IBxl ~: F
Q .
in
if and only if
,
I r ~ R
~(x)}
= (A x F/B)/M
,
by the following extension:
c gO
M = ~ (f(Ar)-l,Ar)
ab*[fe]
[e(Ko) ] = [e(K1) ]
Using 1.2.4 and putting
[el] - [eo]
of A in R, and
Then -
PROOF.
be complements
This shows that the row in the middle
158
represents
[fe]
, whereas
)[~]
the last row now represents
, and
we are done.
By 3.3 the extensions clinic
extensions
helpful,
when
In a d d i t i o n
e(Ki)
of A by Q.
one tries
represent
Hence
to classify
to 4.1 we can show,
4.2 PROPOSITION.
(1)
centralizes
A and
(ll) section
if and only if there ) F/D
(A x O)/O
(i)
if and only if
Ko
in terms
e(Ko)
be
of isoclinism.
e(K1)
are
of F that
KI
generated,
maps
and
an a u t o m o r p h i s m
to
exists
The extensions f = f(Ko,K1)
If ~ is an extension
3.7(ii)
exists
Then
~ Q , which
above might
iso-
and D the inter-
e(Ko)
and
e(Ki)
are iso-
an a u t o m o r p h l s m
of
Ko/D
and n o r m a l i z e s
to
KI/D
.
PROOF.
a(x)
Ki
of strongly
3.4:
F finitely
of all c o m p l e m e n t s
e/D: R/D;
by
and maps
Let A be finite,
morphic,
groups
The extensions
if and only if there F/A
the situation
c.f.
equivalent,
a class
e(Ko)
from
F/A
to A
are equivalent,
from
a suitable
On the other hand,
to a h o m o m o r p h l s m
e(K1)
can be extended
of f, we obtain
:= xf(Bx) -I
and
C/B
to
F/B
automorphism
e of F
each a c o r r e s p o n d s which
induces
.
by
an exten-
sion of f. (ll)
The proof
Y be a finitely m a positive
generated
integer.
by an a u t o m o r p h i s m Proof: nat2:
Consider
Y/mX----@Y/X
GASCH~TZ
of (li)
of
on the following
free abellan
group,
Then
each a u t o m o r p h i s m
Y/mX
.
the natural , and let
[2] we obtain
nat2~ = ~ .
is based
homomorphisms @ := aonat I.
the existence
But then it readily
of
follows
statement:
X a subgroup ~ of
Y/X
nat1:
Y
of Y and
is induced
mY/X
From the results
~: Y that
~ Y/mX Ker
Let
and in
with
~ = mX
.
Hence
159
yields an a u t o m o r p h i s m Let Then
of
Y/mX
, which
Cl,...,c n
be a basis
of
mcl,...,mc n
is a b a s i s
of D.
we obtain Assume
mX = (D x B)/B
that
e(Ko)
isomorphism
and
6: F / K o ~
induces
K o , and m the e x p o n e n t If we put
e(K1)
are isomorphic.
m F/K I , w h i c h maps
8 of
Y = F/B
of A.
,
X = C/B
, and we can a p p l y the s t a t e m e n t
F/C = (F/RJa b , 5 i n d u c e s an a u t o m o r p h i s m by an a u t o m o r p h i s m
a.
F/(D x B)
above:
T h e n we h a v e an
R/K o
to
a of F/C,
Because
,
R/K I which
As is i n d u c e d
of the obvious
iso-
morphisms F/D ~ F / K o J , where
F/(D x B) ~ F / K I ~
in both cases
F/(D x B)
in the fibre p r o d u c t s
above the g r o u p s
identified,
6 and
~ give rise to a suitable
w h i c h maps
Ko/D
to
We shall false,
.
automorphism
F/C
of
are
F/D
[]
see in the f o l l o w i n g
examples
that 4.2(ii)
can be
if we a l l o w A to be infinite.
4.3 E X A M P L E S . R'c
KI/D
,
~ F'
~ Q
F := F' x A
,
R
tions above.
(i)
The E x t - g r o u p . (abelian)
presentation
:= R' x A
,
,
of
and we r e a d i l y this situation.
and
(ii)
we a s s u m e
that
KI
of the e x t e n s i o n s
of 4.2(ii)
over a r b i t r a r y groups.
We put
Let
e(K1)
If
vanishes,
does not h o l d in of
Ext(Q,A)
rings.
e: R ' C
group Q, and
A = M(Q)
runs
4.1 and 4.2 yield
see that this d e s c r i p t i o n
of the (arbitrary)
its c e n t r a l i z a t i o n .
Then
A = Z , the group D from &.2(ii)
One can also
Representation
presentation
in terms
of Q.
s a t i s f y the condi-
Propositions
see that the s t a t e m e n t
can be g i v e n for m o d u l e s
C := R
, whereas
of A in R.
Ext(Q,A)
Q = 2/5
B := A
K o := R'
t h r o u g h all c o m p l e m e n t s
we a s s u m e
that Q is a b e l i a n and
is a free
We put
a description
Assume
~ F'
~ Q
be a free
c(e):
R"
~F
= (R' O [ F ' , F ' ] ) / [ R ' , F ' ]
~G ,
160
B = [F',F']/[R',F'] the representation II.3.4,
C = BR
groups
.
Then the extensions
e(Ki)
are
of Q, and 4.1, 4.2 yield a refinement
of
3.5.
(iii) finite
, and
Finite
isoclinic
stem group,
presentation
whose
p-groups.
Let p be a prime,
order is a power
of the following
of p.
and G a
Hence
G has a
form:
Generators: xi' Yi' Zk Relators: (a)
[Zk,Zk,]
,
(b)
[zk,yj]
(d)
zkP.Uk(Zl,...,Zk_1)
,
(c)
[Zk,Xi]
(i.e. u k is a word in
p
(e)
I 2 yj .vj(y I .... ,yj_1).vj(z)
(f)
4 [YJl'YJ2]'v31,J2(Yl,'",Yjl)'VJl,J2
,
z1~.-.,Zk_ I)
z = (zl,z 2 ..... ) (z)
, Jl < J2
ni
(h)
[xi,Yj].w~,j(yl,...,yj)-w4(z)
(i) [xil,xi2].w~,i2(y~.w~1,12c~ We assume
:
an abelian
[G,G] =
group P with a basis
Let H be the group generated relators
, Z(G) = (z k) al,a2,a3,...
by the elements
(a) - (i) and the following
(J)
[ae,X i]
(n)
alp
,
(k)
[ae,Y j] ,
Furthermore
(1)
, where
we consider m1 fall = p .
xi,Yj,Zk,a I
with the
ones: [al,z n] ,
(m)
[al,al,]
,
m1 .rl(z ) .
From the results
in Section
words
rl(z )
group,
and one obtains
3 we see that for any choice
in (n), H is isocllnic representatives
of the
to G having P as branch of all classes under
factor strong
161
isocllnism.
For a fixed H we consider
generators elements
al, xl, yj, z k .
the free group F' with the
Let S be the normal
of F' given by (a) to (n), i.e.
the normal closure F = F'I~
,
H ~ F'/S
.
of the
By ~ we denote
of the words: (a) - (f) and (h) - (n)
C =
Let
<S,al,Yj,Zk>/~
R =
(S,al,Zk>l~
,
B = (S,al,Yj,Zk)IS
A =
(~,al,Zk>/S
,
K o = S/S
Then
closure
F,C,B,R,A,K O
.
satisfy the conditions
A basis of
Ko
is given by the relators
all groups
that are strongly
isocllnlc
of Propositions
4.1, 4.2.
(g) ; and presentations
of
to H are given by (a) - (f),
(h) - (n), and nl (g')
xiP
where
wl
I .wi(y)'wi(a).w~(z) and
wE
,
are arbitrary
words
results can be worked out for arbitrary summsry
of the observations
above
given by power and commutator
is the following:
and A is finite.
which normalizes
L on the group
KI/D
If a group is
relations
invarlant
and
(D x A)/D
correspond
e/K i , resp.
the action of L "coincides" coset
~ =
A ~ (D × A ) / D
.
Hence,
, i.e. Q is of
By 4.2(ii)
the orbits of
to the isomorphism
classes of ex-
e(K i)
.
which are strongly .
isocllnic
We are going to show that
with the action of
l[e(Ki)]~
of L induces an automorphlsm
of 4.2(ii)
Let L denote the subgroup
in the class @ of extensions
to the extensions
sponding
A very rough
relatlons, one obtains the other Iso-
Let us assume the situation
finitely generated
tensions
Similar
certain power relations.
4.4 REMARK.
Aut(e/D)
z.
finite groups.
clinic groups by keeping the commutator c~anglng
in a, resp.
in CeXt(Q,A)
A(@)
, cf.
on the corre-
3.4.
~ of Q and an automorphism
we have a homomorphism
Each element ~ of
162
~: L " (Aut(O) (4.4)
Im ~
Aut(A),
x
=
while
3.4 and the proof of 4,2(ii)
yield
A(~)
On the other hand, we have the surJection a =
{KI/D:
: [e(Ki)]l
:
{KI/DI----~
~ Cext(Q,A)
and we readily see the following r e l a t i o n (4.5)
~(x)oCKi/O)
Now we restrict additively). A =
= o(x(Ki/O))
for all
, x ~ L :
.
our a t t e n t i o n
to finite abelian groups
(written
Let
u • (a i)
,
v • (tj)
T =
i=1
,
J=l
where the numbers
n i = fail
and
mj = Itj(
Let F' be free abelian with a basis the subgroup generated by
are powers of primes.
bl,b2,...,bv
r1=mlbl,...,rv=mvbv
,
, and let R' be so we have a free
presentation (4.6) Let
R, =
¢ ) F'
w ~T
YJi = ni/gcd(ni'mj) ~Ji :
Hence
T - A ;
,
~ =
Ibj~---~tj}
' and define
t0~
) yjlai
, tk:
~0
for all k # J .
we have:
Hom(T,A)
=
•
J,i
<,jl)
,
i~jii = g c d ( n i , m J)
Let ~Ji :
R' - A ;
rj ~
~a i , rk~--+O
for
Then
Hom(R',A)
=
• (~i)
,
and = ~ji~--~ji~ is an eplmorphism.
:
Hom(R',A)
~ Hom(T,A)
Now we readily see that
k ~ J .
163
Ker ~ = where exact
I ~ [ • e Hom(R',A)
¢*: Hom(F',A)
, w(rj) E mjA for all j I = Im ¢* is induced
- Hom(R',A)
by
¢. Hom(T,A)t
Hom(T,A) (a,~)~
4.5 LEMMA.
,
a e Aut(T)
are
we have a close
(4.7)
fl e Hom(C/B,A) = fi "
e(K1),...,e(Kn)
be a central strongly
we obtain
T h e n 4.1 represent
classes
that the stabilizer
e(Ki)
in Cext(Q,A)
, and
of E x a m p l e
abelian
and
groups,
abelian
on
of a b e l i a n
~l,...,~n
Hom(Q,A)
.
Via
elements
the groups
(4.4),
Ki
(4.5)
by
yield that
classes
groups
isoclinlc
- 4.5
, given by
classes
and
Ext(T,A)
By 4.1
the i s o m o r p h i s m
the i s o m o r p h i s m
on finite
in
Ext(Q,A)
can be g e n e r a l i z e d
extensions.
.
to
Let
~ Q
extension
isoclinic
coset
Then
•
the extensions
Define
.
[]
T = Qab
corresponding
- 4.5,
~ of strongly
~G
groups.
x Aut(A))-orbits
, cf. 4.3,(i).
observation
A~
(Aut(G)
:
the following
In the very easy situation
of the
the map v from
e :
between
x Aut(A)
, ~ e Hom(T,A)
abelian
the case
Let Q and A be finite
representatives
Aut(T)
x Aut(A))-isomorphic.
we can use 4.5 to evaluate
extensions:
certain
~ ~ Aut(A)
K i , the c o r r e s p o n d i n g
Hom(Qab,A)
of
we can deduce
we consider
connection
the c o m p l e m e n t s
This
above,
(Aut(T)
In the f o l l o w i n g
f(Ko'Ki)
,
action
Let T and A be finite
Hom(T,A)
4.3(i),
, Hom(R' ,A) ~ H o m ( T , A )
we have the following
= ~ m -1
the groups
~Ext(T,A)
~* ~ Hom(F',A)
From the c o n s i d e r a t i o n s
and
the
sequences
(4.7) On
¢, and we obtain
where A and G are finite
extensions St[e]
containing
of [e] in
e.
A(*]
and ~ the class
Furthermore satisfies
of
we assume
164
(4.8)
St[el
= A(~)
,
i.e. the automorphisms
of e induce all strong autoclinisms~
Under this assumption,
the action of
equivalent
to the action on
4.6 PROPOSITION. satisfies
(4.8).
Assume
Aut(e)
of
see 3.7.)
PROOF. A(e)
classes
classes
in
C(e)
Aut(Q) readily
.
The group
x Aut(A)
of
Ext(Qab,A)
that the orbits of U on A(~)
by
on
W Ext(Qab,A)
Hom(Qab,A)
Aut(Qab)
Let us again consider
such that
representatives
of
Aut(e)
Let us assume
chosen such that
~ Qab
sions in ~.
~F/B
e(Ki)
with
isomorphic
to
.
Ko
to C(e),
corresponds
[]
Without
loss
of A in R is
corresponding
fl = f(Ko'Ki)
that
F, C and B are
has the form of (4.6).
(as in the case of abelian groups)
fl,...,fn ~ Hom(C/B,A)
correspond
of the orbits under the
action of U, cf. proof of 4.6.
we choose
, and it
As in the abellan case we consider
~l'''''~n e H°m(Qab'A)
C/B •
x Aut(A)
(4.3), and ~ and e as above.
e = e(Ko)
of
of
Hom(Qab,A)
of generality we can assume that the complement chosen
of ele-
By 4.5, we can replace
, which is naturally
with the elements
in •
to the orbits
Ext(Qab,A)
.
e
~ can be replaced by
and it can be shown that the action of U on to the conjugation
classes
, which is a subgroup
induces a subgroup U of
follows
the orbits
A(e)
extension
(For the definition
in • correspond
on 0 by 3.5, and by our assumption,
¥ Ext(Qab,A)
is
of extensions
with the conJugacy
that are contained
The isomorphism
W Ext(Gab,A )
that the finite central
correspondence
ments of
on
0 = [e] + ~ Ext(Qab,A)
Then the isomorphism
are in one-to-one
C(e),
A(e)
cf. 3.7.
to
homomorphlsms
~1'''''~n
represent
Then
"
The exten-
the isomorphism
classes
165 4.7 EXAMPLES.
(i)
Representation groups of finite abellan g r o u p &
From Proposition II.A.13 one sees that the representation groups of finite nllpotent groups are the direct products of the representation groups of their Sylow subgroups. tion to abelian p-groups. Q =
n z/pki X , i=1
where
Hence, we can restrict our atten-
Let k I ~ k 2 ~...~ k n
From II.(4.7) one can deduce
M(Q)
n iZlph i=1
Let F be a free group on the free generators
Xl,X2,...,x n , and
ki R = ~ xlP
, [xl,x j] I i<J )F ~ IF,F] ki
= ( xiP
ki , [[F,F],F],[Xl,xj]P
I i<J )F ~ R .
(The upper index F denotes "normal closure".)
Hence we have a free
presentation
and let
~i = axl'
~1 = R x i '
G = F/~, M = ~ [ ~ l , ~ D ]
Now it is not very dlffucult to show that representation group of Q;
LEMMA.
I i ~ D ) = [G,G]
e: M ~
~ G ----mQ
is a
see TAPPE [3,13.1] for details.
If p is an odd prime, each automorphlsm of Q can be lifted
to an automorphlsm of e.
PROOF.
We firstly show that the statement of the lemma holds for
endomorphlsms of Q. ~i'
and by
m n _pij ~ ~ xj j=l
A n endomorphlsm ~ of Q is given by a map ,
where
k i + mlj >_ kj
166
n xi :
;
pml J
11
xj
J=1
we obtain an endomorphlsm ~ of F, which induces ~ on
G ~
F/R .
W e have ~[[F,F],F]
and
_= R
,
[[F,F],F] _c R
implies
ki ~([xi,xj]P
) ~ R
for a l l
i < ,J •
As p is odd, G is a regular p-group of class 2, what implies ki ~.(xlP
) ~ R
for a l l
i.
Thus we have
~(~) _~ ~
,
and ~ induces an endomorphism
8 of
G = F/R .
As
M = [G,G]
, we
obtain 8(X)
= M .
This finishes the proof for endomorphlsms. automorphism of Q x = [a,o]
81M
and 8 any endomorphism of e lifting ~.
= [Q,c]
n z(a)
of M.
In particular,
Hence
As
,
iS uniquely determined
cf. 2.4.
Now assume that ~ is an
and
is an autocllnlsm of e,
(~,81M)
we obtain that
el M
B must be an automorphism of G.
is an automorphism []
As a consequence of this lemma, we can apply 4.5 in order to describe the isomorphism classes of representation groups of Q. These extensions form a class ~ of strongly isocllnic extensions, where
A(@)
is isomorphic to
Aut(Q)
.
So the study of the iso-
morphism classes "reduces" to the (non-trlvlal) problem of determining the orbits of
A(~)
on
Hom(Q,M(Q))
resp.
Hom(F/R,M)
.
167
Each
y e Aut(O)
(~,a)
~ A(e)
(~,a)f
=
.
af~ -1
induces a unique The action on Let
a E Aut(M(O))
f ~ Hom(O,M(O))
fl,f2,...,fr
, such that
is given by
be representatives of these
orbits, and ft(~i) = wt([xi,xj]) a word in the elements
, i < J , We then obtain representatives of
..rxl,xjl
ki the isomorphism classes by replacing the relators xiP of
ki wt([xi,xj])
Using similar methods,
7/p x Z/p × Z/p
are
xiP
in R by
the representation groups
were studied by PLATH [1], who showed that there
2p+10 isomorphism types.
(ii)
The stem extensions of
Q = Z/p n x Z/p n .
Each stem exten-
sion of O is strongly isoclinic to exactly one of the following groups
F/Rm, m = 1,2,...,n pn
the normal closure of Aut(O)
= GL(2,Z/p n)
M(Q) ~ Z/p n Example
~ x
, where
F = ~x,y) is free, and
pn ,y ,[F,F,F],[x,y]p m r r
, and the elements of
Aut(O)
I •
Rn
is
We have
induce on
the multiplication with the determinant.
As in
(i), we see that the automorphlsms of 0 can be lifted to
automorphisms of
F/R m .
Hence,
Aut(0)
is isomorphic to the group
of strong autoclinisms for all m.
The action on
sponds to the following action of
G = GL(2,7/p m)
a l
~ det(T).T*a
Hom(0,Z/p n) on
corre-
Z/p m × Z/p m :
,
for all
T ~ G , and the star denotes the transpose of the inverse
matrix.
Any two vectors are conjugate under G, if and only if they
have the same additive order.
Hence we have
there are (up to isomorphism)
m+l
order
m+l
orbits,
i.e.
stem extensions of 0 with the
p2n+m .
(iii)
Generalized representation group of dihedral groups with
elementary abelian center.
Let O resp. H be the dihedral group of
168
order
2n
resp.
2 n+1
of Q.
Now it readily
isocllnlc
By 1.3.8(iv)
2 n+m
with elementary
As the automorphisms
GF(2)
of groups,
factor of
which are strongly
center are
~ Z/2
are amalgamate~.
isocllnic
of H, the
to G, is reduced
of the group of strong autoclinlsms
, which is isomorphic .
abelian
of Q can be lifted to automorphisms
to the study of the action
over
representation
,
where the center of H and a direct
Hom(Gab,Z(G))
group
to
G = Hy(~Z/2)
determination
H is a representation
follows that all generalized
groups of Q of order strongly
.
on
to the group M of 2xm-matrices
The action of the autoclinlsms
corresponds
to the
action of the matrices
on M f r o m t h e
left,
on M from the right. provided
m ~ 3 •
and of
This yields exactly
seven isomorphism
types,
169
5. Representations
of Isocllnic
Groups
In this section we study the connection and isocllnlsm.
between
In most cases we restrict
representations
ourselves
to finite
groups.
The existence resp.
of faithful
faithful p-blocks
PAHLINGS
[2].
cllnlsm
irreducibles
in general,
irreducible
representations
for finite groups was studied
Easy examples
ence of faithful
ordinary
of abelian groups or blocks
in GASCHSTZ
show that the exist-
is not an invarlant
but the results mentioned
[I],
of iso-
above can be used in
order to prove:
5.1 PROPOSITION. are not isocllnlc a faithful p-block,
Let G and H be finite to one of their proper
ordinary
irreducible
irreducible
By the results character,
of GaschGtz
resp.
p-block,
socle of G, resp.
Op,(S(G))
class of G.
is the product
S(G)
groups,
factor groups.
representation,
if and only if the same holds
PROOF.
isocllnlc
resp.
which
Then G has
a faithful
for H.
and Pahllngs, if and only if
is generated
G has a faithful S(G)
, the
by a single conJugacy
of all minimal normal
subgroups N
of G, and for all such N we have N ~ [G,G]
or
N n [G,G] = 0 .
As G is not isocllnlc have
N ~ [G,G]
S(G) 2 [G,G]
[G,G]
and
factor group of itself,
we always
, hence and
Now the proposition of
to a proper
Op,(S(G)) follows
[H,H]
[]
E [G,G]
from 1.4(ill)
. , the operator
isomorphism
170
In Section 2 we obtained the following result, cf. 2.5.
5.2 THEOREM.
Let G and H be groups,
~: G/Z(G)2
)H/Z(H)
an
isomorphism, and K an algebraically closed field of characteristic O. Then the following properties are equivalent: (1)
~ induces an isoclinism from G to H.
(li) Each projective K-representation P of lifted in H, if and only if
P~
H/Z(H)
can be lifted in G.
can be
[]
By the results of Chapter II, it suffices to consider in 5.2(ii) the irreducible projective representations, sional if G and H are finite.
which are finite dimen-
Let us now shortly discuss the situa-
tion of 5.2 in the case where G and H are finite ically closed of characteristic p.
If D is an irreducible K-repre-
sentation of G, each normal p-subgroup, contained in the kernel of D.
and K is algebra-
in particular
Op(G)
is
This gives rise to the following
definition:
5.3 DEFINITION.
Let G and H be finite groups, K an algebraically
closed field of characteristic p, and
ep(G)
the following central
extension: ep(G)
:
0 (G)Z(G) POp(G)
c
~ G t~O/pi, O J
We call G and H p-isoclinic,
G
Op(G)Z(G)
if and only if there exists an iso-
morphlsm
: G/Op(G)Z(G):
= H/Op(H)Z(H)
such that each irreducible projective K-representation P of H/Op(H)Z(H) lifted in
can be lifted in ep(G)
ep(H) precisely when
P~
can be
.
Now it is not very surprising that the following result holds:
171
5.4 PROPOSITION.
The f o l l o w i n g
(i)
G and H are p-isocllnic.
(ii)
ep(G)
PROOF: finite
Let
ep(H)
e.(ep(G))
follows
more
Z = Z(G)
Z~
ep(G)
is a
~Y
For
that G and H are isocllnlc
by
, and that G is a stem group. the c o n n e c t i o n
between
the irre-
closed
sake of simplicity,
we write
Z ~ Y , and that
~ induces
, and assume .
D
that G and H are
of G and H over a l g e b r a i c a l l y
than in 5.2.
, Y = Z(H)
embedding
we assume
we try to describe
representations
detailed
of
till the end of this section
~: G / Z ( G ) ~ - - ~ H / Z ( H )
In the following ducible
As the kernel
in the same vein as 5.2 from II.2.20.
Furthermore
an i s o m o r p h i s m
.
_o M(Q)p
As in 5.4 we assume groups.
equivalent:
we have
the result
finite
are
are isoclinic.
Q = G/0p(G)Z(G)
p'-group,
Ker Hence
and
properties
In p a r t i c u l a r
we have
Z = [H,H]
fields
the
0 Y ~ [G,G]
Now we c o n s i d e r L := G A H By 1 . 1 1 L
= ~ (g,h) I g ~ G, h ~ H,
is Isoclinic
are isoclinlc
,
Z(L~
of
~ ~ Hom(Z,K*)
k to Y, denoted ~-luCz,Y)
Hence
can be extended
Z(L)
i (z,z)
Let k-lu
above
I z~
•
an arbitrary, Define
onto G and H
yield:
Z I p ~ 0 .
but fixed
~-lu E Hom(Z
extension
x Y,K*)
by
•
N [L,L] ~ Ker(k-lu)
of
~ .
field of c h a r a c t e r i s t i c
we choose u = u(k)
to L.
extension
=
closed
:= ~(z) - I u C Y )
we have
but fixed
by
Our a s s u m p t i o n s
n [L,L]
Let K be an a l g e b r a i c a l l y For each
= hY
to G and H, and the p r o j e c t i o n s
epimorphisms.
