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Lecture Notes in Mathematics Edited by A. Dold, Heidelberg and B. Eckmann, Z0rich Series: Forschungsinstitut for Mathematik, ETH Z~irich
359 Urs Stammbach Eidgen6ssische Technische Hochschule, ZLirich/Schweiz
Homology in Group Theory
Springer-Verlag Berlin. Heidelberg • New York 1973
AMS Subject Classifications (1970): 20J05 ISBN 3-540-06569-5 Springer-Verlag Berlin • Heidelberg • N e w York ISBN 0-387-06569-5 Springer-Verlag N e w Y o r k • Heidelberg • Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin . Heidelberg 1973. Library of Congress Catalog Card Number 73-19547. Printed in Germany. Offsetdruck : J ulius Beltz, Hemsbach/Bergstr.
INTRODUCTION The purpose braist group
of
may
these
learn
theory,
Notes
something
second,
methods
are
Chapter
I introduces
able
to a c h i e v e the
II w e h a v e
of groups.
Together
Chapters
as
far as
III,
applications extensions central
with
to b e
determined
being
that
tools
are
trary
variety
in all the
by
to
homological
some b a s i c
notions
facts
in g r o u p
about
the
to the
theory.
(co)homology
[43]
this will
(co)homology
theory
for t h e s e N o t e s .
the c o r e but
of this
volume.
not entirely
in a v a r i e t y ,
on central
We present
disjoint
theorems
extensions,
~
the
functors.
The group
[26],
V
areas:
o n the
localization
here
(a g r o u p
to
H2
it
. These
say
lower of n i l -
, H2
make
their
by Hopf
In a c e r t a i n
isomorphic
to)
in o r d e r
the
guide
about
the h i s t o r y
second
H2
appearances
, H2 • of t h e s e as
Eilenberg-MacLane
however
in 1 9 0 4
line
to an a r b i -
functors
first
was
homological
generalize
[46],
to s t u d y
a mild
group
sense
that
of t o p i c s
the main
functors
something
is w e l l - k n o w n
the m u l t i p l i c a t o r
the a u t h o r ,
(co)homology
in p a p e r s
[20].
of
the c h o i c e
of applications
, V
second
functors
for e x a m p l e
introduced
areas
functor
Eckmann
complete;
the p r e f e r e n c e
functors
the p l a c e
homology
in a n y w a y
four
It m a y be
group,
theory
groups.
largely
older;
alge-
field.
introduction
kernel
theorems
see w h a t
VI of Hilton-Stammbach
it is n e e d e d
abelian
the homological of h o m o l o g y
may
the basic
Chapter
form
own
to
in four d i f f e r e n t
We do n o t c l a i m
[25],
in his
complete
IV, V, V I
series,
potent
theorist
reader
with
First,
applications
assembled
as a r e a s o n a b l e
of groups
about
the g r o u p
In C h a p t e r
serve
is t w o f o l d .
they are much
Schur
integral
projective
[72],
[73]
homology representations
iv
of a group. H2(G,A) [71],
Also,
has been known
Baer
[5],
a.o.).
have been k n o w n with
the
logical
equivalence kernel,
functor
for a h o m o l o g y
(a c a t e g o r y
furnished
Subsequently
various
theories:
Andr4
[I],
functors
, V
explicit
(co)homology
of
to the lower [54],
[55]
V
and
is in the
tried
of groups
possible,
the
group
, V
[51].
~
general
In
. A defi-
circumstances
[56]
of S t a m m b a c h
similar [9], R i n e h a r t
to the h i g h e r
none of these In S t a m m b a c h
Later,
appears
authors [78]
and a p p l i c a t i o n s
the c a t e g o r i c a l
[53],
theories
in m u c h detail. [79]. We are m a i n l y
of h o m o l o g i c a l
as possible.
makes
a new
in L e e d h a m - G r e e n
[78],
[~].
not only yield
correspond
are d i s c u s s e d
the use
[50],
Barr-Beck
of groups
applications
V
in d e f i n i n g
[8],
of groups. V
that
later been given by Beck
However,
to be as e l e m e n t a r y
where
in quite
theories
that
are given.
spirit
in p r e s e n t i n g
avoided,
Barr
for
with a variety
has
in L e e d h a m - G r e e n - H u r l e y
Our a p p r o a c h
have
groups.