Z(L) = Z × X
~(gZ)
, what
t~*=u*(~) ~ Hom(L,K*)
proves
that
~-lu
be an arbitrary,
172 Now we c o n s i d e r D: G - GL(n,K)
an i r r e d u c i b l e
of G.
By S c h u r ' s
matrix
representation
Lemma we have
DI z = ~'I n , where
k=k(D) ~ H o m ( Z , K * 7
rank n. =
In
is the i d e n t i t y
F r o m D we obtain an i r r e d u c i b l e {(g,h),
where
, and
) w*(g,h)D(g)~
p* = p*(k(D))
:
matrix
representation
of
of L:
L - GL(n,K),
is d e f i n e d as above.
The d e f i n i t i o n
of
yields X As
:=
{ (z,17
L/X m H
(5.17
g E G
:= u * ( g , h ) ' D ( g )
on the choice
By
Hom(H,K*)
, w h i c h is o b t a i n e d
(5.27 which
b' =
to
, the value of
it as a s u b g r o u p
we denote a t r a n s v e r s a l by e x t e n d i n g
Hab
.
[Y[H,H]/[H,H]Ip,
is the p ' - p a r t
~(gZ) = hY
and r e g a r d
Xl,X2,...,Xb,
Hom(Y[H,H]/[H,H],K*)
D of H:
D(h)
of g.
U = Hom(H/Y[H,H],K*)
Hom(H,K*)
representation
,
Is a n y e l e m e n t w i t h
does not d e p e n d Let
I S Ker(D7
, we obtain an i r r e d u c i b l e
D(h)
where
I z e Z
=
each e l e m e n t
of
to U in of
Thus we have [Y/Zip,
of the b r a n c h
factor
, of H; b' equals the b r a n c h
factor if p = O.
5.5 LEMMA. 91(k) =
lyl
The p r o o f
If k runs t h r o u g h ) U(~7(y).ri(YT}
of 5.5 is easy.
Now we are in the p o s i t i o n ence b e t w e e n (5.37
D
the i r r e d u c i b l e )~
Dx i =
~h,
Hom(Z,K*)
runs t h r o u g h
, then Hom(Y,K*7
[]
to write down the d e s i r e d c o r r e s p o n d K-representations )D(h)Ti(h)}
of G and H:
I i=1,2,...,b'
$ ,
173
and it readily follows (5.4)
DTil Y = 9i()(D)).I n
where
9i
is defined as in 5.5.
5.6 THEOREM. of G.
Let
DI
and
D2
be irreducible K-representations
Then the following properties are equivalent: (i)
Dl~i
(ii)
PROOF.
and
D2Tj
are equivalent representations
i = j , and
DI
and
If
D2
are equivalent,
DI
and
D2
of H.
are equivalent.
and the definitions above show that
D1~i
then and
they induce the same characters
of Y, i.e.
~i(~(D1)) = 9j(k(D2))
X(DI) = ~(D2)
, and again the proof
and
follows from the definition of
5.7 COROLLARY.
Let
DITi
and
D1,D2,...,D m
ducible K-representatlons
of G.
D2Tj
are equivalent,
If
i = J
and
D2Ti
proving that (ll) implies (i).
By 5.5 we obtain
DITi
X(DI) = k(D 2) ,
D2Tj
are equivalent,
[]
be a complete system of irre-
Then
Di~j
, I ~ i ~ m , I ~ j ~ b'
is a complete system of irreducible K-representations
of H.
^
PROOF. alent. 1.17
By 5.6 the representations
Diw j
are pairwise
The number m equals the number of p'-classes mb'
REMARK.
is the number of p'-classes
In all considerations
inequiv-
of G and by
of H, proving the c o r o l l a r y . ~
above,
it suffices to assume that
K is a common splitting field of G and H.
5.8 COROLLARY. let
m1(n )
tations of
resp. HI
Let
m2(n )
resp.
m1(n).IH21p,
H1
H2
and
H2
be finite isocllnic groups, and
be the number of irreducible K-represenof degree n.
= m2(n)-IH11p,
Then
174
PROOF.
Use 5.7 to compare
stem group.
By the results
and
with a common isoclinic
of II.1 we see that the correspondence
the linear irreducible
yields a correspondence G/Z
H~/
[~
5.9 REMARK. between
and
HI
H/Y
between
K-representatlons
of G and H also
the irreducible
K-representations
which can be lifted in G resp. H.
versals to Z in G and to Y in H, different inequivalent
factor systems
~i(~) ~ Hom(Y,K*)
of
G/Z
yield equivalent
factor
moment,
ideal P containing
the representations
p.
above are written
tation D, we denote by D its reduction the following
that for each
acters
and
u(k) = ~
u*(~)
and
5.10 PROPOSITION. properties
For the
which is a
R a valuation
ring in
we can assume that
over R, and for each represenmodulo P.
We also assume
the corresponding
such that
k-~ = k-~
in
char-
implies
Now we can prove
just made,
the following
are equivalent:
(ll)
Tq = r~ , and
and
D2zj
are
of H.
D2
are in the same p-block
We firstly remark that
ri
and
and
DITi
Tj .
coincide
DI
in the same p-block
and
Y[H,H]/[H,H] taining
are chosen
field
Furthermore
Under the assumptions
DITi
Ti
number
~ E Hom(Z,K*)
~i*(~i) = ~J*(~2 ) .
(1)
PROOF. if
on the modular represen-
field for G and H; let p be aprime,
K with maximal
~ all
groups G and H from above.
we assume that K is an algebraic
splitting
for a fixed
yield
systems.
We finish this section with a few remarks tation theory of the isoclinic
If we fix trans-
~ ~ Hom(Z,K*)
, whereas
of
on the p'-part
D27j
coincide,
of the branch
The values of the central resp.
rj
on the classes
characters
of G.
if and only
factor group
of blocks con-
of H which are contained
175
in Y are given by D2~j
~i(~(D1) )
resp.
~j(~(D2))
belong to the same block, then
cide on the p'-part of DITI
and
D2~j
Y/([H,H]
n Y)
~(D 1)
.
If
and
Dl~i
X(D2)
, which implies
and
must coin~
= T% .
are in the same block, if and only if we have irre-
ducible K-representations
FI,F2,...,F n , where
F 1 = Dlml
,
A
F n = D2~ j , and tuent.
Fi,Fi+ 1
have a common modular irreducible consti-
By 5.7 we can assume
K-representations Ti=Ti(1)
D~
F k = D~Ti(k)
of G.
, for suitable irreducible
The remarks above show that
, Ti(2),...,Ti(n)=~j
all coincide.
have a common modular constituent for all k. above yield
~(D~) = I(D~+ I)
tion above that D~
and
D~+ 1
D 1 = D~ and
u*(k(D~))
5.11 COROLLARY. p > 0 .
and
D~+ 1
The considerations
= u*(X(D~+I))
.
Hence
have a common modular constituent, D 2 = D'n
D~
for all k, which implies by our assump-
are in the same block.
The converse follows analogously.
teristic
Thus
it follows that what
proves that
Thus (1) implies (il)
[]
Let K be an algebraically closed field of characThen the correspondence
ducible K-representatlons
(5.3) between the irre-
of G and H can be chosen such that the
following properties are equivalent. (1)
D1~i
(il)
wl = mJ ' and
PROOF.
and
D2Tj
are in the same p-block of H. D1
and
D2
are in the same p-block of G.
We can assume that the representations given here are the
modular constituents of the representations given in 5.10.
In the following we investigate the connection between 5.10 and 5.11 in more detail.
Let
K,R,P
be as in 5.10, let D be an irre-
ducible K-representatlon of G written over R, and T a regular matrix over
R/P , such that
176
Id2
P
dI
T-1D1 T
@ where
dl,d2,...,d k
are p-modular irreducible representations
of G.
(By a theorem in FEIT [1], we can chose K and D such that D is even completely reducible.)
Hence we have
f fl
T-IDol T
=
@ where
fj = djv i , cf. proof of 5.11.
This observation and the
results above yield:
5.12 THEOREM.
Let G be a finite stem group and H a finite group
isoclinlc to G; let b be the branch factor of H, b' resp. bp the p'-part resp. p-part of b, where p is a prime. corresponds to
b'
p-blocks of H.
If a p-block of G contains n
ordinary and m modular irreducible characters, corresponding p-blocks of H contains irreducible characters.
Each p-block of G
n.bp
each of the
b'
ordinary and m modular
If N is the decomposition matrix of G and M
the matrix obtained from N by repeating each row bp-times,
then the
decomposition matrix of H is given by the block matrix consisting of
b'
blocks equal to M.
177
5.13 EXAMPLE.
Let G be the symmetric group of degree 7 and H
the non-spilt extension of 2/2 by G
, which is isoclinic to G:
2 G = group (Xl,...,x 6 I x i = [xl,xi+j+ 1] = (xlxi+l)3 = 1 ) H = group (Y,Xl,...,x 6 I x~Y -1 = [xi,xi+j+l]Y -2 = (xixi+l)3y -I = 1 ) . The decomposition matrices of G for
p = 2 , and
3
are given in
KERBER [I], KERBER/PEEL [1], ROBINSON [1]:
p=2
p=3
1 01 11
1 11 011 1111 0 011 1 111 1 0 I 1 1
1 0 1 0 1 1 0 1 1 1 11 21 11 1
II 11
1
The branch factor of H equals 2. From 5.12 we obtain the decomposition matrices of H ,
see next page.
We close this section with the remark that isocllnlsms of finite groups yield a correspondence between their defect groups, and also behave nicely with respect to the Brauer correspondence between the groups and their normallzers of the defect groups.
178 Decomposition
matrices
of
p=2
p=
1 1
1 11 011 1111 0 011 1 111 1 0 1 1
o
1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1
1 1 1 1 1 1 1 1
H
:
3
1
I 11 011 1111 0 011 I 111 I 0 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1
I
I
CHAPTER
IV.
OTHER GROUP-THEORETIC
APPLICATIONS
OF THE SCHUR MULTIPLICATOR
1. Deficiency
of Finitely
Our treatment
Presented
of the Schur multiplicator
rather abstractly
tion.
used free presentations
so far, except for Section II.3.
already that the size of erably finite)
Groups
group
G
M(G)
is severely
restricted
is known to have a "small"
Here we investigate
this relationship
strate that the Schur multiplicator tion on the possible
There we saw
free presenta-
closer and also demon-
does not contain
free presentations
if the (pref-
full informa-
of a given group,
the latter is finite.
To this end, we give an elementary
of the famous
of SWAN [1].
examples
The general assumption generated
group.
Many open problems
of this section is that
Equivalently
, we assume
G
even if treatment
remain.
is a finitely
the existence
of a free
presentation (1.1) with xi
R c F
;F
P ~ G
free of finite rank,
are called
"generators".
say on the basis
By the Nielsen-Schreier
R = Ker p
is again a free group
finite or
R = 0
many elements group
G
(of infinite
If it is possible
rl,...,r m E F
rank,
subgroup
R of
The
Theorem,
unless
to generate
as a normal
is called finitely presented.
information (1.2)
).
Xl,...,x n .
G
by finitely F , then the
In this case the above
is codified as
G = ( Xl,...,x n : rl,...,r m >p
and the presentation
(In the extended
sense)
is
is abbreviated
as
180
( X l , . . . , x n : r l , . . . , r m ) , or
(1.3')
P =
(1.3")
P = ~ Xl,...,x n
In these and
R
rj
formulas
F
The above rj = I"
The
0
of this topic
following
ators.
can,
Since
it w i t h o u t
proposition
~yl,...,yk ~
a presentation
general. defined
is
with
n
[I;
:= F/R.
"relations
[1 ; chp.
IV]. of " f i n i t e l y
on any finite
set of gener-
in these notes,
(8) p. 124]).
generators
: Sl(Y), .... Sk+m(Y)
The d e f i c i e n c y
def(P)
=
varies
= m - n .
inf
we r e c o r d
and
m of
If the group
relators G
, then
G
and G
also has
(Warning:
be the m i n i m u m
G
presentation
is a finitely
P
as
presented
as
finite p r e s e n t a t i o n s for finite
In the literature,
M
[q
of a finite
is defined
is n o n - n e g a t l v e
as the n e g a t i v e
When
If
>
Idef(P)}
over all
def(G)
1.3 LEMMA. d(M)
G(P)
of the form
def(G)
that
with
that the p r o p e r t y
be tested
relators"
here we rely on the thor-
in C R O W E L L / F 0 X
NEUMANN
then its d e f i c i e n c y
below
of language;
(B.H.
1.2 DEFINITION.
P
the use of the
is any set of g e n e r a t o r s
yl,...,yk
where
~x 1,...,xn~
proof.
has a p r e s e n t a t i o n
group,
is i d e n t i f i e d
this is not used elsewhere
1.1 P R O P O S I T I O N
in (1.3')
G
means
in principle,
on
of the set of "(defining)
most n o t a b l y
some abuse
rm = 1 ) .
as the free group
is specified,
, involves
presented"
closure
terminology,
ough t r e a t m e n t
Y =
is implicit
as the normal
and, u n l e s s
: r I .....
of G
G
.
We shall
see
and an integer
the d e f i c i e n c y
if often
of the above.)
is a finitely
number
generated
of g e n e r a t o r s
and
abelian rank(M)
group, :=
let
in
181
= dlmo(M ® Q) = d(M/Tor M)
the rational rank.
holds for finitely generated abelian groups (1)
d(A × B) ~ d(A) + d(B)
(li)
d(A x B) = d(A) + d(B)
PROOF.
Then the following
A
and
B :
; if
B
is free-abellan.
Easy consequence of the structure theorem for finitely
generated abelian groups.
1.4 PROPOSITION
(P. Hall's Inequality).
presented group, then
Gab
and
M(G)
If
G
is a finitely
are finitely generated
abelian and (1.4)
def G
~
d M(G) - rank(Gab)
PROOF (EPSTEIN [1; of
G
1.3.5,
as
in
(1.3'),
or directly
§I]). determing
by the
P
Given any finite free presentation F
Schur-Hopf
and
R
as above.
Formula,
By P r o p o s i t i o n
we h a v e a n e x a c t
se-
quence 0 Now
)M(G)
) R/[R,F]
~ F a b - - - ~ Gab
>0 .
K := Ker(Fab - Gab ) , being a subgroup of
abelian and therefore
R/JR,F] ~ M(Q) × K .
by elements of the form
fri f-1 = [f,ri]-r i
f e F , the abelian group rl[R,F],...,rm[R,F generated.
] , thus
R/JR,F]
Fab , is again free-
Since with
R
is generated
I ~ i ~ m
and
is generated by the cosets
d(R/[R,F]) ~ m
and
M(Q)
is f i n i t e l y
By the additivity of the rank function on short-exact
sequences and by Lemma 1.3 (ii), we obtain def(P) = m - n ~ d(RI[R,F])
- n = dM(G) + d(K) - n
= d(M(G)) + rank(K) - rank(Fab) = d(M(G)) - rank(Gab)
.
[~
182
1.5 REMARKS.
(a) As
this case P. Hall's def(G)
(1.5)
Gab
is torsion for a finite group
Inequality reduces to
~ d(M(G))
.
But even then the general formula is useful.
For we are often given
a free presentation without knowing whether the group it is finite.
G , in
But one easily computes
Gab .
G
defined by
If this is finite,
P. Hall's Inequality and the comparison with known finite factor groups of
G
may allow one to prove the finiteness of
G .
This
method was behind the reasoning of Example II.5.8. (b) Let
G
presentation ment that
be finitely presented. P
with
n = n(P)
Evidently there exists some
def(P) = def(G)
Can one add the require-
be the minimal number of generators of G ?
This is an open problem,
even for finite groups
G .
Consult
RAPAPORT [1] for results in this direction.
1.6 DEFINITION called efficient,
(EPSTEIN [I]).
A finitely presented group is
if equality holds in (1.4).
B.H. NEUMANN [2] stated (1.5) and raised the problem whether every finite group with trivial multipllcator has zero deficiency. EPSTEIN [I] suspected that
(Z × Z/2) , (Z x Z/3)
is not efficient.
We now give examples of efficient groups and of non-efflclent finite groups.
1.7 EXAMPLES.
a) According to EPSTEIN [I], every finitely gen-
erated abellan group G = Z/c I
x...x
Z/c t
× (Z) r
,
Cll...Ic
t
is efficient by virtue of its "canonical presentation". onical presentation has and
n(n-1)/2
and
M(G)
has
n = t + r
commutator relators. n(n-1)/2
generators,
t
power relators
On the other hand,
cyclic direct snmmands.
This can-
rank(G) = r
(Here we also
183
need that the orders can be arranged Cl,
in a divisor chain:
(n-2) times c2,... , r times ct.) def(G)
The groups ciency
-1
= ~ n(n-1) Z
and
(n-l) times
We find
- r ~ -1 .
Z x Z
; a complete
are the only abellan groups with defi-
llst of those with deficiency
Z x Z x Z , O, Z/c x Z
and
Z/c
for
zero is
c ~ 2 .
b) It will be shown in the next section that all finite metacyclic groups are efficient. c) The group tation,
G
of a tame knot with
called the Wirtinger
x I = Xn+l, x2,...,x n any two consecutive Gab ~ 2 .
and
presentation,
n
relators.
generators
In addition,
quence of the other (n-l) relators.
P. Hall's
Inequality
and the knot group
of
gives G
variants
d) The group
G
(1.6') results,
< x,y where
d(M(G))
the superfluous lar group (1.6") It follows groups are
( x,y
relation
to
y6 = I
= 0
G ; this implies
by deleting
= 0 .
-1
.
one relator, Then
Consequently,
The necessary
M(G) = 0
facts from knot
[1; chp. VI), where certain
presentation
are also discussed.
(~-~)
and
y
to
has been deleted.
free product
(~-~)
and
Then the modu-
is seen to have the presentation .
from (1.6') and (1.6")
= M(PSL(2,Z))
that
x 2 = y3 >
: x 2 = y3 = 1 >
Z/12
express
such that the presentation
corresponds
PSL(2,Z)
generators
is known to be an amalgated
: x4 = 1 , x
in
with deficiency
of the Wirtinger
of finite cyclic groups
n
The relations
Thus,
is efficient.
SL(2,Z)
with
has a presen-
is known to lle in the conse-
theory can be found e.g. in CROWELL/FOX "smaller"
crossings
are conjugate
any relator
we obtain a presentation
n
and
Z/6
that the commutator
, respectively.
by P. Hall's
Inequality
Hence
quotient
M(SL(2,Z))
and the groups are
=
184
efficient. e) Let us look at the infinite dihedral group fashion.
D~
in the same
From the presentation given In Example II.4.7 (b), we find
(D.)ab ~ Z/2 x Z/2
and we regain the result
M(D~) = 0
without any
effort.
1.8 EXAMPLE.
The binary polyhedral groups are, by definition,
the non-cycllc finite subgroups of
SU(2)
ful 2-dimensional complex representation.
Thus they have a faithUnder the celebrated
covering i±Ib ~
~ su(2)
T ~ so(3)
of topological groups, they correspond to the ordinary polyhedral groups, cf. DUVAL [I; chp. 3]. groups of order
4n
For
n ~ 2
the binary dihedral
are known to have the presentation
( s,t : s 2 = t n = (st) 2 >
,
and are commonly called dicycllc.
The binary tetrahedral,
octa-
hedral, and icosahedral groups have the presentations ( s,t : s 3 = t n = (st) 2 > with
n = 3,4,5
, respectively.
A simultaneous elementary treat-
ment of these presentations is found in COXETER [I] who departed from problems posed by Threlfall.
The above presentation of the
binary icosahedral group was already discussed in Example II.5.8. (Actually, we there started with the presentation and showed that its group is a subgroup of
SU(2)
that maps onto
A 5 .)
All binary
polyhedral groups have trivial multiplicator by virtue of the presentations listed.
We d l s g r e s s
to remark that
the concept of deficiency
linked with the study of 3-dimenslonal manifolds.
Let
ls closely M
be an
oriented compact connected 3-manifold without boundary, with finite
185
fundamental group
G , then
M(G) ~ H2(G,Z) ~ H2M = 0 by results from algebraic topology
(e.g. Poincar4 duality).
[1; Lemma 2.7] uses this to prove that
G
EPSTEIN
has zero deficiency.
This explains the previous example since THRELFALL/SEIFERT
[1]
showed that the binary polyhedral groups are fundamental groups as specified above. (Xo+X3i
, x2-xll ~
\-x2-xll of
SU(2)
3-sphere,
(Actually,
the real parametrlsation
, Xo2+X12+x22+x32 = 1
, Xo-X31J
implies that the underlying topological slmply-connected in particular.
space is a
A finite subgroup
G
gives rise to a covering SU(2) whence
M
~M
:= GkSU(2) = U Gx ,
is a 3-manifold as required with fundamental group iso-
morphic to G.)
EPSTEIN [I; Theorem 2.5] proceeds to show that the
fundamental groups
G
of all compact connected 3-manlfolds
(possibly non-orientable,
1.9 PROPOSITION group and
U
infinite
(SWAN [I; §2])
G
allowed) are efficient.
Let
G
a subgroup of finite index.
be a finitely presented Then
U
is finitely
presented and
(1.7) PROOF.
def(U) Let
+
P
1 ~
IC:ul.(def(C)
+
I)
as in (1.3') be a presentation of
def G = def P = m - n , let
i := IG:UI
Schreler process yields a presentation of generators and
m.i
relators,
with
Then the ReidemeisterU
with
see REIDEMEISTER
def(U) < i.m - 1 - (n-1).i = i.(def(G)+l)
1+(n-1).i
[I] or ROTMAN [1].
Hence
m
G
- I
186
Alternatively, one vertex, U
construct n
arcs,
which results
space
C(P)u
sen generator A = (Z/7) k
k , but
k - -
m
2-cells.
CW-complex
to the subgroup
of SWAN [I].
t
of
Z/3
Let
7 = 23 - 1 .) def(G) > 0
for
with
U
of
G ~ ~IC(P)
G k = (Z/7) k ~ Z/3
tat -I = a 2
for
It is asserted
.
, where a cho-
abelian group
a E A
that
(This is an
M(Gk) = 0
k ~ 3 , actually
These were the first examples
of
of the i-fold covering
acts on the elementary i.e.
C(P)
Look at the presentation
from the cell decomposition
by squaring,
action due to all
and
belonging
1.10 EXAMPLES
the 2-dlmensional
def(Gk)
- ~
for for
of finite groups known to be
not efficient.
PROOF.
We have
def A = k(k-1)/2
by Example
def(G k) > ~(def A + 1) - 1 = ~k(k-1)
- ~
>
3
-
-
for
k ~ 3
by (1.7).
the following
subgroup 1.6.8
and
M(Gk)
1.6.9 and for
~tl
is elementary.
Since the isomorphism t
acts on
A A A
.
M(A)
by
Iml
M(A)
has exponent
1.11 PROPOSITION.
Then
,
~m41
We know from Proposition for all
such that
G
By Proposition
and 7-torsion;
by Proposition
m e M(A)
of Theorem
in view of 1.3.14 Finally
seven (exponent
Let
is nilpotent,
is an isomorphism.
Invoking
res: M(A) " M(Gk)
M(A) ~ A A A
Ker(res) as
Gk .
of
w: G " Q
1.6.9 now
is surJective. 1.4.7
is natural,
a 2 A b 2 = 4(a A b)
(b) that
in
m 3 = tm.m-1E
res: M(A) - M(Gk)
one only for
and
vanishes
k = I ).
be an epimorphism
N := Ker ~ ~ [G,G]
.
topo-
is the Sylow 7-
= 0 , there is no 3-torslon. that
M(G k)
A
is a Sylow 3-subgroup
p = 7 , we conclude
of
tools from algebraic
Obviously
has at most 3-torsion
M(Z/3)
3
We are left to show the vanishing
Swan's proof of this fact used advanced logy,
1.7 a) and then
of groups M(Q) = 0 .
187
PROOF. by
We invoke
N ~ G - Q .
N/[N,G]
N
=
= 0 .
The assumptions
imply
Due to the nilpotency
[N,G]
Of course,
the 5-term exact sequence
[...[N,G]...,G]
=
this is a special
see 5TALLINGS
"unknown"
nilpotent
corresponding
~ [...[G,G]...,G]
[1].
Qab ~ Gab
G .
factor group
in the obvious
(il) typically,
0
=
.
Q
Theorem,
r]
can have striking applications presentation
A d d further relators
fashion.
If it happens
unknown group
G
quotient,
that
we have achieved
of some
such that the
has the same commutator
group with trivial multiplicator, (i) the previously
and then
G ,
Start with a "small"
group
= 0
determined
case of the Stalllngs-Stammbach
[1], STAMMBACH
[2] noticed.
Ker(~ab)
of
This innocent looking proposition as WAMSLEY
I.(3.3)
Q
is a known
two things:
has been identified
a smaller presentation
i.e.
as
for the known group
Q ; Q
has been found. (Warning:
Due to Theorem
generators
have non-trlvlal
1.12 EXAMPLE G
1.13,
(WAMSLEY
finite nilpotent
groups
on 4 or more
multiplicator.)
[2; p.135]).
Let
p
be an odd prime and
be defined by
(1.7)
( a,b
: c := a-lb-lab,
This group was introduced and nilpotent,
c-lac = a I+p, cbc -1 = b 1-p ) .
by MACDONALD
but the precise
order remained
factor group subject to the additional 2 ap
= 1 ,
Clearly
Gab ~ Qab ~ Z/p x Z/p.
bp
= 1 ,
and trivial multlplicator. to
open.
Let
Q
be the
relations
2
(1.8)
isomorphic
[1] who proved it to be finite
Q
and
Q
cp = 1 It is claimed
that
This being granted, to be efficient
G
by virtue
Q
has order
p5
turns out to be of (1.7).
188 We are left to prove the claim. exhibit
Q
of order
as a representation group of the non-abelian group p3
and exponent
representation group of The relations for
Q
b-laPb
p
H
and find
and
= (b-lab)P
p-q ~
i
~_I r_~.p
=
(Another
was specified by P. HALL [I; p.139].)
ckbc -k = b 1-kp
,
k ~ Z .
= (ac) p =
= acP.c-(P-1)acP-q
due to
M(H) ~ Z/p x Z/p .
H
imply
ckac -k = a q-kp Next
In the course of doing so, we
....
.c-2ac2.c-lac
= ap
2 and
cp = a p
= 1
Likewise
bp
is central.
i=1 The order of
G
cannot be less than
p5 .
For upon adding either
the relation a p = 1 or b p = I , we obtain a known group of order 4 p , viz. group No. 13 (resp. No. 12 for p = 3) in the llst of HUPPERT [I; Satz III.12.6; Aufg. 29, p.349]; were both
a p = I = bp
and thus
Q ~ H
IQ I = p4
then
would follow, contradiction.
We
conclude
IQI = p5
,
Z(Q)
=
laP,bPl
,
Q/Z(Q)
=_ H
We now invoke the 6-term exact sequence consisting of I.(3.3') and I.(4.2), for the central extension Using
[Hab I = p2
we conclude that Jective.
eH: Z/p ~
and IM(Z/p x Z/p)I = p 8.(eH)
Therefore
~Z/p
x Z/p .
and counting cardinalities,
is biJective and thus
M(H)
~H
is a factor group of
MH = x(eH)
sur-
Z/p x Z/p .
The
corresponding sequence for eO :
Z(Q)
c
~ O
~ ~H
with p-ranks written underneath,
,
is e.(eQ)>
Oab ® Z(O)
"XO) M(O)
4 We first find that
? Wab
~M(H) <2
~ab Z(H) 2
is isomorphic and thus
)Qab 2 8.(eQ)
~ Hab 2
.
surJective -
189 n e x t that tation
e.(eQ)
group
of
M(H) ~ Z/p x Z/p
We are going definition of
Q .
R/JR,F]
of
by P r o p o s i t i o n
.
Hence
~
x
(RP[R,F])
~Q
eQ
and proves
and to this end invoke
of the free p r e s e n t a t i o n
has e x p o n e n t
n (R n [ F , F ] )
at most
p
the explicit
(1.7)
- (1.8)
and
, we conclude
= JR,F]
u, v, w
as a r e p r e s e n -
is surJective.
~free-abelianl
elements
exhibits
II.2.14
7Q = 0
in terms M(Q)
~ M(Q)
This
H
to prove
Since
We define
is isomorphic.
of
R
. by
2 u = ap
v = a-q-Pc-lac
Let us a b b r e v i a t e (i) for
a = l+p
.
w = cp A n easy i n d u c t i o n
c_kack ~ a a k v l + ~ + . . + •
k > 0 .
We examine
the case
p-1 :=
~
k-1
gives
mod JR,F]
k := p
and note
p-1 (1+~+...+~
I-1)
-
i=1
~
i -- 0
rood p
,
i=1
P~l~i :=
= ~(a-1)
mod p2
+ p - p
i=o Thus
a = w-law
= c-Pac p = a l + P ~ v ~
1 + p~ = 1 + p2 rood p3 u m 0
(ii) Redoing
Inserting
from above,
= (b-lab)P
we use
~ ~ 0 mod p
, we conclude
(I) and
(ii) to deduce
= (ac) p
= a . c P . c - ( P - 1 ) a c p-1.
(ill)
a p2 = u
with
mod RP[R,F]
an a r g u m e n t
b-laPb
.
mod JR,F]
....c-lac
- w.a~v ~
mod [R,F]
= w.a p
mod RP[R,F]
a-Pb-laPb
= w = cp
In the same fashion we o b t a i n
mod RP[R,F] (iv)
from
.
,
and
190 a-lb-Pa = ( a - l b - l a ) P (iv)
= (cb-1)P = cb-lc -1.
a-lb-Pab p ~ w = c p
mod RP[R,F]
By the direct description I.(4.3) of by the cosets of
[a,b p]
....cP-lb-lc-(P-1).cP.b
and
-1,
.
~4~ ' its image is generated
laP,b] = [b,aP] -I
alone.
As
~
is
bihomomorphic, we have 2 [aP,b p] ~ [a,bP] p ~ [a,b p ] ~ 0
(V) Next, as
cac-la p-I
thus are central in a
-1b-lab
and
cbc-lb p-I
F/JR,F]
mod [R,F] .
and
[aP,b -I]
lie in
R
and
, we find
c ' c - c -I = [(cac-1 )-1 , ( c b c - 1 ) - l ] =
o
=
[aP -1,b p-I] = a-laPbPb-lal-Pbl-P a-lbPaPb-lal-Pb 1-p
by (v)
= a-lbP[aP,b-1]b-labl-P a-lbP-labl-P.[aP,b -1]
mod [R,F]
Consequently, (vi)
[b-l,a p] ~ [a-l,b p]
Again, as gives
~
w2 ~ 0
and finally
sod JR,F] .
is bihomomorphic, the combination of (iii) - (vi) and then
w ~ 0 mod RP[R,F]
.
We conclude
w ~ [R,F]
MQ = 0 .
It is still not known whether all finite p-groups are efficient. At least, the following famous result implies that p-groups on four or more generators have non-trivial multipllcator and thus need more relators than generators.
1.13 THEOREM (Golod-Safarevic). d(G)
Let
its minimal number of generators.
> d(G)2/4 , consequently (1.9)
def(G) ~ d(M(G)) > ~i(G) 2 - d(G)
G
be a finite p-group and
Then
d(Cext(G,Z/p))
191
GOLOD/SAFAREVIC
[1] proved
d(Cext(G,Z/p))
> [d(G)-112/4
.
We
recommend Roquette's proof which is given in each of the books of GRUENBERG [I], HUPPERT [I], and D.L. JOHNSON [I].
Here we are con-
tent to relate (1.9) to the more customary formulations of the Golod-Safarevi~ Theorem.
By the Burnside Basis Theorem and the
Universal Coefficient Theorem 1.3.8, we have d(Cext(G,Z/p))
= d(M(G)) + d(G)
d(G) = d(Gab)
and
.
This and (1.9) reduce (1.9) to the asserted inequality for d(Cext(G,Z/p)) free group on
Choose a presentation d(G)
generators for the step is
Z[G]-module
r' > d(G)2/4
shown to agree with Finally
generators,
let
R ~ F--~G
r'
Rab/R~b
d(Cext(G,Z/p))
F
the
be the minimum number of .
The first and trickiest
, see HUPPERT [1; III.18.9]. d(R/RP[R,F]),
with
Next
r'
is
see HUPPERT [1; V.25.2 (c)].
= d(R/RP[R,F])
follows from the exact se-
quence I.(2.3) associated with the free presentation and the coefficients
Z/p , trivial action.
It is only fair to say that our knowledge on deficiency is quite deficient.
We draw the reader's attention to WAMSLEY [2]; most
questions of this survey are still open.
JOHNSON/ROBERTSON
listed many finite groups known to have zero deficiency. [4] recalled conjectures llke the one that
A5 x A5
[I]
WIEGOLD
is not effi-
cient, i.e. requires at least three more relators than generators.
Complete results are available for certain modified concepts. For example, a finite p-group erators and
d(Cext(G,Z/p))
G
relators as a pro-p-group;
starting point in GOLOD/SAFAREVIC potent varieties of exponent zero. variety of all nil-3 groups three.
can be presented with
[I].
d(G)
gen-
this was the
Another good case are nil-
For example, let
•
be the
G , i.e. of nilpotency class at most
Whenever (1.2) is interpreted as a ~-free presentation of
192
G , It is understood that
R
need not be listed separately.
contains
[[F,F],F]
The roles of
Gab
, such relators and
N(G)
are
now played by certain "varietal homology groups" - instances of which wlll be treated in Section 6.
Then the analogue of P. Hall's Ine-
quality turns out to be equality by STA~4BACH [2], [3; Cot. IV.6.6].
193 2. Metacyclic
Groups
A metacyclic possesses cyclic.
group is, by definition,
a cyclic normal (Cyclic groups
a trivial
case.)
subgroup G
A
a finite group
such that
are not excluded,
Subgroups
G/A
G
which
is also
though they constitute
and quotient groups
of metacyclic
groups
are again metacyclic.
Metacycllc
groups have well-known
on four parameters.
However,
the group isomorphism pose a new choice
free presentations
it is a difficult
for the fourth one - this parameter
We show that every metacyclic
pute the Schur multipllcator groups,
results
group possesses
turns out to
a metacyclic
among others.
and prove the efficiency
first obtained
Much of the material
We pro-
and thus a group invariant.
tation group with trivial multlpllcator,
ferent fashion.
problem to enumerate
types in terms of these parameters.
be the order of the Schur multiplicator
metacycllc
which depend
by WAMSLEY
represen-
We then comof arbitrary
[I] in a dif-
of this section is taken from
BE"~ [ 1 ] . 2.1 PROPOSITION. with generator
v
Let , let
O = Z/n A
be the cyclic group of order
be a O-module.
A0 _ (I+v+...+Tn_I).A
n
Then
=
u :
Opext(O,A)
the isomorphism
u
being specified
,
in the proof.
This result is a special case of the classical see SCHREIER
[I] or ZASSENHAUS
and 2.5 below, that
AO
[1; III
it is given a different
is the subgroup
§7]. proof.
extension
theory,
For the purposes Recall
of the fixed elements.
of 2.2
from ~(6.4)
194
PROOF. ator
Let
b .
Z
be the i n f i n i t e
We invoke
(2.1) of
Z
en :
Q , with acts on
the o b v i o u s
Re-
.~Z
u(b) = T A
e*(en,A)
by =
and
is a h o m o m o r p h l s m
with kernel
of d e r i v a t i o n s
by Lemma
1.2.2;
thus
d:
Z
and,
~ = bn . by T h e o r e m
HomQ(R,A)consisting
-'.
A
.
0pext(Q;A)
T h e n the g r o u p 1.2.7,
Opext(Q,A)
of all r e s t r i c t i o n s
And
e*(~n,A)
is a q u o t i e n t
is s u r J e c t l v e
g r o u p of
HomQ(R,A)
R -- Z .
A homorphlsm specifying T'~(o)
a: R - A
u=a(~)
= a(To)
parametrlzed has
free on
:
free on the g e n e r -
,
:= Ta
>a.[~n]}
a = dl R
with
R
group,
free p r e s e n t a t i o n
~ ~Z/n
ba = T.a
~a,
cyclic
e A
.
of g r o u p s
It is Q - h o m o m o r p h i c
= a(b n) = a(~)
by
AQ .
is u n i q u e l y
If
.
d: Z - A
d(b)
~(10+~+...+,k-1)'a
for
k > I
for
k = 0
for
k < -1
~_(-1+..+T-k). T h i s is the e x p l i c i t element
2.2 C O R O L L A R Y . ( a,b
the i n t e g e r s
a
description
u e A Q , define
U-q(u + Denominator)
(2.2)
But every
defined
d(b k) =
of
a: R - A
= a.[~n]
(2.3a)
m,n > 0
(2.3b)
rn E I
(2.3c)
mlt(r-1)
, mod m
,
is of the
Pick a r e p r e s e n t a t i v e
~(~) = u
a n d let
g r o u p has a p r e s e n t a t i o n
: a m = 1 , b n = a t , bab -I = a r ) being
~ = dl R
by
u-l: by
a ~ A
are
[7
Every metacycllc
m,n,r,t
~'s
is any d e r i v a t i o n ,
, take the d e r i v a t i o n
by
precisely when
Thus the a d m i s s i b l e
u = d(b n) = ( l + T + . . . + T n - 1 ) - d ( b )
form
determined
,
subject to the c o n d i t i o n s
195
Conversely, (2.3),
the group d e f i n e d by (2.2),
is m e t a c y c l i c
of order
This characterization there a t t r i b u t e d
PROOF.
A
is also cyclic,
p h i s m of r
prime
A
is stated by 2 A S S E N H A U S
be a cyclic n o r m a l m =
of
IAI
G/A
and
.
n =
A
some
t ~ Z .
IG/AI
is a G/A-module.
by
[I;
(2.3c)
III
G
~7S and
such that b E G
determines a~-~a r
that maps
an a u t o m o r -
(2.3b)
, thus
at ~ A Q
G/A
for some i n t e g ~
Condition
bn ~ A
means
of
Choose b
by a).
Of course
Condition
•
has the form
m (A b e i n g g e n e r a t e d
that
subgroup
Conjugation
which necessarily
to
.
to H61der.
Let
on a g e n e r a t o r
m.n
subject to the c o n d i t i o n s
expresses
bn = a t
and follows
for from
a t = b n = bbnb -I = (bah-l) t = a rt We now recall A~
from P r o p o s i t i o n
~ G---~G/A
According Now
G
is i s o m o r p h i c
subgroup
to
along
A~ Z /
in the d e n o m i n a t o r
tional
relator
a-tb n .
normal
subgroup
is cyclic.)
(One a d d i t i o n a l
~ =
~a!
~ (a-kt,b kn)
is just a p a r a p h r a s e
A = £/m
into a £ / n - m o d u l e
forward-lnduced
Here
~ at } I k ~ ~ }.
for the addi-
suffices
since the 2.1,
the
that the c o n d i t i o n s
and the fixed element
at
turn
specifies
a
(2.3c),
viz.
extension.
integer
, and all other such
(m,n)
of the i n t e g e r s
relator
of the facts
is a s m a l l e s t p o s i t i v e
t o = m/(m,r-1)
accounts
On the basis of P r o p o s i t i o n
converse
to .
en
: a m = I, bab -1 = a r )
the n o r m a l
There
from
has the p r e s e n t a t i o n
(a,b while
is f o r w a r d - i n d u c e d
to I.I.5,
A S i
2.1 and its p r o o f that
denotes m
and
as a d e f i n i n g p a r a m e t e r
n
the
t
t
satisfying
are integral
(positive)
greatest
, not both zero.
instead
multiples
of
common d i v i s o r
We choose
X = t/t o
of t, tbls will be v i n d e c a t e d
soon.
196
2.3 DEFINITION• order
m-n
being
Let
presented
subject
:
and
(2.3b).
metacyclic
is inclusion,
w(b)
~brl
B - B
Clearly
1" = 1 .
mands.
By T h e o r e m
we express
in case
Write
by saying:
"Opext
a .
by
The formula
Q = Z/n
Assume
m.a = a r .
rn ~ I Then
.
a~ n
= at
no: R - A
e(m,n,r,1)
the e n d o m o r p h l s m
Irl
= (-I)*
sum-
and
;
~ , let
and r e g a r d is cyclic
A = Z/m A
as a
of order
•
exhibits
p
[e(M,n,r,1)]
from P r o p o s i t i o n
metacycllc
for some Q - h o m o m o r p h l s m
2•1,
extenslon
~: R - A
is
with
m t o
=
be the Q - h o m o m o r p h i s m
of 2.2.
of
that the typical
,
= ao(~n)
cf. the p r o o f
.
n-l)
X[e(m,n,r,1)]
a Xt° =
Let
Z/n
0pext(Q,A)
Recall
to)
mod m
Opext(Q,A)
We l n v o k e t h e d e s c r i p t i o n
A = Z/n
c(a)
of
of
is additive"•
•
=
and
~
with
with generator
= (m'r-1).(m,l+r+..+r
[e(m,n,r,X)]
as a g e n e r a t o r
(congruent
r = ±1..±I
this p r o p e r t y
Let
lal
is even a Q-module.
- 0pext(Q,B)
m
now
B
0pext(Q,B)
h(m,n,r)
PROOF.
, consider
:
= r*
with g e n e r a t o r
(2.4)
B
Z/m =
generator
0pext(Q,(-1)*)
2.5 PROPOSITION.
Z/n-module
where
we here
Opext(Q,r*)
1.2.4,
, the p a r a m e t e r s
;
Q-homomorphlc
and
extension
group
this is a u t o m a t i c a l l y 0* = 0
of
~Z/n
is the d i s t i n g u i s h e d
:
group
Let
group
For any a b e l l a n
~b|
t = ~ mX
Z/m~ ~ j G ( m , n , r , ~ )
be the supporting
2.4 REMARK.
be the m e t a c y c l l c
by (2.2) with
to (2.3a)
e(m,n,r,X)
r* =
G(m,n,r,X)
and
e(m,n,r,~)
Since
~
with = a(~n )
no(a)
= a t° .
Then
by the very definition,
is the c o m p o s i t e
197
ao > A
R
the formula 1.1.11
k* ) A
,
[e(m,n,r,X)] = X[e(m,n,r,1)]
(a) and the addltlvity of
Opext .
is immediate from Proposition 2.1. group of order
m/t o
subgroup of order al+r+...+r n-1
PROOF.
s e Z
with
d
Let
plt
a t°
and
AQ
h(m,n,r)
is the cyclic
and the denominator is the
generated by
for integers
and
s E t
(Z/K)* - (Z/L)*
K,L,t
mod L .
b e the largest factor of
p ~ t .
If
p
dividing
LIK , then
, then
Conse-
is an eplmorphlsm for
denotes the unit group of the ring
For the primes
and
K $ 0
(s,K) = 1
• r+L.Z}:
(Z/K)*
s := t + d-L . cases
generated by
(t,L) = 1
~r+KZl
Here
In detail,
m/(m,l+r+...r n-l)
If
there exists
LIK .
The formula for
[]
2.6 LEMMA.
quently,
results from Proposition
K
Z/K .
prime to
t .
Set
K , distinguish the
~r+KZ :
~ r+L-Z}
is a
(unltal) homomorphism of rings, surJectlve by the first assertion. Alternatively,
this lemma follows from the Chinese Remainder Theorem,
see HASSE [1 ; §4 No.2].
2.7 THEOREM. module. [e2]
Assume that
Q
be an arbitrary group and
0pext(Q,A)
have the same order in
morphism groups
Let
~I
a G1
PROOF.
of
A
with
and
G2
Since
A
0pext(Q,A)
a finite Q-
Whenever
[e 1]
and
, there exists an auto-
e 2 ~ ae I ; in particular,
the middle
are isomorphic.
is finite, there is
in the notation of 2.4.
finite cyclic of some order [el] = S[eo]
m ~ o
The addltlvlty of
m*=0: 0pext(Q,A) - Opext(Q,A)
erator, let
is cyclic.
A
Opext
Consequently, L
and
dividing
with
m .
[e2] = t[eo]
m*=0: A - A
implies
Opext(Q,A) Let
[e o]
is be a gen-
have the same order in
198 0pext(Q,A)
.
both p r i m e
to
t/d ~ r l . S / d m
with
Let
Then
(s,L) = (t,L) = d , say.
As
s/d
and
t/d
are
L/d
, there exists an integer
rI
prime to
L/d
with
mod L/d
r ~ rI
.
Now Lemma 2.6 gives an i n t e g e r
mod L/d and
afr*: A - A
, this is an isomorphism.
1 = u'm + v-r
is additive,
a.[el]
morphism
from S e c t i o n
(a,S,y):
~*r* = 1
(There are i n t e g e r s
due to
when
Assume
m* = 0 .)
W,
Since
Opext
This corollary
[~]
1.1 that an i s o m o r p h i s m in w h i c h
rn ~ I
a,~,?
and
with
is a
isomorphic.
h = h(m,n,r)
as in
are i s o m o r p h i c
group
k' = (X,h)
is part of more c o m p l e t e extensions
of e x t e n s i o n s
e(m,n,r,u)
The m e t a c y c l i c
e(m,n,r,~')
of m e t a c y c l i c
are
mod m , let
e(m,n,r,k)
(k,h) = (u,h)
s u p p o r t e d by
classification
= [e2]
e I - e2
The e x t e n s i o n s
precisely is also
, then
mod L .
we c o n c l u d e
2.8 COROLLARY. (2.4).
t ~ r-s
= rS[eo] = t[eo]
We recall
prime to
, hence w i t h
(m,r) ffi 1
with
r
G(m,n,r,k) dividing
results which
h .
include
up to isomorphism,
the
the
d e t a i l s are in B E Y L [1].
PROOF.
Assume
Abbreviate
that
eI
eI
= e(m,n,r,k)
and
e 2 = e(m,n,r,u)
and
e2
are isomorphic.
By T h e o r e m 1.1.10
a
of
Z/m
Z/n
are a u t o m o r p h l s m s
and
~
of
there
with
[e2] = (~*)-la.[el] (Warning:
Implicit
in the cited result
module
structure
on
Z/m
which
Since
(~*)-la.
is an i s o m o r p h i s m
c l a s s e s have the same order in
is the use of an i n t e r m e d i a t e
is d e s c r i b e d
by a p o w e r
of a b e l i a n groups,
Opext(Z/n,Z/m)
.
of
r .)
both e x t e n s i o n
As the order of
199
e(m,n,r,l)
is
h/(k,h)
(~,h) = (u,h)
•
For the converse,
means
that
there
is a Z / n - a u t o m o r p h i s m
This
[el]
by P r o p o s i t i o n
[e2]
follows
first note
that
have the same order. a
yields an i s o m o r p h i s m
second a s s e r t i o n U :=
and
2.5, we conclude
of
Z/m
(c~,-,I):
(~,h)
By T h e o r e m
such that
e I ~ e2
= (u,h) 2.7
[e2] = a.[e I]
of extensions.
The
from the first by letting
I' = (~,h)
The p r e c e d i n g
(2.5)
corollary
allows
us to impose
the c o n d i t i o n
~ I h(m,n,r)
without
loss of generality.
times p r e f e r
~ = 0
2.9 THEOREM. G(m,n,r,1)
(BEYL [2]).
(m,r-1)
prime
to
m
(b) and
to
with
(m,r-1)
it has trivial
s ~ 1
S c h u r multlplicator.
and
1
with
.
in
and Lemma
2.6 gives
an
s
G(m,n,r,1)
, the f o l l o w i n g
relations
hold:
bn = at
(c)
[b,a s] = a (m'r-1)
bn = at .
and only two
mod m
(b)
G(m,n,r,1)
k
group
mod t , hence w i t h
E s(r-1)
(c) clearly
It is c l a i m e d
integers
t := m/(m,r-1)
[b,a s] = a s(r-1)
The m e t a c y c l i c
w i t h two g e n e r a t o r s
Choose
that we some-
in the split case.
mod m .
= k.m + l(r-1)
is prime
(a) Since
rn ~ I
In particular,
PROOF
1
Let
I = h(m,n,r)
can be p r e s e n t e d
relators.
Then
to
This we now do - except
with
t = (m,~-1) .
that the r e l a t o r s
are c o n s e q u e n c e s We conclude
'
of the d e f i n i n g
of (b) and
from (b) that
at
presentation
(c); this is obvious commutes
with
of for
b , and
.
200
from (c) t h a t
[b,a s]
commutes w i t h
a
Hence
.
a st = b aStb -1 = (baSb-1) t = ([b,aS]aS) t = [b,aS]ta st and
[b,aS]t = I .