series
of v a r i e t i e s
fore have
functors
for a r b i t r a r y
m a n y of the f u n d a m e n t a l
succeeded
[4],
to v a r i e t i e s
in the case
interested
have
for a v a r i e t y
central
contains
a triple)
in h o m o -
, form an a b e l i a n
Knopfmacher
these c a t e g o r i c a l
but also
references
definition
with
, V
definitions
that works
Bachmann
[83]. All
dimensional
theory
basically
fundamental
V
~
associated
authors
[68], Ulmer V
~
~ = ~b
from the b e g i n n i n g
to lie in
already
(Schreier
of a g i v e n g r o u p b y an abelian
sophisticated
[50]
theorists
for they c o i n c i d e
clear
[32],
of extensions,
for the v a r i e t y
functors
It was
the p r o p e r t y
the paper of the
, V
also,
to define
given by G e r s t e n h a b e r
properties
V
and as such are
, Tor ZI
More
classes
to g r o u p
of those e x t e n s i o n s
addition.
particular,
nition
functors
are plentiful.
that have
have been
The
Attempts
classes
Baer
for a long time
Ext~
algebra. V
of e q u i v a l e n c e
for a long time,
functors
varieties
under
as a g r o u p
ideas and
In p a r t i c u l a r ,
of h i g h - p o w e r e d
therewe
categorical
and
v homological this
s u c h as
goal we clearly
probably V
tools,
being
, leaving
had
for e x a m p l e
to m a k e
that we have had
aside
completely
spectral
certain
sequences.
sacrifices,
to r e s t r i c t
the h i g h e r
To achieve
the most
important
our considerations
dimensional
to
~
,
(co)homology
groups. We w o u l d have
Gruenberg,
texts
the
[35] or Within
[ 3 ], o r e l s e
theorems, played
consists second
lemmas,
(2.10). we
cited
The
we use
system
occurs
we will
in w h i c h
the
that
t h a t of p r e -
group
theory.
Theory,
Lecture
1970. in G r o u p
same
the
covered
Theory,
are
in an a r e a
first
refering
formula
to a t h e o r e m ,
system
two numbers
If
series
section,
2 of Chapter which
the
is l a b e l l e d formula,
item
in a d i f f e r e n t
numeral
the
V we have
or a d i s p l a y e d
provided
it o c c u r s
by a Roman
to the
for
for d i s -
of t h e s e
appears
etc.
of enumeration
same chapter.
in
one
the o t h e r
in e a c h
in S e c t i o n
a displayed
the
not covered
of enumerations,
of enumerations
Thus
either
is d i f f e r e n t .
series
the
in t h e s e t h r e e
to a m i n i m u m ;
and c o r o l l a r i e s ,
item.
to r e f e r
in the
preceed
two
of n u m b e r s ,
If we w i s h
simply use
Methods
in our N o t e s
to the p a r t i c u l a r 2.4
in G r o u p
o f the m a t e r i a l
propositions
of a p a i r
Proposition
namely,
in p u r e
Springer
the presentation
chapter
formulas.
voI0~43.
it h a s b e e n k e p t
presented
a given
Topics
Cohomological
trivial,
applications
results
of groups
i972.
intersection
is n o t
(co)homology
of o u r N o t e s ,
Cohomological
A.:
Dekker
on
of h o m o l o g i c a l
K.:
Babakhanian,
texts
to t h o s e
in M a t h e m a t i c s ,
Marcel Although
two
similar
applications
Notes [ 3]
to m e n t i o n
objectives
senting [35]
like
specifying
to b e chapter the
chapter. It is a p l e a s u r e like
to e x p r e s s
to m a k e m a n y my gratitude
acknowledgements. to m y
teachers
and
First
of all
friends
Beno
I would Eckmann,
Vl
Karl
Gruenberg,
volume
would
to P e t e r
not have
Hilton
in the a r e a
the
thank
text
Minzloff illegible
who
Hilton, been
read
without
possible.
the w h o l e
of mathematics
led to n u m e r o u s I also
Peter
as w e l l
whom Very
the w r i t i n g special
manuscript; as
in the
his
area
of t h i s
thanks expert of
are
due
advice
linguistics
improvements.
the
in t h e i r
editors series.
for h a v i n g manuscript
done into
of Springer Finally,
Lecture
my
thanks
s u c h an e x c e l l e n t a neat
Notes
for a c c e p t i n g
are d u e
job
to F r a u
in c o n v e r t i n g
typescript.
Eidgen~ssische 8006
ZUrich
Juli
1973
Technische
Hochschule
Eva my
T A B L E OF C O N T E N T S
I. V a r i e t i e s
of Groups
I. Some D e f i n i t i o n s 2. D e f i n i t i o n
in Group
Theory
of a V a r i e t y
3. Free Groups in a Variety~ in a V a r i e t y
II.
Elements
of H o m o l o g y
i. D e f i n i t i o n
of
the C o p r o d u c t
II
Theory
iI
(Co) H o m o l o g y
12
2. D e r i v a t i o n s 3. The
5-Term
4. E x t e n s i o n s 5. U n i v e r s a l
with Abelian Coefficients;
6. The M a y e r - V i e t o r i s Coproduct Theorem
III.
Extensions
in
V
5-Term
the K ~ n n e t h
Sequence
V(Q,A)
17
Kernel
with Abelian
=
i. The Groups 2. The
15
Sequences
Theorem
and the
30
Kernel
37 38
, V(Q,B)
41
Sequences
43
3. The G r o u p of E x t e n s i o n s 4. The C a t e g o r y
6. The C o p r o d u c t 7. The Change
IV. The Lower C e n t r a l
2. Free
Exact
Theorem
of V a r i e t y
8. The U n i v e r s a l in a V a r i e t y
1. The
Kernels
45
Sequences
51
of A b e l i a n
5. The C o e f f i c i e n t
for
3. S u b g r o u p s
Coefficient
Series
V
53
Sequence
55
and
Exact
Sequences
58
63 64
Theorems
of ~-free
V
Exact
Basic T h e o r e m Subgroup
25
Groups
67 70
viii 4. Splitting 5. P a r a f r e e
Groups
73
in
76
Groups
82
6. The D e f i c i e n c y 7. Groups
Given b y Special
8. The H u p p e r t - T h o m p s o n - T a t e
V. C e n t r a l
89
Presentations
92
Theorem
95
Extensions
97
I. G e n e r a l i t i e s 2. The Ganea 3. Various 4.
104
Term
Classes
of Central
iII
Indecomposables
5. Stem
Extensions
6. C e n t r a l
Extensions
8. T e r m i n a l
iO. T h e o r e m s
w i t h Local
3. U n i q u e
Roots
5. P r o p e r t i e s
129
Groups
135 140
Type
147
Groups
148 151
Homology
153 of N i l p o t e n t
of L o c a l i z a t i o n
6. L o c a l i z a t i o n
126
of Schur
Groups
2. Groups
4. L o c a l i z a t i o n
119
Groups
of the Schur M u l t i p l i c a t o r
of N i l p o t e n t
±. Local A b e l i a n
Bibliography
of the T h e o r y
of Hall's
Localization
7. A Result
o f Perfect
and U n i c e n t r a l
9. On the Order
115
and Stem C o v e r s
7. A G e n e r a l i z a t i o n
VI.
I08
Extensions
of N i l p o t e n t
of N o n - N i l p o t e n t
on E x t e n s i o n s
158
Groups
Groups
of H o m o m o r p h i s m s
Groups
164 170 173
179
CHAPTER
VARIETIES
In t h i s used
chapter
in the
of
notions
about
varieties
ble,
the
this
chapter
language
of
the
unfamiliar
chapters
that
to
the
denote
has
by
free
unique
group
to
Some
the
and
homomorphism
Most
those
even
point
of
F
of
f'
become
of
facts
possi-
contained
familiar
with
here
in t h e
are
introduce
whenever
adopted
great
that
will
in group
possi-
subsequent
advantages.
in G r o u R _ T h e Q K z
It
is w e l l
free
function
f
f
)
on
the
: S ~ G
extending
S
the
vaguely
has
we
use,
clear
groups.
: F ~ G
We
view
is c a l l e d
to e v e r y
theory
In p a r t i c u l a r
approach
category
in g r o u p
groups.
Definitions
A group
G
of
It w i l l
categorical
G{
objects.
to e v e r y
some.
notes.
theory.
functorial
1.1.
We
known
GROUPS
definitions
these
category
are well
However
be
some
parts
basic
bly
assemble
OF
later
the
theory.
we
I
f
known set
S
that ¢ F
, there
G{ if,
exists
a
.
G
/7 (I.I)
N F
Given
the
set
group
on
S
shows
that
presented Every G
up if
this to
S =
as w o r d s
group
~ F/R
S
. The
G
is
,
/
/ /
f'
/
universal
property
isomorphism.