(d)
am
Since
Invoklng
(c), we obtain
a (m'r-1)t = [b,aS] t = 1 .
=
(s,m) -- I , there is
v e Z
with
sv-= I
[b,a] = [b,a sv] = baSVb-la -sv
The f i n a l
(baSb-1)Va -sv = ([b,aS]aS)Va -sv
=
[b,aS]Va sv-sv = aV(m, r-l)
assertion
Then
by (d)
=
= aVS(r-l)
mod m .
= a r-1
by
(c)
by
(d),(a)
.
f o l l o w s from P r o p o s i t i o n 1. 4 (P. H a l l ' s
Inequality).
2.10 T H E O R E M metacyclic groups.
[I]).
E v e r y metacycllc
group with trivial m u l t i p l i c a t o r
If
rn ~ I
G(m,n,r,k) in
(BEYL/JONES
mod m
group has a
among its r e p r e s e n t a t i o n
, then the Schur m u l t i p l l c a t o r
is cyclic of order
(X,h(m,n,r))
with
of
h(m,n,r)
as
(2.4). PROOF.
Without
loss of g e n e r a l i t y
be subject to (2.3a), prime to
m/(m,r-1)
(t,m) = I Put
s
~
r
Consequently, G(m,n,r,X) (2.6)
m.~
Therefore
rood
and (2.9).
Since
, Lemma 2.6 gives an integer
and
s := t(m,r-1)
(2.3.b),
let the p a r a m e t e r s
r-1
t E ~
m mod (m,r-1)
m,n,r,X
r-I/(m,r-1) t
is
with
"
+ 1 , then m
.
(m,r-1) = (m,s-1)
~ G(m,n,s,~)
I m.h(m,n,s)
.
and
h(m,n,r)
= h(m,n,s)
and
Due to (2.5) and (2.4), we have
I (s-l)(1+s+.-.+sn-1)
G := G(m.~,n,s,1)
= sn - 1
is defined as a metacyclic
group of
201 order
m-n.l
on the generators
o: G " G(m,n,s,t) This
0
a
and
b , cf. Cor. 2.2.
be the eplmorphism with
0(a) = a
is well-defined because the relations of
a .
and
G
0(b) = b .
are respected; in
m.~ (m.X,s-1)
the latter is obvious except for the relation with the exponent of
Let
However,
t h(m,n,s) i (re,s-l) and our choice of
t
imply
(m.X,s-1)l(m.(m,s-1),s-1)
=
(m,s-1)(m,t)
=
Counting orders, we conclude that the kernel of ~ m } ~ Z/I .
It is claimed that
(2.7)
I~m~ ~
~ :
;~ = G(m'~,n,s,1)
is a representation group of
[~,a~=] = ~ - l E - m
abellanized
g r o u p s have t h e same o r d e r
Jective.
central.
A g a i n by
is a stem extension and Now
G
.
coincides with
0=G(m,n,s,~) .
First,
= a~l~(S-1) = 1
e
e
0
G(m,n,r,l) ~ G(m,n,s,X)
shows t h a t
Thus
is
(m,s-1)
(m'X,s-1)
= (m,s-1)
n.(m,s-1)
, the
and c o i n c i d e .
~.(e): MG(m,n,s,X)
- Z/~
is sur-
has trivial multiplicator by Theorem 2.9.
The
claim and the remaining assertion now follow by Proposition II.2.14.
[] 2.11 COROLLARY.
PROOF.
If
All metacyclic groups are efficient.
G(m,n,r,~)
(~,h(m,n,r)) = 1
has trivial multiplicator,
by Theorem 2.10 and
then
G(m,n,r,~) ~ G(m,n,r,1)
Corollary 2.8, the latter has deficiency zero by Theorem 2.9. the other metacyclic groups, the presentation
2.12 EXAMPLE. of order
2m .
Consider the dihedral groups We easily compute
by For
(2.2) is efficient.
D2m = G(m,2,-1,0)
202
h(m,2,-1)
= [1
1 Thus
M(D2m)
Then our
= 0
method
M(D2m) ~ Z / 2
.
If we c h o o s e group
for m o d d , 2
for
m even
whenever
m
replaces Which
is
X = 0
odd. by
F r o m now o n ,
X = 2
representation
s = r = -1
G(2m,2,-1,1)
.
s = m-1
h(2m,2,m-1)
l e a d s to the s o - c a l l e d
2.13 LEMMA. group
G(m,n,r,~) a
where
to to
such that in
rn ~ 1
m/(m,r-1) mlr~-I
(Z/m)*
and
PROOF.
D2m
and because
quasl-dlhedral
The c e n t e r
Theorem 2.107
quaternlon
.
If
41m
,
of group
of the m e t a c y c l l c
by
no
is the s m a l l e s t p o s i t i v e
In other words,
The central
G(to,no,r,O)
from
.
mod m .
is g e n e r a t e d
be even.
n b o
and =
g r o u p of
is also p o s s i b l e
~ G(2=,2,m-1,0)
Let
result
, then we o b t a i n the g e n e r a l i z e d
as a r e p r e s e n t a t i o n
G(2m,2,m-1,1)
m
, we c o n c l u d e
groups
then the c h o i c e = 1
let
no
integer
is the order of
factor group is the m e t a c y c l l c
r + mY group
•
The element
z = aib j
with
J ~ 0
lles in the c e n t e r
if, a n d only if, [z,b]
= a i(1-r)
and
[z,a]
= a (rj)-I
are the u n i t element.
This amounts
free p r e s e n t a t i o n
G/Z(G)
for
2.14 P R O P O S I T I O N . w: G - Q=G/Z(G) Then
M(w):
sentation
Let
G
group of
G .
toll
and
nol j .
The stated
is immediate.
be a m e t a c y c l l c
be the n a t u r a l
M(G) ~ M(Q)
to
projection
vanishes
and
eG
in
g r o u p and eG
, cf.
III.(1.1).
is a g e n e r a l i z e d
repre-
203
PROOF.
From
M(a)
I.(3.3')
M(~)
we have an exact sequence
M(Q)
--Z(a) n [G,a]
where the final arrow is the restriction Let
G = G(m,n,r,~)
this lemma
of
, invoke the notation
Q = G(to,no,r,0)
e.(e G)
onto its image.
of the previous
; the multiplicator
lemma.
By
of this group has
order dividing t I = (~-~
, r-l)
by Theorem 2.10 and (2.4).
On the other hand,
at e
Z(G)
O [G,G]
with t = lcm(to,(m,r-1)) thus
Z(G)
O [G,G]
= m/t I
;
has order at least
e.(eG)
is monomorphlc
follows
from Proposition
2.15 COROLLARY.
and thus
tI .
M(~) = 0 .
11.2.13.
We conclude
that
The final assertion
[]
A n y two metacyclic
groups with isomorphic
central
factor groups are isoclinic.
PROOF.
Let
G
and
tral factor groups. H---~H/Z(H) G/Z(G) III.2.4
~ G/Z(G)
be metacyclic
By Proposition
(ill).
Finally
111.1.4.
2.16 COROLLARY. G(m,n,r,l)
PROOF.
G
and
groups with isomorphic
2.13 both
are generalized
Thus these extensions
Proposition
groups
H
representation
are isoclinic H
G~G/Z(G)
cen-
and
groups of
by Corollary
are isocllnic
groups,
cf.
[]
For fixed
m,n,r ~ N
with
lle in the same isoclinism
Combine Corollary
rn ~ I family.
2.15 with Lemma 2.13.
mod m , all
204
3. The Precise Center of an Extension
Group and Capable Groups
We recall that the idea of isocllnic P. HALL [1]. in Section
The concept
III.1,
formulations
of isocllnism
of extensions,
basically has the advantage
of the results.
group isocllnlsm
groups was invented
can be attributed
with regard to homomorphlsms
, epimorphlsms
to the fact that the Baer sum of two strictly not strictly (3.1)
central
e = (K,~)
is, by definition, utilize
in general. :
the strictly
central
we obtain various of the form
if
for some
esting in its own right. [1],
and
central extensions
~(A) = Z(G)
from
.
is
Cext(Q,A)
In the process
fundamental
i.e. groups
The latter problem
The principal
clas-
for selecting
of capable groups,
G .
In order to
for the isoclinlsm
we need and now give criteria
characterizations
Inaut(G)
BEYL/FELGNER/SCHMID
central
extensions
excepted,
~Q
results on isocllnism
of groups,
of the center
Here an extension
>G
strictly
our pre~ious
sification
A~
smoother
in the case of
to the bad behavior
f: G - H
as introduced
of allowing
The added difficulties
by
is inter-
source for this section is
papers on the capability
of
abelian groups being due to BAER [2], [3].
3.1 DEFINITION is capable, unlcentral
(cf. HALL/SENIOR
if it is isomorphic if
~Z(G) = Z(Q)
3.2 DEFINITION.
N
xeJ
G/Z(G)
<x>
[1]).
A group
for some group
for each eplmorphism
For each generating
let
Wj(G) =
to
[I] and EVENS
subset
J
Q
G , and
,: G ~ Q .
of the group
G ,
205
In other words, J
or
x e Wj(G)
x = 1 .
runs through
subgroup
W(G).N/N
e
subsets
of
G
W(Q) S ,Z(G) ~ Z(Q)
G
.
of
x
Wj(G)
Obviously
contain
where
J
W(G)
is a
and N ~ G .
MILLER
be any central .
of
for
(cf. G.A.
as in (3.1)
w h e n the roots
as the Join of all
S W(G/N)
3.3 P R O P O S I T I O N Let
W(G)
all g e n e r a t i n g
characteristic (3.2)
Define
precisely
[1; p.359] extension
Thus a group
Q
or P. H A L L
by
with
Q .
W(Q)
[I; p.137]).
Then
~ 0
is not
capable.
PROOF.
Let
is central, Ej while J .
J
be a g e n e r a t i n g -1((x))
E x :=
:= ~-Iwj(Q) G
Wj(Q)
3.4 EXAMPLES.
A
Q
and
x E J .
As
e
Clearly
N Ex , xeJ Ex .
It follows
= ~(Ej) _c ~Z(G)
W h e n applicable,
the c o n d i t i o n
a) Let
of
is abelian.
is the Join of the
Finally
However,
=
subset
W(Q)
be a finite
A ~ Z/n I x ... x Z/n r
= 0
Ej _c Z(G)
for all
.
Proposition
3.3 is quite handy.
does not imply the c a p a b i l i t y
abelian with
of O.
group, nrJ...Jn I
,
tI
being a g e n e r a t o r of the i-th factor, of order n I . T h e n one n2 finds W(A) = (t I ) of order n l / n 2 . (Read n 2 = I in case r = 1 .)
Sketch
~t; 2) S W(A) J = when r = 2
of proof:
results
I t 1, t2"tl 1,
is incapable
for
r ~ 2
by hand;
the formula
from .--
r ~ 3 , we can derive by invoking
Do the cases
,tr't; 1 1
;
the reverse
inclusion
(3.2)
for s u f f i c i e n t l y
n I ~ n2
We will
from the case
many projections.
see in Example
4.11
that
Thus A
A is
206
capable
when
nI = n2 .
b) Let let
Z/2
Q8
be g e n e r a t e d
group w i t h
J =
t .
, where
p5
p
[1; E x a m p l e
capable.
:= i2 = j2
Indeed
p
and
G
We are going
minimal
generating
say
J ,
follows
from
(-1,t 2) e Wj(G)
G .
[I; p.355]
to compute
and
is an incapable
This
in 4.16 that
sets,
i
the extraspecial
, cf. HUPPERT
It will be shown
thus not capable. to consider
generates
be an odd prime
5.13].
G = Q8 x Z/2
~- 2/2 x 2/2
-1
and exponent
group w i t h g e n e r a t o r s
Then
~(i,t2),(j,t2),(i,t)~
c) Let order
by
Z(G) ~- G/2(G)
(-1,t 2) e W(G) where
be the q u a t e r n i o n
W(G) J =
p-group
of
or W A R F I E L D G
is unicentral,
= 0 .
It suffices
~Xl,...,x4~
.
Assume
we had
for
m,n e ~ .
divide
m
x P = x P2 = I , it would
Since
and
xI e
~x 2)
3.5 PROPOSITION. G
such that all
capable.
If G/N i
PROOF.
that
= ~g:
contradiction
is a system
are capable, G
G/N
Consider
This
~Ni~le I
Consequently
the p r o p e r t y
.
admits
follow
that
proves
of normal
p
Wj(G)
doesn't = 0 .
subgroups
then
G/( ~ N i) is also ieI a least normal subgroup N with
is capable.
the "diagonal"
~IgN ill :
C-
n
map
(Q/N i) •
ieI Since
Ker
show that central
A =
O Ni iEI
A(G)
H ei leI
strictly
is capable.
~(G) ~ G/( N N i) i~I
By assumption,
and w i s h to
there are strictly
extensions
e i = (~i,~i) Then
, we have
:
A i~
as in I.(2.3),
central.
~E i
,~- G/N i
but w i t h
The r e s t r i c t i o n
index
e =
set
I , is again
~ ellA(G) ieI
is a central
of
207 extension central
of
NAi
and yields
3.6 THEOREM. and
by
@
A(G)
Let
e=(~,~):
~ = q[Q,Q]
(3.3)
e.(e)
It is easily
the c a p a b i l i t y
its "commutator
element
.
form"
of
Qab
of
At
A(G)
[~
~Q
>G
as in R e m a r k and
be a central 1.4.9.
z e Z(Q)
7~(q ® z) = ~(q,z)
to be strictly
checked
extension
For the typical
,
e A
holds and (3.4)
~Z(G)
=
~ x e Z(Q)
Note that the final
(3.5)
I V ~ E Qab
conclusion
: 7~(q ® x) e Ker
depends
8.(e)
j .
only on the subgroup
U(e) := Ker 8.(e) S M(Q)
rather
e
than
itself
Theorem
III.2.3.
central
extensions
PROOF.
In particular,
Recall
x = ~(g)
and
for all
y ~ Q
- the same was true U(e)
for isoclinism
must be "small"
in
for strictly
e .
that
@(~,~h)
y = ~(h) and all
= ~-l[g,h]
commute
in
z e Z(Q)
Q .
is defined
Hence
In view of
whenever
~(ytz) ~Z(G)
is defined S Z(Q)
we
find ~Z(G) Thus
(3.4)
=
is r e d u c e d
R ~ F---~Q be evaluated. diagram :
i z e Z(Q)
I V y E Q : ¢(y,z)
to the formula
be the free p r e s e n t a t i o n F r o m 1.3.6 and
[~A,G]
(3.3) of = O
Q
= I ~ A
i •
which we now prove. at w h i c h we obtain
M(Q)
Let
shall
the c o m m u t a t i v e
208
~O
F.
R
where
F = F/JR,F]
projection. f,w e F Ganea
xo(q
Pick
with
Q
and
Q/Z(Q)
@ Z) = ~[f,w]
e.(e) As
= g
eQ: Z(Q)"
~
characterizes
ill
II.2.13,
=
~Q (b)
Z*(Q)
representation
(c)
than
Recall
and
that
and then ~
I.(4.3),
is the w i t h the
Q , we obtain
= ~ ¢(q,z)
(3.3).
subgroup
of
representation
the p r e c e d i n g
theorem
I V q ~ Qab
Characterizations = Ker(~Q:
Z(Q)
the adJolnt is the image
group,
z*(Q)
G
,(v) = z
Applying
= [g,v]
we conclude
{ x e Z(Q)
denotes Z*(Q)
.
is the natural
M(Q)
and
groups draws
of
Q
U(e) = 0 , see
our a t t e n t i o n
to
subgroup
3.7 COROLLARY:
where
rather
and
= v .
r Q---~Q/Z(Q)
the g e n e r a l i z e d
Z*(Q)
(a)
~(w)
is the smallest
the c h a r a c t e r i s t i c (3.6)
and
9: F - F
~(g) = q
xQ(~ ~ z) = sv[f,w]
0 =
Proposition
with
and
and thus
is monomorphic,
Since
~Q
R = R/JR,F]
g,v e G
~(f)
map of
groups
and
~
~G
e.g.
= n ~Z(G)
of
: ~Q(q ® x) = 1 } .
Z*(Q)
- Hom(Qab,M(Q))) homomorphlsm of the center
of a c e n t r a l i z e d
of
,
~Q .
of any g e n e r a l i z e d free p r e s e n t a t i o n
of
,
e
where
e
ranges
In particular,
over all central Z*(Q)
contains
extensions W(Q)
.
by
Q
as in (3.1).
Q.
209
PROOF.
Statement (a) reformulates the definition of
Z*(Q)
Now combine Theorem 3.6 with Proposition II.2.13 to obtain the assertions
(b) and (c).
The final remark follows from 3.3.
In the case of finite groups Z*(Q)
Q , READ [1] already arrived at
via representation groups.
3.8 COROLLARY.
For a group
(i)
O
(il)
z*(o)
(ill)
~e = o :
Q , the following are equivalent:
is unicentral, = z(o)
,
Qab ® z ( Q )
-X(Q)
[]
Consequently perfect groups and groups with trivial multiplicator are unlcentral.
Combining Corollary 3.8 with the Ganea sequence of
Theorem 1.4.4, we find the following formula for the Schur multlplicator of a unicentral group
N(Q) ~ K e r ( e . ( e Q ) :
Q :
X(Q/Z(Q)) - z ( Q ) )
(3.7) = ~er(X(QIZ(Q))--.,Z(Q)
3.9 PROPOSITION. Z*(Q) = O .
(a)
A group
Q
n [Q,Q])
is capable precisely when
A capable group is (isomorphic to) the center factor
group of any generalized representation group of (b) Q/N
Z*(Q)
is the least normal subgroup
is capable, cf. Proposition 3.5.
N
Q . of
Q
In particular,
such that Q/Z*(Q)
is
capable. (c)Let with
M
be a normal subgroup of
M n Z*(Q) = 0 .
PROOF.
If
Q/M
Q
outside
Z*(Q)
is capable, then so is
Q .
Part (a) is immediate from Corollary 3.?(b),(c);
second half was already proved as Proposition III.2.9.
, i.e.
the
Concerning
210 (b), we note that
Q/Z*(Q) ~ G/Z(G)
tion group
G , by Corollary 3.7(b); thus
G
of
ble.
Now let
say.
Then there is a commutative diagram ey :
e :
N ~ Q
for any generalized representa-
be such that
Z(E):
. E~
Z(E) ~
; E
with exact rows and ~$Z(E ~) = 0
t
~
hypotheses of (c).
is capable,
Q/N ~ E/Z(E)
)~Q/N Since
~"
Z*(Q) S SZ(E ?) E N .
Then
is capa-
~Q
Ker ~ = N .
and thus
Q/N
Q/Z*(Q)
is surJectlve, we find Finally, assume the
Z*(Q) = M 0 Z*(Q) = 0
by (b), thus
Q
is
capable by (a).
3.10 PROPOSITION. ~: Q
DQ/N
then
N ~ Z*(Q)
and
res=M(1): M(N) - M(Q)
a)
extension
M(9)
tion of
pre-
denotes the inclusion map into
Q ,
vanishes.
~;Q/N , vlz.
M(~)
, M(QIN)
in Theorem 1.4.4(ii)
~Q(x @ n) = I
to
N S Z*(Q) Since
N ~ Z*(Q)
is monomorphlc.
is inJectlve precisely when
~(e)
vanishes.
~ Q
x(e) ~ M(Q)
to
b)
i
Then
Q , let
We invoke the exact sequence I.(4.2) for the central
e=(i,~): N c
Qab ® N Thus
be a central subgroup of
M(~): M(Q) - M(Q/N)
If
PROOF.
N
denote the natural projection.
cisely when b)
a) Let
for all
x ~ Qab
x(e) = 0 .
exhibits and all
Now the defini-
x(e) = 0
as equivalent
n ~ N , in other words,
. M(0) = 0 , the composition
M(N) - M(Q) " M(Q/N)
Thus b) follows from a).
The preceding results yield many easy-to-check conditions which imply that a group is not capable.
W e can only list a few of these.
211
3.11 PROPOSITION.
z(o)
a)
If
b)
If one of
Assume
Hom(Gab,M(G)) Gab
G
is a capable group and
= 0 , then
or
M(G)
If
Gab
has finite exponent
is torsion, then
a prime.
Z(G) = 0 .
is bounded and its exponent divides c)
p
n , then
n .
TorpZ(G)
~ 0
implies
TorpGab ~ 0 .
d)
If
"M(a)
is torsion, then
TorpZ(G) ~ 0
implies
TorpM(C) ~ 0 . e)
If
pk , for
Gab
is torsion and there exists
k > I , then the Sylow p-subgroup
does not have exponent
Here
TorpA
, for
p
A
x ~ Z(G) S
of
of order
Gab/(X[G,G]>
k-1
abelian,
denotes the subgroup of the ele-
ments of p-power order, i.e. the (unique) Sylow p-subgroup of
PROOF.
Both (a) and (b) follow from the observation that
is isomorphic to a subgroup of
Hom(Gab,M(G))
together with Corollary 3.7(a).
If
T
A .
Z(G)
, by Proposition 3.9(a)
ks any torsion abelian group,
then TorpT = 0 where
iff
p*(x) = x p
TorpF(T) = 0
p*
is isomorphic,
as in Remark 2.4.
Therefore
for any additive functor
F
TorpT = 0
from the category of
abelian groups to itself (terminology as in Remark 2.4). (c), if
Gab
is torsion with
= 0
The proof of (d) ks analogous.
let us assume the hypotheses of (e) together with q = pk-1
.
Z*(G) = 0 .
We are going to derive Now
Gab = P × 0
Concerning
TorpGab = 0 , then
TorpZ(G) ~ TorpHom(Gab,M(G)) by the preceding arguments.
implies
where
x q ~ Z*(G) P
Finally,
q*(S) = 0 , where
, a contradiction to
is the Sylow p-subgroup and
212 Q the Hall p'-subgroup of y e Q
Gab .
~G(y ® x) = 0
~G(X m ® x) = 0
by order reasons and
I.(A.3).
Clearly
for all
by the formula
Therefore
= lY~ factors over
>XG.(y ® x ) } Gab
~G(Gab ® x q) = 0
3.12 REMARKS.
~P
:
~S
~ab " M(a) This implies
.
q*.~ = 0
and thus
by the bilinearity of
~G •
a)
that possesses a faithful
A finite group
Q
irreducible complex projective representation
(FICPR), must be
capable: llft the FICPR to a linear representation of a representation group of
Q
and apply Schur's Lemma.
The converse rarely
holds, metacyclic groups constitute one of the exceptions - see Corollary 4.20. b)
The result of PAHLINGS [1] quoted in Remark II.3.13 implies
the following:
A finite group
there is a group
G
such that
Q
admits a FICPR precisely when
G/2(G) ~ Q
join of the minimal normal subgroups) single conJugacy class of
3.13 PROPOSITION. exp(Q).exp(Z(Q))
of
and the socle (i.e. the G
is generated by a
G .
If
Q
is a finite capable group, then
divides the order of
Q .
This result is based on PAHLINC,$ [1; Satz &.9].
PROOF.
of
Let
O , then
Since divides
e=(a,~):
a(A) = Z(G) 0 [G,G] IQI
~]
~ G
Z(G) = aA ~ M(Q)
~ Q
be a representation
, we conclude that
Z(Q)
group
by Propositions 3.9(a) and II.2.14.
by a result of ALPERIN/KUO
Finally the exponent of
3.11(b).
A;
exp(~A)-exp(Q)
[1], cf. also BRANDIS [1].
divides that of
M(Q)
by Proposition
213
4. Examples of the Computation of
Z*(G)
Recall from Proposition 3.9 and Corollary 3.7(c) that the abelian groups
Z*(G)
and
Z(G)/Z*(G)
being capable resp. unicentral.
measure how much
G
deviates from
One may ask whether a particular
group construction preserves capability or unicentrality.
In gen-
eral, a positive answer will require certain additional hypotheses, a typical verification being based on the computation of
Z*(G)
In the course we obtain
G
special p-groups, groups).
(extra-
Most of the results are taken from BEYL/FELGNER/SCHMID
~Gi]i~ I .
If all
N Gi imI Pi "
PROOF.
Let
Let Gi
By definition,
tlons
for many specific groups
finite metacycllc groups, various types of abelian
4.1 PROPOSITION.
product
Z*(G)
.
G
G
Ei
be a subdirect product of the groups
are capable, then
G
is capable.
is a subgroup of the (unrestricted)
such that
[I].
Pi(G) = G i
be groups with
direct
for all the natural proJec-
Ei/Z(Ei) m G i .
Then we obtain
a morphism of extensions A
g
c
-~)G
~E
Z(Ei ) e
~ g
El
~:. g
where
A
is the center of
Since
G
is a subdlrect product and
a subdlrect product of [ti,E i] = 0
or
DE I
Gi
and
~Eille I .
t i e Z(EI)
E
~-I(G)
E ~ Ker ~ , the group
Hence
for all
is defined as
~tll e Z(E)
i E I .
E
. is
implies
This means
Z(E) = A .
D
214
The preceding proposition
(4.1)
z*(
Gi) ~
x
z*(oi)
x
iEI
implies
l~I
for the restricted direct product
× G i , while equality does not i~I The minimal counterexample is given by I = {1,2}
hold in general. and
G I = G 2 = Z/2 .
is capable.)
Z/2
is unlcentral while
Assume
Z*(Gi) ¢
G =
x Gi i~l
~ Gi
ab
and let
D(Gi)a b
be the obvious homomorphisms.
Then
those
that satisfy
z=~zilie I E
(4.2)
Z/2 x Z/2
The following result gives the precise conditions.
4.2 THEOREM. ~i :
(Note that
× Z*(Gi) i~I
Z*(G)
consists precisely of
gi @ ~j(zj) = I ~ G i ® Gj = (Gi)ab ® (Gj)ab
for all
i $ ~
and all
cides with the subgroup
gi ~ (Gi)ab × Z*(GI) iEI
"
Consequently
of
Z*(G)
coin-
G , if and only if the homo-
morphlsms
(4.5)
I ® ~j:
s i ® z*(oj) - G i ®
are trivial for all
PROOF. e
of
G
:
a)
i + j ~ I .
Choose a representation
A :
~
~ H
note that
We invoke
,-1(z) e Z(H)
gi E G i , i ~ I . central in defined maps
G .
group
~ ~ G
with commutator from
H i = ~-I(G i) .
(Cj)ab
Since
¢
as in 1.4.9 and 3.6, put
Z*(G) = ~Z(H) precisely when [Gi,Gj] = 0
for
Hence the commutator
form
@iJ: Gi × Gj " A
phic, thus yield homomorphlsms
form Corollary 3.7(b) and ¢(gi,z) = I
for all
i ~ J , we have ¢
[HI,Hj]
restricts to well-
which by Lemma 1.4.1 are blhomomor@lj: Gi @ Gj " A
with
215
~ij(~(hi)[Gi,Gi] for
hi E H i
and
® ~(hj)[Gj,Gj])
=
hj ~ Hj , whenever
~-l([hi,hj]) i $ J .
Write
~j = gj[Oj,Gj] . b)
Assume that
z e Z*(G) hi e Hi
.
Fix
and
z
i
for the moment and let
tj ~ Hj
for almost all
satisfies the conditions
with
J e I .
As
z i ~ Z*(G i) , it follows Lemma 1.4.1(b)
exhibits
for finitely many
¢(gi,zj) = for
i ~ J c)
d)
~(gi,z)
®
gi ~ Gi "
and
Choose
w(tj) = zj , tj = 1
is a central extension and
[hi,tl] = 1
Again by
[Hi,H j] E Z(H)
as a product of elements
,
~(gl,zj)
Since
zj) = ~lj(~i ® ~jzj) =
by (4.2), we conclude I i ® ~j = 0
I
~(gi,z) = 1 , thus
for all
i ~ j .
z ~ Z*(G)
Then all of
.
X Z*(Gj)
(&.2), thus equality holds in (4.1). Given
each pair and
elG i
J $ i .
~lj(~i
Assume
satisfies
n(hi) = gi
(4.2), to be shown
z=~zi~ ~ Z*(G)
i,J ~ I
p: G - Q
with
, we are going to prove (4.2) for
i ~ J .
To this end, let
be the natural projection.
Q = G i x Gj
Formula (4.1) for the
decomposition o
yields
o ×
=
Gk
pZ*(G) ~ Z*(O)
representation
groups
, thus ei
invoke Theorem II.4.11(c) Let
@
implies
~(gl,zj) = 1 :
with
ej
of
Gi
and
@(gl,zi)
~ .
Again, as
= 1 , thus
Now choose
Gj , respectively;
to obtain a representation
be the commutator form of
z i e Z*(Gi) to
and
p(z) = (zl,z j) ~ Z*(O)
group
~IG i
@(gi,p(z))
~
of Q.
is central,
= 1
reduces
As in step (a), ~ yields a homomorphism
G i ® Gj " Kernel(~)
~(gi @ gJ) = ~(gi'gJ ) •
~ M(Q) This description of
agree with the definition of the monomorphlsm
~
is seen to
~''(~1 @ ~2 )-I
of
216 Theorem II.4.11(d).
Consequently
(1 ® ~J)(gi ® zj) = 1 ~.(1 e)
® ~)(gi
is monomorphlc,
® Zj) = ~(gl,z~)
for all
i ~ J , by the previous step.
This means the vanishing of all maps
groups G i
are also fulfilled if , and
G Gp
i.e. satisfy
=
I g ~ G
I g
Hirsch.)
i ~ J .
Z(Gi) ~ [Gi,Gi]
is the Sylow p-subgroup of G
has finite order.
has p-power order
is taken as a definition, G = XGp .
for
r7
.
They
is a nilpotent torsion group,
assume that every element of Gp
I i ® vj
Conditions (4.3) are trivially fulfilled if all
are stem groups,
I = ~primes Pl
= 1 .
if equality holds in (4.1), then we must have
gi ® vjzj = I E G i ® Gj
4.3 EXAMPLES.
and
follows from
® Z~) = ~ ( g l
Finally,
~
then
Gp
G .
(Thus we
If
I
is a normal subgroup and
See e.g. WARFIELD [I; p.19] for these results of Baer and The point is that
infinite) p-group and
B
A ® B = 0
whenever
a q-group for primes
A
is a (possibly
p ~ q .
In par-
ticular, all finite nilpotent and all torsion abellan groups included.
Consequently,
a nilpotent torsion group
resp. unlcentral precisely when so is each
4.4 PROPOSITION. in
G
PROOF. U
U
Then
Let
~x1,...,xm} -I gk--Xkg.xk~g E U
e ~ Cext(U,A)
Gp .
m: Gab - Uab
is sur-
be a generalized representation group
f , define
e'--Cor2(e) ~ Cext(G,A)
be a right transversal of
U
in
G
in the notation of Corollary 1.6.12. m
~(y[G,G]) =
is capable
Z*(G) D U = 0 .
with factor system
Let
are
be a capable subgroup of finite index
such that the transfer (Verlagerung)
Jectlve.
of
Let
G
G
l'I Yk[U,U] k=l
and Then
217
for R~
y ~ G . = xk .
For
g ~ U n Z(G) S Z(U)
we have
Choosing the factor system
f'
of
y-~ = y e'
and
as suggested by
Corollary 1.6.12(b), we obtain m
f'(y,g) =
m
n f(yk,g) k=1
and
f'(g,y) =
n f(g,yk ) . k=1
We next recall from Remark 1.5.9(b) the formula -I e.(e') ~G(Y[G,G ] ® g) = f'(y,g).f'(g,y) and a similar formula for
e .
Since
XU
is blhomomorphic,
m
e,(e') XG(Y[G,G ] ® g) =
n f(yk,g).f(g,yk )-1 k=l
= e.(e) XU(k~ 1= Yk[U,U] ® g) = e.(e) XU(~(y[G,G]) Now let even
g ~ U n Z*(G)
y ~ G , while
e.(e)
.
Then
® g)
.
WG(Y[G,G ] ® g) = 1
for all
is monomorphic by Proposition II.2.13.
Consequently ~U(~(y[G,G]) Since
~
® g) = 1
for all
is epimorphlc by assumption,
4.5 COROLLARY.
Let
N
G/N
are capable, then
PROOF. 3.9(c).
g ~ Z*(U) = 0 .
be a normal subgroup of
index such that the transfer and
y ~ G .
Gab - Nab G
G
of finite
is surjectlve.
If both
N
is capable.
Proposition 4.4 gives
Z*(G) n N = 0 .
Apply Proposition
[3
4.6 EXAMPLE. is capable and
Let N
N ~ G
~Q
be a group extension such that
is centerless perfect.
Then
capable and the transfer is trivially surJective. involving symmetric and alternating groups, let N = An x 0
for
n ~ 5 , thus
S n x Z/2
N
is clearly For an example
G = S n x Z/2
is capable.
Actually
and
Q
218
Z*(S n x Z/2) = 0
holds already
The Ganea map of an a b e l i a n induces a natural The c o n n e c t i o n
for
n ~ 2
group
isomorphism
S
Combining
this w i t h
If
A
~ x ~ A
4.8 EXAMPLE.
I V a ~ A
then c l e a r l y
comparing
that an a b e l i a n
:
group
M(A)
A
A
m A ^ A ~ 0
is r e s u m e d
4.9 P R O P O S I T I O N . ted direct product) d e n o t e the t o r s i o n of
If
G
.
Let
I .
torsion
and thus
precisely
or
Z*(A)
= A.
we see
w h e n it is abso-
3.8 t o g e t h e r w i t h 1.4.5 gives a dif-
in E x a m p l e
is e q u i v a l e n t
to
M(A) = O.
6.17(b).
G =
x (Z/n i) be a d i r e c t sum (restrlciEI of cyclic g r o u p s (n i = O allowed). Let T
subgroup
= 0
Z*(G)
b)
Suppose
for
and
r = dim~(G
@ Q)
the r a t i o n a l
rank
r ~ 2 .
r = 1
is of finite
is cyclic; of
.
Then
a)
T
1.4.7.
then
in 3.1 and 1.3.10(c),
is u n i c e n t r a l
Now Corollary
group,
is d i v i s i b l e
ferent p r o o f of the fact that the l a t t e r This topic
a,b ~ A
a ^ x = 0 e A ^ A
the d e f i n i t i o n s
group
l u t e l y abellan.
for all
is an a b e l i a n
If the a b e l i a n
cyclic,
Actually,
by T h e o r e m
and
(3.6), we obtain
4.7 P R O P O S I T I O N .
locally
M(A): A ® A " M(A)
is
MA(a ® b) = Mo(a ^ b) ~ M(A)
=
is
Mo: A ^ A - M(A)
(4.4)
Z*(A)
by T h e o r e m 4.2.
If
exponent
a generator
is
xm
T
is u n b o u n d e d ,
m , then
then
Z*(G) = G m =
for any i n f i n i t e
Z*(G) = O { gm
direct
.
~ g ~ G
summand
}
{xl
G . c)
of p r i m e s
Suppose
r = 0 , i.e.
G = T
torsion.
for w h i c h the S y l o w p - s u b g r o u p
Gp
Let
P
is bounded.
be the set F o r each
219 p ~ P
let
cyclic.
n(p)
the least power of
(In other words,
from the top of
(4.5)
Z*(G) =
PROOF. each
Gp .)
n(p)
is
0
convenience.
if necessary, we may assume that
or a prlme-power. xi
for all
Z/(nl,n j)
(ni,n j) I Xi
•
is
Then
We use additive notation for
be a generator of
~.7, the typical element
ator of
(Op) n(p)
x (Op) n ( p ) I:~P
Let
~ixl ^ xj = 0
such that
is the second "torsion coefficient"
By further decomposing,
ni
p
z = ~ J ~ I .
, thus
A/n i .
By Proposition
Xix i
lles in
Z*(G)
For
i $ j ,
x i A Xj
Xix i A Xj = 0
precisely when iS a gener-
iS equivalent to
The discussion of the various cases is now
straightforward.
In particular, whenever
implies
for all
n i I ki
4.10 COROLLARY groups as above.
i ~ l\lJJ
(BAER [3]). Then
G
Let
nj = 0 , then
, i.e.
G
z ~
z A xj = 0
~xJl
be a direct sum of cyclic
is capable precisely when one of the
following holds: (i)
the rational rank
(il)
r = I
(iii)
G
p-subgroup of
Gp
Tor(G)
of
G
exceeds
is unbounded;
is torsion and for each prime
Gp
I ; or
or p , either the Sylow
is unbounded or the two highest torsion coefflclents
agree.
4.11 EXAMPLE.
Consider the special case that
generated abelian. cyclic,
and
r
indeed
G
is finitely
Then Proposition 4.9(c) implies that
Z*(G) = W(G)
as in 3.4(a).
obtain the well-known result that
A
Z*(G)
is
In particular, we
is capable precisely when the
two highest torsion coefficients agree.
220
4.12 REMARK. groups
An explicit
description
seems to be difficult,
(direct limit) example
of all capable abellan
mainly because a directed union
of capable groups need not be capable.
is the unicentral
group
directed union of the capable
G = Z(p ®) x Z(p ®)
subgroups
which is the
Z/p k x Z/p k , k e N .
A result of MOSKALENKO
[1;
p-group
if, and only if, no non-zero
G
is capable
§4]
A counter-
implies
that an unbounded abellan element of
has infinite height.
A bounded abelian p-group
finite cyclic groups,
this case is covered by Corollary
4.15 PROPOSITION. r = 1
Let
for the rational
not capable.
PROOF.
If
rank
A
generated
free subgroups.
and thus
M(A) = 0 .
abellan
inclusion Indeed,
bases
J: B 1 =
by
of
present
isomorphism Let
r = 0
J
between with
B1
and
[1;
of rank two with
2 ~ m ~ n
and
is locally cyclic
means the trivial group. B a runs through all freeWe claim that any M(J): M(BI) - M(B2).
free-abellan
ni ~ Z
groups can be
with respect to suitable
{yl,...,yn ~
of
B2 .
to the companion
(Apply el-
matrix of
and
ni ~ 0 .
a E B
In the
, the claim is immediate.
Since
1.5.10,
M(B) - M(A)
j ,
In view of the natural
, then there exists a free-abellan
direct limit argument a monomorphlsm
A
§76] for an explicit procedure.)
M(B i) ~ B i A B i
a ~ Z*(A)
is unicentral
induces a monomorphlsm
row and column operations
situation,
A
If
is capable.
A = UB awhere
J(x i) = nlY i
see SEIFERT/THRELFALL
, then
abellan group.
of rank at least two.
~B 2
of
A
r = 1 , then
The case
A
{Xl,...,Xml
ementary
of A
If
the homomorphlsm
described
r
4.10(ill).
is the directed union of its finitely
r ~ 2 , write
subgroups
is a direct sum of
be a torslon-free
r $ 1 , then
We use that
Now assume
A
G
subgroup
M(A) ~ dlr.llm.M(B
we conclude
that
J: B c
Now the commutative
B
) by the >A
diagram
induces
221
B ®B 0®0
~(B)
IM(J ) x(A) ..- M(A)
t
A ®A yields
M(J)
b e Z*(B)
,M(B)
XB(a @ B) = XA(a ® JB) = 0 , thus
= 0 .
XB(a ® B) = 0
In the last step, we also u s e d P r o p o s i t i o n
and
4.9(a).
[] 4.14 P R O P O S I T I O N . subgroup
of
A
t h e n so is
A
A
be an a b e l i a n g r o u p and
, e.g. Tor(A).
If
U
and
A/U
U
a pure
are b o t h capable,
.
By d e f i n i t i o n n~
Let
U
is p u r e in
A
, if
U n An = Un
holds
for all
~.' .
PROOF. Z*(A)
Let
c U .
M(~)
~: U - A
Let
~u(a
be the embedding.
a E Z*(A)
® U) = ~A(a
u ~u,
x(u)
.
Corollary
A s in the p r e v i o u s
3.9(b)
implies
proof,
we have
embeds
M(U)
® ~U) = 0 .
,M(u)
~®~I I M(~) A ® A - y(A) )M(A) We now invoke M(A)
from BEYL [3; Thm.
as a p u r e subgroup.
1.6] that
(The p o i n t
d i r e c t limit of split e x t e n s i o n s exhibits
M(U)
the d i r e c t and
as a d i r e c t
i e I .
~A
M(Ai)
.
Consequently
~A/U
into is a
The s p l i t t i n g
Finally
M(~)
is
xu(a ® U) = 0
= 0 .
4.15 THEOREM. groups
U ~
;AI----~B i .
of
limit of m o n o m o r p h i s m s . )
a E Z*(U)
is that
U ~
summand
M(~)
IGiliE I .
G i v e n an a b e l l a n g r o u p Assume
IIl ~ 2
T h e n the c e n t r a l p r o d u c t
and G
A
and a family of stem
~i: A ~ Z(G i)
of this family
for all
is u n i c e n t r a l .
222 The central product precisely
the centers
definition N
G
is
=
PROOF.
Z(Gi)
G = X/N
I Izll ~
Let
is that quotient
are identified.
x
Z(GI) S X
~: X " X/N = G
Z(Gj) ~
has image
Z(G)
e = (~,w)
I
of
G ; let
form" of
e
x Gi i~I
in which
More formally,
~ Gj ~
-1 ~i (zl) = I ~ A
n
the
M>
I
denote the natural p r o j e c t i o n .
~X
We
map
°~G
Choose any (generalized) :
X =
with
first observe that each composite 9j :
of
>E
representation
group
~G
E i = ~-l(Gi)
for
i ~ I , let
as in 1.4.9.
Given any
#
be the "commutator
gl e Gi
' zj m Z(Gj)
and use
~jzj = ~kZk
for
i,J ~ I , we claim (4.6)
#(ogl,gjzj)
If
i = j , pick -1 z k = ~k~i (zi) Since
[Gi,Gj]
[EI,[Ej,Ej]]
e
with
k ~ i
Thus we also assume
by the T h r e e - S u b g r o u p s on
~ [Ei,[Ej,Ej]]
Gj
~-l(gjzj)
E Z(E)
Let
p
by C o r o l l a r y
3.7(b).
with
Z(G) = [G,G] ~ ~/p
G
one has
IGI = p2k+1
with
central in
E
and
But
, hence
~gi
generate the group zj ~ Z(Gj)
,Z(E) = Z(G)
.
Thus
G ,
From this G
is
[]
be any prime number.
G
(4.6) is proved.
Lemma 1.4.3.
, for each
and the initial remark, we conclude unicentral
until
with
•
is central and the elements
(4.6) implies
i ~ J
[EI,E j] S ~M
by the a s s u m p t i o n
~(,gl,gjzj) Since
k ~ I
= 0 , we have
= 0
zj ~ [Gj,Gj]
= I ~ M .
A n extra-speclal
and
Gab
k ~ I
of exponent
p-group is a group p .
For finite
223
4.16 COROLLARY: a) p2k+l
For
Multlpllcators of extra-special p-groups.
2 ~ k ~ - , every extra-speclal p-group
is unlcentral and not capable.
Is elementary abellan of p-rank b)
The quaternlon group
is unicentral.
of order
p3
A/p x ~/p
Z/2 .
Q8
For
and exponent
The Schur multipllcator of
has trivial multlpllcator,
p
p
of order
D8
p2
thus
of order 8 is capable and
odd, the extra-special p-group
H
is capable and has multlpllcator
, whereas the extra-speclal p-group of order
exponent
G
2k 2 - k - 1
The dihedral group
has multlpllcator
G
has trivial multlplicator,
p3
and
thus is unlcentral.
Part (b) corrects errors in OPOLKA [I; Lemma (1.4)] and In BEYL/ FELGNER/SCHMID [1; Proof 8.2 p.174], in the latter case read "stem extension"
instead of "stem cover".
PROOF.
a)
xy ~ yx .
Since
Let
We claim that and that
G
G1 = G1
Ix,Yl
and
and
G2
G2
M(Gab) ~ Gab A Gab Theorem 1.4.7.
G
are stem groups wlth
is elementary abellan of rank
Finally (3.7) exhibits
M(G)
G1
with in
G .
Z(G1) = Z(G) = Z(G 2)
is unicentral by Theorem 4.15.
GI
and
G2 .
Now
k(2k-1)
by
as a subgroup of
wlth rank defect one.
We are left to prove the claim. p-group of order at most In particular,
[x,yJ
G1
Z(G 1) = [G1,G1] = [G,G] = Z(G)
generates
g = x~.yk.z [G,G]
(The flrs~ step is to select We conclude
Since
G = G1.G 2 .
is a non-abellan
p3 , it must be extra-special of order
may be written In the form because
x,y e G
the centralizer of
is (isomorphic to) the central product of
This being granted,
M(Gab)
[G,G] ~ 0 , there are elements
~
wlth
, has order such that
.
A n arbitrary
p3 . g e G
z E G 2 , basically p , and is central.
x-~g
commutes wlth
This and the definition of
G2
imply
y .)
224
G I n G 2 = Z(G) IG21 > p .
.
By order reasons
Now
IZ(G) I = p
proper inclusion.
Thus
G2
implies
G2
is not contained in Z(G 2) = Z(G)
is not abelian and
By the previous steps the multiplication map Jective with kernel isomorphic to central product of b)
GI
and
[G2,G2] = [G,G]
G 1 x G2 - G
, exhibiting
G
.
is suras the
D8
have been obtained in
The reader checks that
D8
is the central factor
proved in Example 1.12. G(p2,p,1+p,O)
or one can prove
rather than
and
group of the dihedral group
group
and
G2 .
The multiplicators of
Example II.3.8.
Z(G)
G1
Q8
D16 .
The assertions on
The remaining group
G
H
have been
is the metacyclic
and has trivial multiplicator by Theorem 2.10;
W(G) = Z(G)
directly.
We mention that LEWIS [1]
actually determined the integral cohomology of the non-abelian groups of order
p3 ; the multipllcators can also be computed from his
results.
[]
4.17 REMARK.
By the preceding corollary there is precisely one
capable extra-special p-group for each prime
p .
These groups do
possess a faithful Irreduclble complex projective representation (FICPR)
, as was noted by NG [2; Prop.
(1.2)].
Alternatively,
the criterion of Pahllngs (as stated in
3.12) is satisfied in each case. Socle(D16) ffi Z(D16) ~ order
p3
Z/2
and exponent
No. 12, call it = Z(H) ~ Z/p .
(7.6)] and OPOLKA [1; L e n a
First,
cycllc. p
As t h e
For central
D 8 ~ DI6/Z(D16) p
wlth
odd, the group factor
of
group of group
H , in HUPPERT [I; Satz 111.12.6] with (Our assertion is also valid for
H
Socle(H) =
p = 3 , but then
has exponent 9 rather than 3.)
4.18 defined is
the
PROPOSITION. in 2.3 with cyclic
Let
G = G(m,n,r,k)
rn = 1
group of order
mod m
and
~ . ( m , rk- 1 )
be the
metacyclic
XIh(m,n,r) generated
.
Then by
group Z*(G)
bk , where
225 k
divides
n
mk/(m,r-1)
a n d is the s m a l l e s t p o s i t i v e
divides
PROOF.
Let
constructed
1 + r +...+ r
V: G = G ( m . ~ , n , s , 1 )
- G
t i o n s of w h i c h are p r e s e r v e d .
Then
The c e n t e r of
by
is g e n e r a t e d
is the l e a s t p o s i t i v e Now
lln
~i
We recall
follows
.
2.10,
the n o t a t i o n
Z*(G)
= pZ(G)
~a m ~ / ( m ' s - 1 )
integer with
Hence
that
I S~-1
sI = I
Z*(G)
s-1
~
group
and a s s u m p -
by C o r o l l a r y
and
~I
mod m.~
where
3.7. 1
, see Lemma
2.13.
is the cyclic g r o u p g e n e r a t e d
= t
iff
( m , r -m1 ) / ~
Iff
( m m. , r - 1~)
4.19 COROLLARY. parameters
is p r i m e to
such that
m
divides
m .
Then
by
k = 1
The c o n d i t i o n
= (m,r-1) = (m,r-1) conditions
.
of
is
G
For the converse, rn E 1
mod m
[]
G = G(m,n,r,~)
, w i t h the
A I h(m,n,r) , is c a p a b l e
l+r+
... +r n-1
is the s m a l l e s t p o s i t i v e .
implies that m =
e(m,n,r,~)
as in 2.3
IM(G)I. I[G,G]I
and
.
Z*(G) m
+r~-I
n
We see that then
T h i s implies
imply
and
+s ~-I)
"'"
and
condition
are i n v a r l a n t s
group
mod m
~ = (m,r-1)
is the split extension.
PROOF.
I 1+r+ . . .
r n -- I
The first n u m e r i c a l
IGl/m
I t-(l+s+
The m e t a c y c l l c
satisfying
if, and only if,
n =
be the r e p r e s e n t a t i o n
from
m'k
integer
such that
by (2.6) a n d thus
iS a p o w e r of bI .
~
~-I
in the p r o o f of T h e o r e m
G
integer
= 0 , which becomes
I 1+r+
... +r n-1
and
n = k h(m,n,r)
and =
first note that the l i s t e d n u m e r i c a l and
k = h(m,n,r)
.
[~
226
4.20 COROLLARY.
A metacycllc group has a faithful irreducible
complex projective representation precisely when it is capable.
Actually NG [2; Theorem (4.4)] obtained the metacycllc groups with an FICPR without noting the connection with capability, his method was to construct an FICPR explicitly.
PROOF. 0: G " G
Assume that
G = G(m,n,r,X)
is capable and again let
be the representation group (2.7).
by Proposition III.2.9 or Corollary 3.9. subgroup of
G
lles in
First
2(G) = Ker
We claim that every minimal
{a} ; this means that the socle of
G
is
cyclic and the criterion of Pahllngs applies (see 3.12).
On the
contrary, assume that there is a minimal normal subgroup
N
taining
z = ~i~J
with
0 < J < n .
~.z.~-1.z-q = ~(1-sJ)
~
we conclude
G
inside
N , hence
sj ~ I
mod m.X
By Proposition 4.18 (cf. the last step of its proof)
b j e Z*(G)
4.21 PROPOSITION. when
Then
I~I n N
generates a normal subgroup of by mlnlmallty.
con-
, contradiction.
[]
A metacycllc group
G
is unlcentral precisely
M(G) = 0 .
PROOF.
Since
x(G): Gab @ Z(G) - M(G)
tion 2.14 and Theorem 1.4.4(ii), M(G) = 0 .
Invoke Corollary 3.8.
x(G) []
is eplmorphic by Proposi-
vanishes precisely when
227
5. Preliminaries
on Group Varieties
While the basic reference H. NEUMANN
for group varieties
[1], our view is close to STAMMBACH's
order to keep these notes self-contalned,
F®
be the free group of countable
Xl,X2,X3, . . . . any group,
group generated
The group variety
G
belonging •
to
fully invariant),
then
V
If V
W £ F®
determines
V = WF.
; then
W G = VG
~[x1,x2]l
,
F®
ties such that
5.1 REMARK. Whenever V(G/VG) variety
V2 =
the variety of
~ .
to us.
F®
If
G
is
is defined as the sub-
and
~
V E W
Let
V = VF®
I
VG = 0
The
(in other words,
G .
~[x2,x32],[x5,x7]l
if
V
is
, and
V 3=
it, viz.
[F=,F=]
with group varie-
~ ~ ~ , cf. H. NEUMANN
be the variety belonging
each of
The fully invarlant
correspondence
is the largest
Every
For example,
of "all" abellan groups.
one has
of
of which are the
a least closed set containing
is eplmorphic, G/VG
if
the objects
for each group
iff
~
V
v e V
is defined as the full subcategory
are in blJectlve
h: G - H = 0
some no-
is called a closed set of laws.
subset
subgroups
G
In
rank on the generators
f: F® - G , all
of "all" groups, V .
defines
of
I].
by
satisfying
VI =
VG
is said to satisfy the laws
the category groups
subgroup
I all homomorplhisms
~
relevance
A set of laws is Just a subset of
the verbal
I f(v)
[3; chp.
we here establish
tation and present a few facts of particular
Let
is the book by
[I; Thm.
14.31].
to the laws
V .
VII = h(VG)
factor group of
. G
Thus in the
228
5.2 DEFINITION. If
e: R "
;F
Q e ~ , then (5.1)
Let ~Q
:
be the variety belonging
is a free presentation
VF ~ R
e/VF
U
F
c
•
~ V'-F
the groups
F/VF
free in the category set".
F .
as in 1.3 and
~
;~ Q
of
Q .
lle in
b y 5.1 and are U-free,
U
with respect to the functor
Every ~-free group is isomorphic
free group
Q
V .
and
is called a U-free presentation
Indeed,
of
to the laws
See STAMMBACH
to
F/VF
i.e.
"underlying
for a suitable
[3; I §3] for the details
of this
aspect.
5.3 BIRKHOFF's
THEOREM.
Every non-empty
that is closed under subgroups, morphic
to factor groups),
group variety,
epimorphlc
full subcategory images
and (unrestricted)
of
(i.e. groups iso-
direct products,
i.e. defined by some set of laws.
The converse
is a is
also true.
PROOF,
cf. H. NEUNANN
[1; Thm.
15.51].
tion is straightforward.
Now let
properties
For each
as specified.
f: F® - Q , let
Kf = Ker f ~ F®
(5.2)
O Kf . all f
V
= def
Since the totality
of normal
section is well-deflned, set.
If
G e •
Ker f ~ V ; thus m
If morphic
Q
and G
satisfies
Hence it suffices
F/VF
Q ~ U
and each homomorphism
and
of
F®
is a set, this inter-
even if "the totality
f: F= - G
of all f" is not a
is any homomorphlsm, the laws
the laws
F/VF e U
then
V .
for some free group
to prove
asse~
be a subcategory with closure
subgroups
is any group satisfying image of
~
Indeed the converse
V , then it is an epiF
for such
of infinite F .
rank.
Consider
Ker f
22g
for all
G e •
these normal select
s u b g r o u p s of
Gi ~ • W
As
a n d all h o m o m o r p h l s m s
Gi
=
and
F
g r o u p a n d p r o d u c t closed,
lle in
=
V , we h a v e all
W = VF
, thus F
~
i ~ VF
F/i ~ Im fl ~ Gi
.
Since
•
is sub-
and then
an embedding
k: F , - F
homomorphlsm
~: F
with
x ~ W
.
- G E S
To t h i s
x m V
of
G
I
and
and then
x E F®
k(x) = x .
can be e x t e n d e d to
On the b a s i s of the p r e c e d i n g , ~G
One t h u s f i n d s
as a free f a c t o r w i t h
We conclude
• -verbal subgroup
F/VF ~ S .
and o b s e r v e that o n l y f i n i t e l y m a n y b a s i s
B y the v e r y d e f i n i t i o n of
= f(x) - I .
~: F / V F - P
is m o n o m o r p h l c a n d
e l e m e n t s o c c u r in a n y single
Moreover,
i e I
K e r fl W A - = V-~ " leI VF
fix a b a s i s of
f.k = ~ .
for e a c h
K e r fl = i , p u t
T h e r e is an o b v i o u s h o m o m o r p h l s m
Ker ~ =
end,
set,
The t o t a l i t y I of
H F / K e r fl iEI
• .
We claim
such t h a t
.
"
s a t i s f i e s the l a w s
P
is a n o n - e m p t y
fi: F - G i
A K e r fl leI
f: F - G
W
Every
f: F - G
such that
, we h a v e
x = k(x) G V F
we feel free to c a l l
and
~(x) = .
VG
the
a n d to use s i m i l a r c o n f u s i o n s below.
B i r k h o f f ' s T h e o r e m e l u c i d a t e s h o w a n y c l a s s of g r o u p s
g e n e r a t e s a v a r i e t y w i t h o u t e x p l i c i t m e n t i o n of laws. n a m e a v a r i e t y a f t e r the g r o u p s in it, i.e.
We typically
"the v a r i e t y
~
of
a b e l l a n groups".
5.4 LEMMA. Then
Let
W G = [UG,VG]
PROOF.
Clearly
U
and
V
be c l o s e d sets of l a w s a n d
for all g r o u p s
G .
W G H [UG,VG]
Recall that
s u b g r o u p g e n e r a t e d by all
Ix,y]
.
with
x G UG
[UG,VG] and
W = [U,V].
is the
y ~ VG
.
There
230
suffice x
finitely many
ui
Xl,X2,...
involves
and
U
x = f(u)
ferent
ui
is closed,
- idea:
u
and
[x,y] = h([u,v]) f
resp.
v
set of laws
g
W
.
W G E 2(G)
groups
31 = S
and
hand,
the totality
equivalently, )G
A
is called the variety [W,W]
the variety and
~1+1
according
G
~ ~
[F ,W] .
or, equivaof
WG
G
For example,
groups of class at most c ~ 1
On the other
abellan
form a variety;
Q ~ ~
5.4.
.
groups;
Therefore
the variety
it is defined by the
For example,
let
1 .
~l Then
denote 61 =
1 ~ 1
For every group variety
S , the following are
equivalent: (1)
The variety contains
(il) S (ill) S
contains
c .
supported by an extension
of soluble groups of length at most
5.6 REMARK.
agreeing
[~
G/Z(G)
for
with
of abellan-by-~
for
h
from Lemma 5.4 that the variety of
abellan and
= abellan-bY-~l
Then
this is called the variety
of nilpotent
to Lemma
v ~ V
defined by the closed
with
~c+I = center-bY-~c
with
for dif-
and
in common.
generators.
G
we deal with groups
2Q
xi
is defined by the laws
of groups
f: F® - G
and a homomorphism
form a variety; It follows
and
g: F® - G
be any variety,
denote the variety
Then
laws
[u,v] ~ W
~
u ~ U
for some
Then the groups
center-by-m
A ~
to exhibit
sets of generators
have no generator
Let
groups.
Sc
y = g(v)
on the relevant
center-by-m
let
- G
there exists
choose disjoint
with
5.5 DEFINITION.
lently,
fi: F
only finitely many of the generators
Likewise,
such that
with
and
n +I n fi(ul)i=1
=
Since each
with
ui ~ U
the integers
Z ;
all abelian groups;
is defined by commutator
laws,
i.e.
V S [F ,F ] .
231
Such
variety
a
is said to have
exists an i n t e g e r
q > 0
lles in
S
S ; then
Actually, Z e •
and
to each
xi
must vanish, Z/n i
zero.
Finally,
is g e n e r a t e d
[I; Thm.
exponent
0
of
Z if
v ~ [F=,F®]
q .
.
are trivial.
If
finite of
has e x p o n e n t
q
V 0 [F=,F®]
If
sum w i t h r e s p e c t •
order,
(as a s u b g r o u p ~
and
Every variety
containing ~
by the laws
[F®,F=]
in the v a r i e t y e
contains then
•
also
P ), h e n c e has and
V = ~F®
, then
, cf. H . N E U M A N N
determines
This variety
a smallest
is g e n e r a t e d
Z , we call it varo(~)
.
variety
of
by the It is d e f i n e d
n ~F=
We
say that an e x t e n s i o n
S , if
lles in
~
S .
together with
5.8 D E F I N I T I O N .
tainly
but no l a r g e r cyclic g r o u p
exponent
large
there
12.12].
5.7 REMARK.
groups
x qI
by
Z/q
Otherwise,
~ , then the e x p o n e n t
hence
and
zero.
(iii) = (ii) ~ (i)
of a r b i t r a r y
P = n Z/n i
exponent V
is said to have
is any law of
cyclic g r o u p s contains
such that
the i m p l i c a t i o n s v
exponent
G ~ S .
If
S , we w r i t e
e ~ eI
e e ~ .
e: N ~ and
For
~ G ---~O
lles
e I m S , then cerO ~ S
and
of
by
A e ~ O • , let Cexts(O,A)
~ Cext(Q,A)
denote the set of c l a s s e s lie in
S .
formula
I.(2.2)
a functor
If
It f o l l o w s that
of c e n t r a l
extensions
A
from the direct c o n s t r u c t i o n s
Cexts(O,A)
is a s u b g r o u p
of
O
that
of 1.1 a n d the
Cext(O,A)
and
~op × (~ 0 S) - S .
v=v(xl,...,Xn)
ments
(variables)
then
v(gl,...,gn)
and
E F=
is a w o r d in the first
gl,...,g n
designates
are e l e m e n t s
the element
n
b a s i s ele-
of the g r o u p
f(v) e G
where
G ,
232
f: F= - G
ks the h o m o m o r p h i s m
f(xj)
for
= 1
results
(5.3)
J > n .
If
with
f(x i) = gi
a: G - H
i ~ n
is any homomorphlsm,
the formula
v ( a g l , . . . , a g n) = a v ( g l , . . ' , g n )
for
•
and there
233 6. Central Extensions and Varieties
The basic theory on the problem whether the middle group of a group extension lies in a given variety STAMMBACH [3].
S , can be found in
In this section we obtain rather explicit (comput-
able) criteria for this question, provided
S
is of exponent zero
and the extension is central.
The reader should consult Section 5 for our terminology on group varieties, a knowledge of STAMMBACH [3] is not required. cipal sources for this section are BEYL [4], [5]. cussed can be viewed as a group-theoretlc
The prin-
The concepts dis-
interpretation of varietal
(co)homology groups, this aspect is pursued by LEEDHAM-GREEN [1] and STAMMBACH [ 2 ] .
6.1 DEFINITION. ~
;~
T~ Q
Let
S
be a variety,
a S-free presentation of
(6.1)
KS(O) = ImiM(~):
(6.2)
l~(O)
= Coker M(T) =
e2
of
and
K~(Q)
el : RlC
i, R2
M~(Q)
~
F2 e S
and
Fq
> F1
~1 ~ Q
,,"
";2 ~ Q
F2
are
Given ~-free
Q , there exists a morphism
extensions because
e2 :
Q , cf. 5.2.
• , and
Define
M(Q)IK~(Q) .
of the choice of S-free presentation. and
a group in
M(~) - M(O)}
We convince ourselves that
eI
Q
is
II
~-free:
(a,B,1):
independent
presentations e I
"
e2
of
234
The naturality assertion of Proposition 1.3.5 yields a commutative diagram
M(~ I )
M(g I )
~ M(Q)
M(~)t M(T2) , M(Q) II M(~ 2) , whence
Im M(~ 1) E I m
M(~ 2)
In the same vein as
M(~)
The other inclusion holds by symmetry.
was obtained in 1.3, K~(V)
for each homomorphism
?: Q1 " Q2
every automorphlsm of
Q
K~(Q)
is a submodule of
in
~
is defined
and is functorial.
Since
can be lifted to an endomorphlsm of M(Q)
F ,
with respect to the Aut(Q)-action.
For clarification we here assume that one free presentation F
~Q
has been chosen for each group
Q ; the ~-free presentation
above may be related to it as in 5.2 (this is often convenient), but need not be. K~(Q)
The coordinate isomorphisms of 1.3 handle
M(Q)
and
simultaneously.
6.2 REMARKS.
a)
Proposition 1.3.5 gives a "relative Schur-Hopf
Formula" M~(Q)
~ R n _
[F,~]
[~,~]
Thus this group is isomorphic with the varietal homology groups • I(Q,Z) b)
of LEEDHAM-GREEN [I] and
V(Q,Z)
In this context the group
the varietal cohomology group the notation of 6.1, :
V(Q,A)
of STAMMBACH [3; III.1].
Cext~(Q,A)
V(Q,A)
of 5.8 appears as
of STAMMBACH [3; III.I].
In
is defined as the kernel of
Cext(Q,A) - Cext(F,A)
.
We extract the proof of this remark from STAMMBACH [3; III.3], who actually treats the more general results of Knopfmacher.
Let
235
[e] ~ C e x t ( Q , A )
e'~
:
A"
e :
A>
ly
if
S-free,
lles An
6.3 T H E O R E M . by the l a w s
V
S
image
, t h e n so is
, i.e. G
e~
are in
G~ ~ G × F .
splits. S .
Since
ConverseF
ks
a d m i t s a splitting.
Let
S
be a g r o u p v a r i e t y of e x p o n e n t zero,
, a n d let
e = (~,~)
the v e r b a l
~*[e] = 0
a n d its e p i m o r p n i c
"o: G~ " F
(6.3)
~Q
First assume that
G~ ~ A x F
is d e f i n e d by the d i a g r a m
~-~
~G
Then
e
~*[e] = [e~]
-~ G ~
as in I.(1.7).
,
Then
:
subgroup
Q ~ s .
A:
VG
defined
For any c e n t r a l e x t e n s i o n
~G
;;G
,
is the image of the c o m p o s i t e map
e.(e) KS(Q) " Thus
~ M(Q)
e ~ S
PROOF.
> A
precisely when
Ker
VF c
8,(e)
e'
:
e
:
(6.4)
) F
Q ~ S
R ~
)F
A)
and
We evaluate
O : F
M(Q)
=
KS(Q)
~ Q
of
.
Q
a n d con-
f r o m it,
~Q
~P
II ~
S and
) G
~
~'Q
has e x p o n e n t M(V/VF)
Im M(~)
=
0
, we h a v e
VF ~ R n IF,F]
at the free p r e s e n t a t i o n s
by (6.4) a n d o b t a i n
KS(Q)
contains
;" F / V F
to Since
)G .
C h o o s e a free p r e s e n t a t i o n
s t r u c t the c o m m u t a t i v e d i a g r a m
(6.4)
~
VF.[R,F]
JR,F]
exhibited
236
and then by 1.3.6 and 5.1: ~e.(e) K~(Q) = 0(VF'[R,F]) = 0(VF) = VG .
6.4 COMMENTS.
a)
only on the subgroup
The question whether U(e)
of
M(Q)
e ~ •
, cf. (3.5).
any two generalized representation groups
G1
erate the same variety of exponent zero. Theorem 6.3 for tion II.2.13. b)
~1 = var°(G1)
resp.
or not, depends We conclude that
and
G2
c)
gen-
For the proof, combine
~2 = var°(G2)
with Proposi-
(This topic is resumed in 7.26.)
K~(Q)
that do not involve free presentations.
Again by Theorem 6.3,
K~(Q)
essentially is the verbal
subgroup of any generalized representation group of mation on
K~(Q)
extensions by
internally,
Q .
The infor-
so obtained can then be used for arbitrary central
Q .
We are going to discuss situations where
K~(Q)
can be described
thus eliminating the use of free presentations.
6.5 PROPOSITION. ~
Q
In some important cases (see 6.5 and 6.12 below) we have
formulas for
let
of
Let
~
be a variety, defined by the laws
be the variety of center-by-~ groups and
Q e ~ .
W ,
Then
K~(Q) = Ker~M(nat): M(Q) - M(Q/WQ)~ = Im ~(e') where
e': W Q ~
~Q
~Q/WQ
.
Under the same assumptions
PROOF.
We start with a free presentation
~-free presentation of diagram
M~(Q) ~ Im M(nat) = Ker 8.(e')
Q
F~Q
, obtain a
as in (5.1) and construct the commutative
237
I
[I t
R t~
))0
"F
II i
) F
R.WF c
The definitions of
)> O/WO
M(~)
and
M(nat)
in terms of these free pres-
entatlons give Im M(~) = (VF 0 [F,F]).fR,F]
Ker M(nat) = R N [R.WF,F]
[R,F]
[R,F]
Under the present assumptions,
VF = [F,WF] E R .
(VF n [F,F]J.[R,F] = VF.£R,F] ~ R 0 [R-WF,F] S [H-WF,F] H [R,F]-[WF,F] = VF.[H,F] by-~ group,
e'
~ Ker
while Theorem 1.4.4 gives
6.6 COROLLARY and
8.(e')
0 is a center-
,
Im ~(e') = Ker M(nat)
(cf. EVENS [I; 43]).
central, then
rn+IG = ~ e.(e)K .
rather
precisely when
Here
Finally, as
Let
K = Ker~M(nat): M(O) - M(O/rnO) l .
(n+1)
R 0 [R.WF,F]
and
is central; now Proposition 1.3.5 yields
M~(O) = M ( O ) / K s ( O )
n
Hence
rio = o , r2o = [o,o]
central series.
Thus
G
O If
.
~]
be nilpotent of class e
as in (6.3) is
is nilpotent of class
Ker e.(e) ~ K .
~]
, r3o = [r2o,O] .... denotes the lower
The assumption of the corollary is
FnO $ 0 ,
Fn+lO = 0 .
In his study of the Dimension Conjecture, PASSI [I; p.27] introduced the notion of induced-central extensions.
6.7 DEFINITION. central extension
Let
Q
el: Q/Z)
n
be a nilpotent group of class ) M----~Q
n .
A
is called induced-central,
238
if there
exists a central
morphlsm
f: A " Q/Z
"induced-central"
with
rather
may be c o n s i d e r e d f-1
induced-central
eralized clearly
valid
K + Ker
If
e
e.(e)
e,f Ker
versely,
We regard
class eI .
A
(If we wish,
by
when
precisely
~A
and
given
eI
f
.
Then
f: A - G/Z
= f.g = 8.(e I)
and G/Z
6.9 COROLLARY.
; this
choose
any
~A = Fn+IG
is divisible.
by
8.(eI)
.
.
as
map.
e.(e I)
By the Uni-
and
g
Moreover,
- Hom(M(Q),O/Z)
We conclude
Con-
Let
e ~ Cext(Q,A)
.
e ~ Cext(Q,O/Z)
Q
be n i l p o t e n t
with is epl8.(fe)
is i n d u c e d - c e n t r a l
=
is an isomor-
fe ~ e I
of class
and
= f 8.(e) ;
or, that
[]
Let
is
implies
M(Q) - O/Z
Ker g = Ker and
then,
8.(e I) = M(Q)
is the inclusion
1.3.8,
(Our p r o o f
is i n d u c e d - c e n t r a l
e.(e 1) = M(Q) e.(el):
[1] who gen-
B .)
implies
K + Ker
8.: Cext(Q,G/Z)
is induced-central.
.
is
.
extension
eI
e I ~ fe
Decompose
8.(e)K = A
group
that
and
By construction,
hence
Q/Z
e.(e)K = A
K + Ker
Theorem
of
n ~ I
e e Cext(Q,Q/7)
by VERMANI
is any central
when
8.(e)
with
where
Coefficient
abelian
of class
8.(e) = M(Q)
instead
Now assume
8.(e 1) 2 Ker
since
K + Ker
B
group
Then
has also been obtained
be as required.
8.(e) = g .
every
- M(Q/FnQ) I
= M(Q)
8.(e I) = f.g
morphic,
M(Q)
as in (6.3)
A = Im 8.(e 1) ~ O/Z
phism,
replace
be a n i l p o t e n t
for any divisible
~A = Fn+1G
versal
map:
Q
it to other kernels
PROOF.
thus
fe ~ e I
then of the extension
Let
precisely
This p r o p o s i t i o n
let
and
and a homo-
.)
K = Ker~M(nat):
6.6,
as in (6.3)
of the congruence
an inclusion
6.8 PROPOSITION. and
e
~A = Fn+IG
as a property
[el] ~ Cext(Q,G/Z)
by
extension
n ~ I
if and only if,
Then
e1
239
M(nat):
M(Q)
PROOF. M(Q)/K
-
If
- O/Z
non-zero
M(QlrnQ) K ~ M(Q)
tion 6.8, none
of
class
O/X
n
(PASSI
Applying
e
K
mK = 0
imply
suggests
n = 2 ,
D
= me.(e)
= 0 .
M(D) ~ Z/2 M(Q)
in
~ M(Qab)
in the exact
Q
be n i l p o t e n t
m .
K
Then
of
m[e] = 0
as in P r o p o s i t i o n
6.8.
' we obtain
, Now
K + Ker
Since
e.
8.(e) = M(Q)
and
is an i s o m o r p h i s m
whether
Cext(Q,G/Z)
the i n d u c e d - c e n t r a l
Although
Corollary
6.10
is negative.
the subset
of (congruence
Cext(Q,Q/Z)
group
denote
from E x a m p l e
by the S c h u r - K t ~ n e t h orders
of
In general,
~ (Z/2) 3
is induced-
= [me] = 0 .
the a n s w e r
extensions
and
open p r o b l e m
be the dihedral
~: Q - - - a ~ Q a b
= f
By Proposi-
[e] e Cext(Q,Q/Z)
~ = ~n-1
m[e]
Let
exponent
m .
of 6.8,
the opposite,
induced-central
finite
dividing
form a subgroup
6.11 EXAMPLE.
4.2]).
extensions
It has been an a p p a r e n t l y extensions
We obtain a
fl K = 0 .
e.(e)
with
Qab ® Yn Q " M(Q)}
8.(me)
as in the p r o o f
has
6.5 for
has exponent
(divisible).
with
be i n d u c e d - c e n t r a l
K = Iml~(e'): thus
[1; Thm.
FnQ
Proposition
homomorphlsm
is immediate.
for all i n d u c e d - c e n t r a l
Let
" O/Z
e e Cext(Q,~/Z)
such that
PROOF.
is a n o n - t r l v l a l
is inJective
f: M(Q)
The converse
6.10 C O R O L L A R Y
the zero-map.
, there
, because
homomorphism
central.
is
classes
is not a subgroup.
of order 8, Q = D x Z/2
the abellanlzatlon.
II.3.8(iv).
of) Let
, and
We use
Then
~ (Z/3) 3 Formula
11.(4.7)
and T h e o r e m
sequence
I.(3.3')
belonging
to
1.4.7. ~
Counting , we obtain
240
K = K e r M(.) ~ Z/2 - Z/2 ~ : G / ~
.
with
One easily fl + f2 $ 0
K A K e r fl = K O K e r f2 = 0 . 8*(el) = fi
for
by P r o p o s i t i o n
i := 1,2
6.8, but
6.12 P R O P O S I T I O N . let
S
Then
PROOF.
Again,
VF c
l R c
R •
with
F
= Imlres:
and
and
Ker(f1+f2)
Choose Then
~
eI
~ F
and
e2
are i n d u c e d - c e n t r a l
- M(Q)}
d e f i n e d by the laws
groups,
with
and let
II
"F
A~Q
; R .WF
-" W Q
R.WF
free and
= [WF.WF].[R,F]
,
Im(res)
F
S-free.
VF : [WF,WF] E R , thus
and
(R n [R.WF,R.~])-[R,F]
Ker Here of
is a c e n t r a l
= (R A [ R . W F , R ' W F ] ) - [ R . F ]
(1+I)
Q
[WF,WF]'[R,F]
S (RA [R-WF,R.WF])-[R,F]
~ [R-WF,R.WF].[R,F]
is solvable
extension,
precisely
then
of l e n g t h G
~ [WF,WF].[R,F].
I ~ I
is s o l v a b l e
and
e
of l e n g t h
[]
as 1
when
8.(e) _o I m l M ( D I _ I Q ) - M(Q)}
DI_IQ Q .
If
By 1.3.2:
JR,F]
Now
r a t h e r then
.
diagram
JR,F]
in (6.3)
,
Q e S .
res = M(Incl)
a commutative
W
~
4
6.13 COROLLARY.
with
is not.
be a variety,
we c o n s t r u c t
~ K
ei ~ Cext(Q,O/Z)
of a b e l l a n - b y - ~ M(WQ)
free, h e n c e
Im M(~)
fl,f2: M(Q) -
e I + e2
Let
be the v a r i e t y K~(Q)
.
finds h o m o m o r p h i s m s
[]
denotes
the last n o n - t r l v i a l
term of the d e r i v e d
series
241
For finite groups, is due to YAMAZAKI
a cohomologlcal
[1;
6.14 EXAMPLE.
and
by
~ = ~2
G
Let
G
be a group such that
is solvable
' then
Let
is called absolutely-S, by
Q
again lles in
6.16 THEOREM.
e.g.
S
[G,G] ~ 0
cyclic.
of length two.
Km(G) = 0
6.15 DEFINITION.
of this corollary
§3.2].
abellan with trivial multlpllcator, extension
formulation
take
~ = ~1
6.13.
be a variety and
if every central
Then every central
Indeed,
by Corollary
is
Q e S
extension
.
Then
A ~
Q
~G
~Q
S .
Let
s
be a variety
of exponent
0
and
Q e S .
Then the following are equivalent: (1)
Q
is absolutely-S
(il)
all generalized
(lii)
some generalized
(iv)
~(Q)
=o
variety
0 .
representation
However,
m = varo(S)
absolutely-S an arbitrary
of exponent
by the theorem
for
~
VG ~ e.(e)Ks(Q)
hence
implies
e.(e)
follow easily.
the formula
lles in
S ; S ;
variety
of
S
determines
S the
Then one has
K~(Q)
is described
.
by Theorem 6.5.
then
theorem
Q
as in (6.5) be a central
tation group, G e S
group of
0 , cf. 5.7.
Q ~ S ; the influence
e
lle in
groups can exist only if
for
Let
Q
representation
Ks(Q) = N~(Q)
PROOF.
groups of
.
By the very definition, has exponent
;
If
Then
is a generalized
is a monomorphism
KS(Q)=O.
follows
Q e S from
represen-
by Proposition
The remaining assertions
Finally let
K~(Q) = K~(Q)
e
extension.
and
II.2.13,
of the
~ = varo(S)
WF = VF 0 IF,F]
.
Then
for free
242
groups
F , cf.
6.17 Q e •
5.7.
EXAMPLES. with
[3
a)
Let
~
M(Q) = 0 ; then
be a variety of exponent Q
0
and
is absolutely-S by Theorem 6.3.
The groups of Example 6.14 are absolutely-~ 2 . b)
By various reasons (use Corollary 1.3.9 or Example 4.8 or
Theorem 6.16) an abellan group when
M(Q) = 0
Q
is absolutely-abelian precisely
VARADARAJAN [I] classified the abelian groups with
trivial multlplicator in a topological context, while MOSKALENKO [I] classified the absolutely-abelian groups as such.
Finally BEYL [3],
among other things, obtained the absolutely-abelian groups via the multiplicator.
The abelian groups
Q
in question are characterized
by the total of the following properties p-subgroup of
the rational rank of
(ii)
the reduced part of
(iii) for each prime
6.18 EXAMPLES.
Q
has
denotes the Sylow
Tor Q ) :
(i)
[Q,Q] ~ Z*(Q)
( Gp
.
a)
Q
does not exceed one ;
Qp
is
p , Q/TorQ
A group
Q
0
or cyclic
;
is p-dlvislble unless
Qp = 0 .
is absolutely-~ 2 precisely when
Indeed, by 3.7(b) every representation group
G/Z(G) ~ Q/Z*(Q)
G
of
abelian; the assertion follows by Theorem
6.16. b)
A nilpotent torsion group
Q
if, so are all its Sylow p-subgroups. Z*(Q)
Let
For
Q
and
[Q,Q]
and
by Theorem 4.2 are the restricted direct products of their
Sylow subgroups, c)
is absolutely-~ 2 if, and only
see also 4.3.
We describe the metacyclic groups G = G(m,n,r,~)
(2.3) and (2.5). generated by
G
that are absolutely-~ 2 .
as in (2.2), the parameters being subject to
By Proposition 4.18
b k , where
k
Z*(G)
is the cyclic group
is a certain integer dividing
n .
243
Since
[G,G]
0 < v < n
is generated by
and
m (m,r-1)
a (m'r-1)
kl(m,r-1) , the condition
I (m,r-1)
resp.
and
b ~ ~ (a)
[G,G] ~ Z*(G)
~ I 1 (m,r_l)2
for amounts to
244
7. Schur-Baer Multiplicators and Isologlsm.
Here we demonstrate that some methods of Sections 1.3 and III.2 easily adapt from isocllnlsm to isologlsm.
The isologlsm relation
puts groups resp. extensions into the same class roughly if they equally deviate from lying in a given variety
S .
For the moment,
a reasonable level of understanding is achieved by assuming that is defined by a single law v = [Xl,X2]
defines
v e F®
S , while the corresponding concept of
logism agrees with isocllnlsm.
In general, the center
replaced by the marginal subgroup which are "not noticed" by multlpllcator
For example, the law
SM(Q)
.
v*(G)
v , M(Q)
S-lso-
Z(G)
is
of those elements of
G
is replaced by the Schur-Baer
This section owes a great deal to P. HALL
[1], [2] and LEEDHAM-GREEN/McKAY
[1].
We hope that the present
treatment may serve as an introduction to the latter paper which eventually focusses on certain varietal cohomology groups (different from those considered in Section 6). limited insofar as
Se. , the analogue of the projection map An the
Universal Coefficient Theorem, LEEDHAM-GREEN/McKAY
7.1 DEFINITION and
v .
(cf. P. Hall [2]).
x
subset
be a set of laws V , with
n
depend-
G , consider the subset of
G
given
= v(gl,...,gi,...,g n)
gj ~ G , all places v*(G)
V S F
satisfying
V(gl,...,gl.x,...,gn) for all
Let
be a typical element of
For every group
by the elements
is not surJectlve in general, cf.
[1; II ~3].
v = V(Xl,...,Xn)
ing on
Our elementary approach is
i .
is a characteristic
V-marglnal subgroup of
G
One easily verifies that this subgroup of
is defined as
G , cf. (5.3).
The
245
V*G
=
N
v*(G)
.
v~V By the very definition functor
~
of
V*G
, every law as above determines
from groups to set functions which assigns
to
G
a
the
function ~G
:
given by
(GIV*G)n
:= ~ (GIV*G)
v .
7.2 DEFINITION. (7.1)
~ VG
e :
A group extension
N~
~ ~ G
is called a V-marginal and every law
v
in
~ ~Q
extension, V
if
~N S V*G
.
Such an extension
give rise to a set function
~e: Qn , VG
natural with respect to morphisms
of V-marglnal
extensions.
arbitrary
the subgroup
T
extension
e , consider
of
VG
,
For an generated
by all v(gl,...,gi'~(x),...,gn)'V(gl,...,gl,...,gn with ,
v ~ V
, gj ~ G , x E N
maps every generator
or simply by [NV*G]
of
, all places T
onto 1.
Hence
e
i .
)-1 Then
We denote
is V-marglnal
T ~ ~N
T
by
since
[NV*G]~
precisely when
[NV*G] K = 0 .
7.3 REMARK. W = VF uously
It is quite practical
is the associated
closed
speak of the B-marglnal
variety defined by the laws
PROOF. = V*(G) group of f
Clearly where F
U
set of laws.
subgroup
.
is the subgroup
be any endomorphlsm
by of
of
V*G = W*G
where
Hence we may unambig-
G , where
S
is the
V .
W*(G) ~ V*(G)
generated
to know that
V . F®
It is easy to show
U*(G) =
rather than fully invarlant
Let
v = V(Xl,...,x n) ~ V
We consider
the law
sub-
and let
246
w = f(v) = v(f(xl), cf.
(5.3).
Let
all v a r i a b l e s = f(x n)
m
be large
that appear
; we treat the
Xl,...,x m
Recall
projection
G
f(Xn))
....
and
w
V*(G) ~ G
~ G/V*(G)
t
enough
!
such that
~Xl,...,Xm}
in any of the words
yj
gl,...,gm,gl,...,g m m G
,
as words
from 7.1.
Yl = f ( x l ) ' ' ' ' ' Y n
!
__
gi = gl
=
in the v a r i a b l e s
Appeal
shows the following: and
contains
to the natural If
mod V*(G)
for
i = 1,...,m
,
then !
!
Y j ( g 1 ' ' ' ' ' g m ) e YJ(g1' .... gm ) for
j = 1,...,n
.
The defining '
w(g
,...,g~)
mod V*(G)
property
V*(G)
2 w*(G)
In the same vein, glnal
extension.
(7.2)
,
laws.
~
of
G
speak of a S - m a r g i n a l
iff
•
rather
than V-mar-
Wlth
S
~*G = G
section,
is the group
The following
yielding
G ~ ~
of this
a group v a r i e t y
7.4 LEMMA.
= w(gl, .... gm )
, what was to be shown.
we may
iff
For the r e m a i n d e r
lently,
now gives
Also
~G = 0
are given and
V*(G)
= V(yq(gq,...,gm),...) = v(Y1(g I .... ,gm),...)
This means
of
variety
let us assume defined
is given and
is rather
immediate
the n o t a t i o n
a commutative
of 7.2,
V
by
that the laws V
or, equiva-
is a g e n e r a t i n g
set of
from the definition:
[NV*G]
is a normal
subgroup
diagram
(7.3) m(e) where
:
N ~-I[Nv*G]
the vertical
~- ~'
G [NV*G]
maps are natural
V
~' . $ Q
projections
and the h o r i z o n t a l
247
maps are induced ginal.
by
~
Any morphlsm
uniquely
over
.
is S-mareI
(~,8,~):
factors
e I " e2
, functorlally.
is the least normal
is S-marginal;
m(e)
extension
any m o r p h l s m
m(e I) - m(e 2)
[NV*G]
e/T
The e x t e n s i o n
in a S - m a r g l n a l
Hence
(am,.,?):
Consequently
~ .
e - eI
e - m(e)
ducesamorphism
such that
and
in-
[~
subgroup
T
of
in view of 7.3 the n o t a t i o n
G
INS*G]
is justified.
7.5 P R O P O S I T I O N be any v a r i e t y [V*G,WG]
and
= O .
(7.4)
(LEEDHAM-GREEN/McKAY
Let
consist
of the c e n t e r - b y - ~
= 0
W = ~F
S
Let
groups.
m
Then
.
and
w=w(xl,...,Xn)
v(xl,...,Xn+ I) = [Xn+1,w(x I, is a law for
1.1.3]).
In particular,
[W*G,WG]
PROOF.
S
[1; Prop.
by 5.5.
Now let
e W
, then
Xn)]
. . . .
x e V*(G)
and
gl,...,g n e G
.
Then v(gl,...,gn,l.x) This means
that
x
= v(gl,...,gn,1) commutes
w i t h all elements,
of
implies
.
W * G E V*G
7.6 E X A M P L E SG = Fc+1G
and
Z2(G),...
S*(G)
denotes
PROOF.
of
generators,
.
Formula
(7.4)
holds
[2]).
Let
S = ~c
with
= Zc(G ) .
G .
Here
central
VG = Fc+IO
V*(G) S Zc(G)
elements
with all subgroup
the upper
The formula
first prove arbitrary
(P. HALL
WG
= I
•
Let
Then
Zo(G)
series
follows x e V*(G)
since
= 0
of
G
hence
~ ~
o ~ I .
Then
, ZI(G)
= Z(G)
,
.
from 5.5 and 5.4. , while
gl,g2,..,
We are
248
[gl,...[gc,l.x]...] implies
= [gl...[gc,1]...]
[g2,...[gc,X]...]
= I
e ZI(G ) , then
[g3,...[gc,X]...]
and finally
x e Zc(G )
We are left to prove
case
c = 1
is clear, we proceed by induction
that
rc+lF . , as a fully invariant subgroup of
e Z2(G),
Zc(G) S V*(G) for
c Z 2 .
F
.
The
Note
, is generated
by the single law
(7.5)
Vc(Xl,...,Xc+ I) = [Xl,...[Xc,Xc+1]...]
cf. HUPPERT
[1; p.257].
Let
x e Zc(G)
,
, thus
x2(G) e Zc_I(G/Z(G)).
Then V c _ 1 ( g 2 , . . . , g l . x , .... go+l) ~ Vc_1(g2,...,gl,...,gc+1)
mod Z(G)
by the induction step, hence Vc(gl,...,gi'x,...,gc+l) for
2 ~ i ~ c+1
.
= Vc(g 1, .... gi,...,gc+l )
The missing case
consequence of Vc(X,g2,...,gc+l) -1 Y := g2 and z := [g3,...,gc+l]
i = 1
= 1 .
is by Lemma 1.4.1(a)
From 1.4.1(e)
resp.
z := g3
a
with
for
c = 2 , we
obtain V c ( X , g 2 , . . . , g c + 1) = y - l x [ [ x - l , y ] , z ] . Y - l z [ [ z - l , x ] , y ]
.
The first factor on the right side vanishes by induction [x-l,y] e Zc_I(G ) . hypothesis
for
The second factor vanishes
G/Z(G)
7.7 EXAMPLE. Vc(Xl,...,Xc+l)
implies
A g a i n let
[x,z] ~ 1
~ = ~c
for
as in (7.5) generates
since
since the induction
mod Z(G)
c ~ 1 . ~F
The fact that
, also implies
[N~*a] = [ O , . . . [ a , N ] . . . ] with
c
entries
that
[N~*G]
O , for
contains
N ~ G .
[G,...[G,N]...]
one may proceed by induction and a typical a r g u m e n t
It is clear from the definition
on
c .
is as follows:
For the proof of equality
The tool is Lemma 1.4.1(a),(b) x ~ y
mod [G...,N]
implies
249
[g,x] ~ [g,y]
mod [G,[G...,N]]
We are now going group
to define
Q , possibly
in analogy
not in
to P r o p o s i t i o n
7.8 PROPOSITION. extensions (7.1)
multiplicator
S , and to e s t a b l i s h
of any
an exact
sequence
1.3.5.
There
is a functor
to that of central
:
n VG
~N
c
>
[NV*G] and to m o r p h i s m s (1,8,1):
s'
extensions
e I " e2
In this sense,
PROOF.
from the c a t e g o r y
w h i c h assigns
e I - e2
The functor
gruence,
s'(e)
to
e
of as in
of
on
so is then ?
Given
subgroups.
= ~
If
eI
has the form
first
Since
(I,B,1)
NV*G2]
g e G I , with
V(GI)/[NIV*GI]
If
= s'(e,8,?).
of functors:
(~,~,~),(a',~',~):
for all
If
s'(1,8,1 ) . s'(a',8',~)
by (7.4).
[NV*G1]
maps.
only.
to the V-verbal
~1
,
induced
is a composite
is central
B'(g) = ~2(ng)'8(g) generator
are given,
s'
then one shows
is a congruence.
then
depends
then the r e s t r i c t i o n
~*=VQ
the obvious
is a congruence,
s'(a,8,~)
is V-marginal,
VG [NV*G]
of e x t e n s i o n s
(a,~,?),(~',8',?):
and
the S c h u r - B a e r
the extension
s'(e)
7.4,
by 1.4.1(b).
, thus
m
of
m(e)
is a cons'(1,~,1)
e 2 , then
-
ng e N 2 .
A typical
x = v(gl,..,gn).[NIV*G1],
B'(v(gl,...,gnD. 8(v(g I .... ,gn)) -1 = v ( ~ ' C g l ) , . . . , B ' ( g n ) ) .
•v(B(g I ) ,..., 8(gn)) -I e [N2V*G2] same map.
Hence
8'
and
8
induce
the
[]
7.9 DEFINITION. the group
.
If
e: R ~
Q , then define
at the coordinate
system
= (R n VF)/[RV*F]
.
~ F
the value e
~ Q
is a free p r e s e n t a t i o n
of the S c h u r - B a e r
as the a b e l i a n
G i v e n any h o m o m o r p h i s m
and free p r e s e n t a t i o n s
e
and
e'
of
Q
group
and
Q'
multiplicator
SM(Q) e =
~: Q - Q'
of
of g r o u p s
, respectively,
250 define
SM(~)ele,
= s'(a,e,~)
By Proposition 7.8,
for any choice of lifting (e,8,y)
~M(~)ele,
is well-deflned and the analogue
of I.(3.1) holds for
SM(-)
mations.
~M(Q)e = M(Q) e , as defined in 1.3.2.
Certainly,
; there are unique coordinate transfor-
dentally, the same argument gives the invariance of := VF/[RV*F]
.
up to coordinate Isomorphisms.
are special Baer-invariants,
Both
SP(Q) SM(Q)
see LEEDHAM-GREEN/McKAY
Inci:= and
SP(Q)
[1; I §I] for
more details.
Every extension
7.10 PROPOSITION.
e
as in (7.1) determines an
exact sequence (7.6)
BM(G) SM(~) >SM(Q)-
se,(e) ~
N
~*
x-I[Nv*G]
G >-VG
~*
Q > ----+0 VO
which is natural with respect to morphisms of extensions. and
..
are induced by
= Se.(e')
If
G = F
x
and
. ).
is free and
If ~
e ~ e' , then
,
(Here
~.
SS.(e) =
an inclusion, then
SS.(e)
is Just the inclusion BM(Q) e = (N n VF)/[NV*F] r
In the special case that
e
>N/[NV*F]
.
is S-marglnal,
i.e.
[NV*G] = 0 ,
FROHLICH [I; Thm. 3.2] obtained the sequence (7.6) in the context of algebra varieties.
(Actually,
our proof can be easily adapted to
varieties of algebras.)
SKETCH OF PROOF. 1.3.5.
The proof is parallel to that of Proposition
Starting with a free presentation I.(3.4) of
e , we consider
the sequence
s nv(F) [SV*F] With
,R nV(F) [RV*F]
e'~
R
SM(Q)e, = (R O V(F))I[RV*F]
N/~-I[Nv*G]
, we d e f i n e
_ ~ ~
[RV*F]-S
Se.(e)
and
S-V(F)
a.:
= o..e':
, F__[__ R.V(F)
R/([RV*F].S) SM(Q)e, -
N/.-I[NV%]
251
One shows that d i f f e r e n t tions
of
Se.(e)
the modular
.
yields
S ~ R
(a)
the f o l l o w i n g
presentation (~,8,1):
of
Q
e - e
of 1.(3.4)
The exactness
law from
7.11 REMARKS.
choices
for
also
e ~ e'
R
iq=IXJ
~qxq-ll:
If
e
map
on
Q
Q - Q .
mM(Q)
(a,~,iq):
by
e
of
by
Q
a free p r e s e n t a t i o n as a subgroup
of
Hence
by with
B
~M(G)
obtain
and
ie.(e)
= ~e.(e')
of
(7.7)
Ker ~e.(e) r where
K
e
of
~F
~M(Q)
from
for
q e G lq
of
and
~M(-)
,
y E ~M(Q)
can be lifted
an automorphism;
via
Now let
We obtain
by the f u n c t o r l a l i t y
~M(Q)
Ker ~e.(e)
F/[RV*F]
~e.(e)
by inner a u t o m o r p h l s m s
thus
to a
Ker se.(e)
.
with
,
~Q
is normal
(7.6)
= ~M(Q)
rather
m(~)
with
, and
in
regard
Then
Ker ~e.(e I) = K .
N/O
Then
as the range
and 5.1 the exact
sequence
e,V(G)
~0
)V(Q) of
K
Since
F/[RV*F]
be ~-marginal.
than
is the r e s t r i c t i o n
start w i t h
= (R 0 VF)/[RV*F]
as in (7.1) N
is a ~ - m a r g l n a l
To this end,
, form
~M(Q)~ K
, there
= K .
extension
e
= O , and we regard
of groups,
be the free
~-I[Nv*o]
G , then every
is a ~ - m a r g i n a l
7.12 REMARK. [NV*G]
on itself
eo: R •
is a Q - s u b m o d u l e
e I = m(e)/K
e
7.10
N
= ~8.(m(e))
For every Q - s u b m o d u l e
extension
[~
is the composite
~m
q.y = M ( i q ) y
e - e
is a Q - s u b m o d u l e
K
acts
is an e x t e n s i o n
(c)
by
.
The group
acts
~e.(e)
follows
be evaluated,
Se.(e)
descrip-
of P r o p o s i t i o n Let
[RV*F]
implies
(b)
Q
shall
then
[aV*F]
~ R n V(F)
se.(e)
SM(Q)
I.(3.5);
sequence
assertion
of
~e.(e)
Naturallty
[RV*F]
The n a t u r a l i t y
at w h i c h
like
of the above
and
description
give compatible
,,~e.(e):
~M(Q)
of
- N - G .
252 Tbls
sequence
sions. • e.(e)
is natural
Consequently,
if
~-marglnal
group
, then
Then
if there are i s o m o r p h l s m s
n
(7.8)
~1
of ~ - m a r g i n a l
G ~ •
precisely
exten-
when
~i
denotes (~,~)
an i s o l o g l s m
Moreover,
if
as in (7.1)
> V(G2)
function
E v e r y group
; G
;~G/V*(G)
of
are called ~-iso-
and
ei
~: V(GI)
for all
of Def. e2
groups
G
- V(G2)
v ~ V ,
and
KIN I = V*(G1)
v e V
, all
.
Then
G
when
a ~-marginal
and eG
and there
is
~-isologlsm.
H
exten-
are @-isologlc
and
eH
G2
, then
and
as ~ - m a r g l n a l
are ~ - i s o l o g l c
are ~-iso-
q E Q , then
functions
~
If
is a ~ - i s o l o g l s m
Indeed,
q e vV*(G)
) = ~(ql ..... q i ' ' ' ' ' q n )
vV*(G)
~wIV*(GI) = v 2 V * ( G 2 ) , w h e n c e
and determine between
eI
and
the a s s e r t i o n s
extensions,
as groups.
~2N2 = V*(G2)
ql,...,q n E Q , and all p l a c e s detect
QI = Q2
(see 7.2).
7.13.
G1
~(ql'''''qi'q'''''qn
If
i = 1,2
determines
are ~-isologlc
is ~ - m a r g l n a l
for
a special
[2] p r e c i s e l y
when
(~,~)
be
'
(a)
and
then the middle
i = 1,2
~
of P. Hall
eI
for
e2
is c o m m u t a t i v e
, this is called
in the sense If
and
~: QI " Q2
diagram
the w o r d
(~,1)
eG: V*(G) ~
(b)
eI
is called a ~-isologlsm.
7.14 REMARKS.
in the sense
~ Qi
) V(GI)
1~n n ~2 Q2
The pair
loglc
to maps
el: N i ~ - - - ~ G i
extensions.
Q1
sion
Q e •
Let
such that the following
where
respect
= 0 .
7.13 DEFINITION.
logic,
with
i .
if
e
precisely
for all laws Thus the w o r d
G/V*(G)
~ Q/~V*(G)
e 2 , then follow.
.
253
7.15 LEMMA.
Let
f: G - H
(Ker f) Q VG = 0 .
Then
= f.: G / V * G - Q/V'H)
PROOF. let
Certainly
y E V*H
and
be an e p i m o r p h i s m
f-I(v*H)
= V*G
and
of g r o u p s w i t h
(~ = flVG,VH
:
is a S - l s o l o g l s m .
~
is an i s o m o r p h i s m
x E G
with
and
f(V*G) ~ V*H
f(x) = y , let
v
.
Now
be any law of
V.
-1 Then
v ( g l , . . . , g l x , . . . , g n ) - V ( g l , . . . , g i , . . . , g n)
for all
gl,...,g n ~ G
f-I(v*H)
= V*G
of (7.8),
and
and all p l a c e s
~ = f.
eG
and
An eplmorphism
is c a l l e d a S - l s o l o g i c
i , hence
is an isomorphism.
for the e x t e n s i o n s
7.16 D E F I N I T I O N .
~ (Ker f) 0 VG = 0
eplmorphlsm
eH
x E V*G
with
of groups.
Thus
The c o m m u t a t i v i t y
, is obvious.
f: G - H
.
[~
(Ker f) n VG = 0
Let
"i ei
:
Ni~
be S - m a r g l n a l morphlsm = Ker
~ Gi
~ Qi
extensions
8: G 1 " G 2
for
with
i = 1,2 (Ker
~I , then the u n i q u e l y
c a l l e d an S - i s o l o g l c ( ~ = ~ I V G I , V G 2 ; ~ = ~) Q1 = Q2
and
.
If there
~)Q VG = 0
and
induced morphism
eplmorphism
V = 1 , we call
8-1(Ker
(a,8,?):
of extensions,
is a S - i s o l o g l s m
exists an epi-
e I " e2
is
since
of extensions;
(a,8,?)
~2 ) =
if m o r e o v e r
a special S - l s o l o g l c
epl-
morphism.
7.17 D E F I N I T I O N . if
~(N)
and
e
n V(G)
If
, let
Q ~ S , then
G e S .
then
e
S-isologic
as in (7.1)
e'
e
the c o n g r u e n c e
as in (7.1)
Q E S .
eplmorphism;
then
definition
is of S - t y p e
N e S .
is also of S-type.
Cexts(Q,A ) = EXTs(0,A)
T h u s the p r e s e n t
that in 5.8 for
e
is of S-type,
E X T s ( Q , N ) S EXT(Q,N)
A ~ SOS
If
= 0 .
is of S-type,
denote by For
A group e x t e n s i o n
If
classes
n Cext(Q,A)
If
e ~ e'
N E S , we of S-type. .
is of S - t y p e p r e c i s e l y w h e n of
An e x t e n s i o n
Cexts(Q,A) of S - t y p e
it is S - m a r g i n a l
by Lemma
agrees with
is e s s e n t i a l l y 7.15.
The
a
254 totality of ~-isologlc eplmorphisms onto a given group metrized by the epimorphisms in
7.18 PROPOSITION. arbitrary group. nat*
:
cf. 1.1.4.
nat*
Then
nat
~ EXTs(Q,N)
0 ~
N;
#(e) : Due to
1.4].
For
~ VG
~
-
[e] e EXTs(Q,N) ~
.'; Q
[[
nat I
~.
nat I
N
) GIVG
VG c V E x
maps both
VG
m
and
eft: N ~
.~ QIVQ .
~E~Q/VQ
VQ c 0 x VQ
t(e) in
and thus
KerlG - El ~ VG
subgroups are actually equal.
is an extension. • , then
1.2.4.
$(e) .
e = (el)nat
~N N VG = 0 .
isomorphlcally onto
It is now clear that
are mutually inverse mappings, up to congruence. then so is
construct
VQ
~
satisfies
is a subgroup of
.
G
given
an
;
Cext~(Q,N)
~N 0 VG ffi 0 , the bottom row
Conversely,
Q
induces a blJection
Cext~(Q/VQ,N) ~ Cext~(Q,N)
:
Q/SQ .
induces an isomorphism
PROOF, cf. LUE [1; P r o f .
e
is para-
be a group variety, N e ~ , and
N e ~ 0 ~ , then
and :
~
with range
nat: Q - - - ~ Q / V Q
EXTs(G/VQ,N) If
Cext(Q,N)
Let
•
Q
If
t
Since VQ , these
and e
nat*
is central,
The remaining assertion follows from Theorem
A special case of this proposition was already observed by
YAMAZAKI [1; Prop. 3 . 4 ] . The main result of this section is the following
255
7.19 THEOREM.
Let
S
el: Ni~----*Gi----~Qi
be a variety of groups and
be S-marginal extensions for
i := 1,2 .
Then the following are equivalent: (i)
eI
and
e2
are S-lsologic extensions;
(il)
there is another S-marglnal extension
together with S-lsologlc epimorphisms
e: N;
(oi,~i,~i):
~Q
;G
e - ei
for
i := 1,2 ; (ill) there exists an isomorphism SM(~) Ker S8.(e I) = Ker Se.(e 2) If
G1
and
G2
in
are finite, then
~: Q1 " Q2
such that
SM(Q 2) . G
in (ii) may be chosen
finite, too.
By this theorem and Remarks 7.11, the set of ~-isologism classes of S-marginal extensions with factor
Q
is in biJective corre-
spondence with the Aut(Q)-orblts of the Q-subgroups of
~M(Q)
.
As
an immediate corollary, we can also decide whether two groups are S-isologic or not.
LEEDHAM-C,REEN/McKAY [1; pp. 113-114] have
further results related to
7.20 LEMMA. ~: Q " Q2
N~
~ ~G
(1,1,~-I):
morphisms and ~e = ~e ''~n
PROOF.
e
e' - e
.
consider
~" ~ Q 2 and
(1,1,~): e - e'
Ker se.(e') = SM(~) Ker Se.(e) for all n-letter laws
The existence of (1,1,~)
SS.(e) = SS.(e') SM(~)
7.10.
~(iii)
as in (7.1) be a S-marginal extension and
an isomorphism,
e' : Then
Let
(i)~
are S-lsologic .
epi-
Moreover
v E V .
is obvious and implies
by the naturality assertion of Proposition
[]
This lemma allows one "to pull an isomorphism into the extension"
256
and reduces the proof of Theorem 7.19 to the following
characteri-
zation of special isologism.
7.21 THEOREM. for
i = 1,2 .
Given S-marginal
eI
(ii)
there is a U-marginal
and
e2
are special U-isologic
epimorphisms
(iii) Ker Ue.(el) If
G1
finite,
and
G2
ei : consider
and
extension
(oi,Ti,1):
are finite,
e - ei
then
G
extensions;
e: N~ for
= Ker ~e.(e 2) 2 ~M(Q)
=
Given S-marginal
Ni~
~ Gi
e=(el,e2):
~G
~Q
with
i := 1,2 ;
.
in (ii) may be chosen
GIAG 2
U-marginal
(7.9)
~ Q ,
=
=
oi(nl,n2)
with
~ (gl,g2) e G I x C 2 i -1gI = -2g2 e Q } and
~(gl,g2)
Ker ~e.(e I)
:
= .Ig I .
Then
e
is
N
Ker Ue.(e 2) .
= ni
(el,e2> " e i and
are evaluated componentwise, is analogous
of I.(3.3').
Ti(g1'g2 ) = gi " cf.
(5.3),
to Lemma III.2.2,
e
Since words in is S-marginal.
G 1 x G2 The last
with (7.6) employed instead
Indeed (7.9) yields
Ue.(e i) = oi.UB.(~e I,e2>) This and
~;~Q
There are obvious morphisms (~i,~i,1)
assertion
G
and
Ker Ue.(e)
PROOF.
extensions
N I x N2 ; ~
~(nl,n2) = (nl,n2)
with
~Q
too.
7.22 LEMMA.
C
el: Ni~----*G i
Then the following are equivalent:
(i)
@-isologic
extensions
•
(Ker a I) n (Ker o 2) = 0
imply the formula for Ker Ue.(e).
[]
257
7.23 PROOF of Theorem 7.21 Clearly
(ll) implies
morphisms
(1).
(oi,~i,1):
(completing
the proof of Theorem 7.19).
We now prove (iii) from (ll).
e - ei
of ~-marginal
R e m a r k 7.12 give commutative
Given
extensions by
Q ,
diagrams
Ker ~e.(e) c
; ~M(Q)
Ker Se.(el) ~
;~M(Q)
e
> VG
~-VQ
~0
)VG i
~ VQ
> 0
(7.10)
where
ai
is an inclusion and
ing that
(oi,~i,1)
that
and
~1
62
Ker ~e.(el) Finally,
are isomorphisms.
We invoke Clearly
-I ~1 ( ~ 1 N 1 )
~I
Ker ~e.(el) by Lemma 7.22.
and
~2
If
Thus
~I
z e VG
Thus
eI and
and
e2
we conclude
by
(oi,~i,1):
Q
as in
e - ei
from
are surJective with "
(iii).
epimorphisms
under
Then
is the identity map in (7.10),
and finally
(Ker ~I ) O VG = 0 . Now assume
then
~I
~2
is a
Likewise
(i) rather than (iii).
We again
To this end (cf. the p r o o f of 5.4) every
has the form
= (V(Xl,...,x n),v(yl,...,yn
, X l , . . . , X n e G 1 , yl,...,y n ~ G 2 , and
z e Ker ~1 ' then
hence
Assum-
= Ker ~e.(e) = Ker ~e.(e 2)
epimorphism.
V e VF
~2
First assume
z = v((x 1,yl),...,(xn,yn)) with
epimorphisms,
are special ~-isologic
(Ker ~1) 0 VG = 0 .
element
extensions
e = ~el,e 2)
~I
and
either assumption.
claim
~i "
Consequently,
-I = N1 x N 2 = ~2 ( x 2 N 2 )
We claim that
• -isologic
of
= Ker e I = Ker e = Ker e2 = Ker me.(e 2) .
(i) or (ill).
is isomorphic
the r e s t r i c t i o n
are special ~-isologic
given U-marginal
Lemma 7.22.
8i
v(Yl,...,yn)
V(Xl,...,Xn)
= 1
= ~2(,2Yl,...,~2Yn)
(Ker Zl ) Q VG = 0 , by symmetry
and
)) ~lXl = ~2Yl
~l(~IXl,...,,lXn)
= 1 ,
= ~ ~1(~IXl,...,~IXn ) = I (Ker z2 ) n VG = 0 .
.
258
7.24 EXAMPLE
Let
S = ~
then S - l s o l o g l s m
•
C
"n-lsocllnlsm"
in the sense
of BIOCH
two groups
and
G
H
~: Fc+IG - Fc+IH
[2]
are n - i s o c l i n i c
and
is the same as
'
.
In view of Example
if there
~ : G/ZcG - H/ZcH
exist
7.6,
isomorphlsms
such that
~c(~g1,...,~gc+1) = ~ ~c(gl,...,gc+1) By T h e o r e m group
K
7.19,
n-isoclinic
and n - i s o c l i n i c
groups
G
and
epimorphisms
H
determine
~I: K - G
and
another
~2" K - H
with (Ker ~1 ) O Fc+IK = 0 = (Ker ~2 ) O Fc+IK (This p a r t i c u l a r
result
is due to BIOCH
Example
7.7, n - l s o c l i n i s m
trolled
by certain
with common
subgroups
.
[2].)
quotient
By T h e o r e m group
Q
7.19 and is con-
of
R O Fc+IF • cM(Q) in terms
=
IF,..[F,R]...]
of any free p r e s e n t a t i o n
and the dihedral have computed
group
D8
~2M(D8)
of
Q
.
In the case of
c = 2
of order 8, L E E D H A M - G R E E N / M c K A Y
~- ~/4 x Z/2
which
carries
[I;p.115]
a non-trivial
D8-action.
7.25 EXAMPLE. sition
The reader
III.2.6(a)
does not hold:
class that does not contain as a ~-marglnal Let
~
extension
extension;
such that
of a b e l i a n
Q = Z/p
that the analogue
we give an example
a S-stem e
be the variety
n = p2 ~ 1 , let
is warned
.
Then
~
e
as in (7.1)
of a ~ - i s o l o g l s m
the latter
~8.(e)
groups
of Propo-
is surJectlve.
of exponent
is defined
is defined
dividing
by the law
n
Xl[X2,X3] when
e
, an extension is central
w i t h the obvious ~-marglnal loglsm
and
of
has exponent
free p r e s e n t a t i o n
extension
class
N
m(ep)
m(ep)
is S-marginal
by
contains
Q .
~
P
dividing as in (2.1)
Let us assume
a S-stem
extension
n .
precisely We start
and obtain
the
that the ~-isoe .
Then
259 Remark
7.11(a)
gives ep
rise to the commutative :
-m(eu)
~ 2 ,
~Q
~ 2/p3Z
~Q
p Z '~
- pZ/p3Z ~
e:
N>
diagram
II
II
~" G ,
We conclude
~M(Q) = K e r l p Z / p 3 Z and
Ker
Theorem
8.(m(ep)) 7.19
direct
= 0 .
~e.(e)
summand
of
and
generated
~ .
pY/p3Z
b)
If
G1 - Q
S
and
yarn(G1) Part
as required,
If
G1
GI
and
•
if
GI
and thus
then by
SM(Q)
be a
Contradiction.
and
G2
agrees and
are ~-Isologlc,
then the
with that g e n e r a t e d
G2
are isoclinic,
by
G2
then
.
G2 - Q
of exponent
are S - c o v e r i n g
0
and
groups
Q
of
is in
•
, and if
Q , then
= yarn(G2) (a) and its p r o o f below
is a v a r i e t y
G1,G 2 e S
PROOF.
[I; Thm.
e
by T h e o r e m
a)
lies in
4.5]
of exponent
a stem e x t e n s i o n
S
existed
~ Z/p 2 .
is a variety
to L E E D H A M - G R E E N
and
by
= yarn(G2)
e
a)
In particular,
yarn(G1)
If
= p2Z/p3Z ~ Z/p
w o u l d be isomorphic
7.26 PROPOSITION. variety
- Z/VZI
Every
by
Q
- the latter
NEUMANN
for finite
O , then a S - c o v e r i n g such that
Ker
[1], part groups.
group
If
of
Q
.
Note
8.(e) = K~(Q)
(b)
is that
6.3.
law
v
VF m , hence
as in the ~ - i s o l o g l c
are due to P.M.
group
for the variety
~
is e v a l u a t e d
G2
GI
Since
in ~
generated
by
G1
in the same way
is the trivial
function
260
(GI/W.GI)n ,
on
the law also holds in
For the final assertion that
varo(G 1)
b) ~=
GI
and
Let
of infinite
the same variety as
PROOF. R
and
G2
G
is S-isologlsm
and
G1
and
G
and
By Proposition
var(H)
the laws
and
Fm-L
erated by
G
and
If
by Theorem
divides
by
varo(G)
= varo(H)
H , respectively. by Remark
0 [F®,F®]
5.6.
= L .
number and
cf. H. NEUMANN
Q = Z/p × 7/p .
representation
groups of
p3 .
One is of exponent
group
G = G(p2,p,1+p,O)
ticular
G , var(G)
because
computations
and
and the
Then
Likewise,
var(G)
has
Then the variety gen-
L.F --R
Fn c Fm cm
--
Let
~
p
Q , viz. p , H
the non-abellan
[]
•
be an odd prime
There are two isomorphism
types of
groups
of order
say, and the other is the metacyclic
of exponent
p2 by
H
By Corollary
7.27 for
and
In par-
Z/p 2 .
.
See the papers by LEEDHAM-GREEN/McKAY successful
of
[I; p.176].
is also generated
var(H) ~ var(G)
L
has the laws
law being applicable
7.28 EXAMPLE,
generates
Let
varo(H)
CO
the modular
H
do.
n (IF ,F ].F")-- ((Fm.L) n IF ,F ]).Fn =
and
III.2.3 with
groups of exponents
n , then
sets of laws defining
(Fm.L)
Z/n
be isocllnlc
m
Z/n
R = Fn.L
, and
H
7.26(b)
generated
L = R 0 [F®,F®]
H
S .
(Set exponent = 0 if the group contains
order.)
denote the (maximal)
variety
by
are isocllnic
and
n , respectively.
elements
.
[]
7.27 COROLLARY. m
All told, G 2 E ~
recall that isoclinism
is the variety generated
The groups
IQ .
G2
of
~M(Q)
[1] and MOGHADDAM
More needs to be done!
[1] for
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INDEX OF SPECIAL SYMBOLS
Section
Symbol
Short explanation
logical equivalence monomorphism epimorphism _~, ~---~
isomorphism
[]
end of proof resp. lack of proof natural numbers
Z
integers rational numbers real numbers
IV.2.3
IV. 2.&
C
complex numbers
GF(q)
field with q elements
D
summation sign
(m,n). gcd(m,n)
(positive) greatest common divisor
0
group with one element
Dn
dihedral group of order n
Sn
symmetric group on n letters
G(m,n,r, k)
metacyclic group
Z/n
cyclic group of order n
O: G ' ~ H
trivial homomorphism
1: G " G
identity homomorphlsm
mw
m-th power map
1
neutral element (of any group) -1 = xyx
Xy
272
Section
Symbol
Short explanation
[x,y]
= xyx-ly -fl
IGI IG:UI
cardinality
[A,B]
mixed commutator subgroup
z(a)
center of the group G
z*(c)
obstruction to capability in Z(G)
Gab
commutator quotient group
Tor(G)
torsion subgroup (when defined)
N~Q
semidirect product (N normal)
GxH
direct product of groups
II.4.1
G*H
free product of groups
11.4.10
GeH
tensor product of groups
1.3.12
QAQ
exterior square
1.2
IV.(3.6)
index of subgroup
x
Gi
restricted direct product
II iei
ai
(unrestricted)
•
Gi
direct sum (of modules)
direct product
I.I .9
D e r ( Q , A , ~)
derivations
1.1.1
tt(Q,N,e)
factor sets modulo transformation
Ext(Q,A)
set of (congruence classes of) abelian extensions
Cext(Q,A)
dto. of central extensions
Opext(Q,A,~)
dto. of Q-extensions of (A,m)
1.1
congruent e~
backward induced extension
Ge
forward induced extension
II.3.1
e/U
factor extension
1.1.8
elP
extension restricted to subgroup
sets
273
Section
Symbol
Short explanation
eI
morphlsm of extensions
-
e2
I. (2.2)
e 1 x e2
product extension
1.1
a., ~°
middle maps in induced extensions
1.2.1
~ab' ac
homomorphisms induced by a
IV. 7.4
am
homomorphism induced by a
AG
fixed elements
1.3
M(G), M(~)
Schur multlpllcator
1.2.7
A*(e,A)
some connecting homomorphism
1.3.5
8.(e)
some connecting homomorphism
IV. (3.5)
U(e)
kernel of e.(e)
1.4.4
wG' ~(G)
Ganea map
IV.5.7
varo(G)
variety of exponent zero
Section
Special extension
1.3.3
e(Q) :
III.(1.1)
eG :
Z(G)~
IV.7.13
eG :
V*(G) e
II.(1.2)
~v :
K*:
~ ~GL(V)
IV.(2.1)
en :
nZ •
~Z
RQr-----,FQ---~Q
,G ---~GIZ(G) :G---,~G/V*(G)
~ ~PGL(V)
~ ~I/n
Cross references with Roman numerals refer to that chapter, without Roman numerals to the local chapter. are cited by NAME.
Items of the bibliography
SUBJECT
INDEX
abelian-b~-W abelian groups with trivial multiplicator abelianization absolutel~ abelian absolutel~-V additive associated cohomology element associated factor system autoclinism strong __
230 35,218,2~2 20 36,242 241 196 70 22 124,139 1~7
Saer-invariant Baer sum binary icosahedral group binary pol~hedral groups Birkhoff's Theorem branch factor group
250 21 118,184 184,185 228 144
calculus of induced extensions capable center-by-~ centralization closed set of laws codiagonal cohomology element commutator form commutator map commutator quotient
14 l Z ~ 2 , 2 0 ; , 209 230 20 227 21 70 36 45 19 5 17 61
congruence, congruent copair corestriction
Darstellungsgruppe > representation ~roup deficienc~ __ of manifold groups of letaczclic groups > Swan's examples, Wamsley's example derivation diagonal - - m
2
180 185 201
direct limit argument
13 22 18q. 57
efficient elementary matrix
182 120
dicyclic group
275
epimo~phism isoclinic isologic __ equivalent projectiwe representations exact sequence > fiwe-term exact sequence exponent {of variety} extension abelian central congruent s induced > induced extension in variet~ isomorphism of s marginal __ metacyclic morphism of s of V-type Q-extension split __ stem strictl X central -_ V-stem exterior s~uare extra-special p-group factor set principal normalized factor s~stem faithful irreducible faithful p-block famil~ maximal FICPE
finitely presented five-term exact sequence free presentation standard V-free presentation fundamental grou~ Ganea map generator generalized reeresentation group generalized A-representation group Golod-~afarewi~ Theorem group > see under indiwidual qualifiers
126 253
68 231 5 7
8 5 230 6 2~5,246 196 6 253 5 6 37 204 258 36,43 222 7 7
8 7 99,212 169 125 143 212 179,180 25a 3 1 , 5 0 , 2 5 0 20, 180 31 228 114,117,185 41 179
78,85 77 190
276
groups with trivial multiplicator 86,183ff,199 > abelian groups with trivial multiplicato£ Hall-Formulas P. Hall's Ine~ualit~ hinreichend erg~nzte Gruppe > generalized representation homology sphere homolog~ transgression homomorphism over a group induced extension backward__ calculus of s forwardinduced-central extension irreducible __ projective representation quotient __ subgroup __ isoclinic extensions strongly isoclinic groups p-isoclinic groups isoclinic homomorphism isoclinic morphism isoclinism n-isoclinism strong __ isologic isologism special isomorphism of extensions isomorphism over a group
152ff 181,94a 192 3 group 11~ 32 91 12 10 14 11 237 72 133 133 124 145 123 170 126 126,137 123,124 258 145 252 252 252 6 91
lifted lling in variety
73,77 230
marginal subgroup metac~clic extension metac~clic group Milnor's K~ modular representation multiplicator > Schur multiplicator > Schur-Baer multiplicator aorphism of extensions > isoclinic morphism
244,245 196 193 120 17qff
6
277
170 113
p-isoclinic groups perfect cover > universal perfect cover perfect group presentation > free presentation product central free netabelian nilpotent __ subdirect tensor projective representation e ~ u i v a l e n c e of _ _ __s faithful irreducible irreducible linearly eguivalent __ s unitary pure subgroup
221 101 106 106 213 107 67 68 99,212 72 68 100 221
~uasi-dihedral
202
group
r e d u c e d form representation > modular representation > projective representation representation group of a b e l i a n groups on 2 g e n e r a t o r s of a l t e r n a t i n g g r o u p s _ _ - _ of d i h e d r a l g r o u p s _ _ _ _ of g r o u p s of o r d e r p3 - - - - of m e t a c ~ c l i c g r o u p s - - ---- of s y m m e t r i c g r o u p s A-representation group minimal A-representation group > generalized representation group relation, relator restriction Schur-Baer multiplicator of d i h e d r a l g r o u p Schur-Hopf Formula relative Schar-KQnneth
Formula
113
102
78,86,92 95 96 95, 201 188 200 96 77 87
180 12., 3 9 , 2 1 0 , 6 0 249 258 30 234 109
278 Schur multiplicator of abelian groups of direct product of extra-special p-groups of generalized ~uaternion groups of infinite dihedral group -- __ of knot groups of metacyclic groups of SL(2,Z) of PSL(2,Z) of unicentral groups > Schur-Baer multiplicator > trivial multiplicator semidirect product set of laws situation of the centers situation of the commutator guotients socle special isologism Steinberg group stem extension V-stem extension stem group strictl~ central strong autoclinism strong isoclinism strongly isoclinic sum of extensions Swan's examples
30 44 108 223 95 105 183 200 183 105,183 209
5 227 146 146 212 252,253 120 37 258
1.34,216 204 147
145 145 21,22 186
Three-Subgroups Lemma transfer transformation set trivial multiplicator > groups with trivial multiplicator > abelian groups with trivial multiplicator twisted group algebra
41 66 8 94,187 69
unicentral gniversal Coefficient Theorem Iniversal perfect cover
20q,209 34 115,119
Verlagerung variety belonging verbal subgroup
66 227 227
U a m s l e l ' s exa=ple
to laws
187