The
(x i)
, then
i.e
, i ~ I .
finite
isomorphic
corresponding
(1.1)
actual the
sequences
short
exact
the
construction
of
free
elements
F
may
in
to a q u o t i e n t
characterizes
x I. of
and
some
sequence
of
-I xl
free
groups be
re-
' i £ I
group
of g r o u p s
free
F
. ;
R k-gF--~G
is c a l l e d
a
Given
and
G
central
(free)
(q)
presentation
a nonnegative
series
{G~}
G ~q
(1.2)
where the
we
use
subgroup
: G
for
any
of
G
as
q
that
= 0
if
we
N
simply
G oI = G
(1.3)
We
call
a group
Gq = e n
. The
Gq ~ e c
'
Gq c+i
i.l.
it
nilpotent
PROOF.
Let
potent ly, nent
and
let q c2
G
G
= e
Let
G
q
every be
in
"
> 0 of
nilpotent
of
c
the
symbol
y,z
is
the
lower
i = i ' 2 .....
G
~
q
N
to
denote
[G,G~]
,
if
qrQup
G
q
#
N
q
. Note
also
that
for
i
there (q)
=
....
exists
group
(O)
1,2
is
n ~
is
with
characterized
just
i.s n i l p o t e n t
I
by
nilpotent.
(q)
if
and
only
if
exponent.
of
~ G
class
has
class
= cl-c 2
G
~ N.
series
nilpotent
(q) x
Set
=
finite
element
, say.
, so
(q)
that
nilpotent
define
'
,
a nilpotent
. The
may
G ql
central
G °i + l
of
Note
£ G
G
lower
,
c
and
be
x
nilpotent
class
LEMMA is
the
we
by
,
get
#q
N ~ G
generated
normal
q
= G
subgroup
is
.
follows.
G;q+~
,
G
integer
xyx-ly-lz q
Note
of
cI
. It
c
exponent , say, is
then
. Then
clearly
dividing and
of
clear
q
G c
finite that
is n i l -
. Converseq
G qc + l
expo= e
.
The
upper
centr~1~
(1.4)
ZoG
where
ZG
then
Zc_IG
The
denotes ~ G
derived
,ZIG
the
center
, ZcG
luble
{G n }
is c a l l e d
lenqth
~
of
series
is as
ci ~ i
. We
setting
follows.
for
(1.6)
row
~
(c I ..... c n)
P
be
= e
. Note
of class , ~-times,
a property
group
G
exists
a normal
perty
P
Given
the
= Z(G/Zn_IG)
G
is n i l p o t e n t
of
class
is d e f i n e d
by
[Gn-I,G n-l]
exists
n
lower
by
central
Gn = e
. The
G~ ~ e
, G ~+I
= e
series
( c l , c 2 ..... c n) subgroup
with
be
and
the
a sequence
G(cl+l,c2+l
s__oo-
derived of
' .... C n + l )
integers of
G
i ~ k ~ n
G ( C l + 1 .... ,Cn+l)
Let
Let
follows
conversely.
G
the
a normal
If
(2,2 ..... 2)
. If
is c h a r a c t e r i z e d
G(cl+l,c2+l
nilpotent
G
if t h e r e
of both
define
, and
as
, ZnG/Zn_IG
of
, Gn =
G
is d e f i n e d
= ZG
of
soluble
A generalization
just
= G
G1 = G
A group
[ZnG }
= e
series
(1.5)
by
series
is c a l l e d
of
..... C k + l )
, then that
(G(cl+l,c2+1
is c a l l e d
polynilpotent
~ c
, and
is j u s t groups
that
N 4 G
if with
class
of
for
of
class
(c)
is
of class
row
row ~
polynilpotent
soluble
P
..... C k _ l + l ) ) C k + l
polynilpotent
of
(nilpotent,
res.~dually
subgroup
G
=
length
~
finite,
etc.).
every
x ~ N
e ~ x
and
~ .
~ G
G/N
Then
a
there
having
pro-
.
semi-direct
group
G
product
and
the
(left)
by
A ~ G
. The
G-module elements
A of
we
shall
A 3 G
denote are
their
pairs
-
4 (a,x)
, x
~ G
, a
and
direct
free
product
product
by
of
two
A variety under
V
GI * G2
of
taking
products law
groups
, a,a'
GI,G 2
is d e n o t e d
is a
full
quotient
of
of
objects
arbitrary
products
in
"F
, the
f
: F
~ G
Given the
a
THEOREM
We
said
v
~
(see
continue = ~
V
= N
is a a
free
in a g r o u p
laws,
hold
[64])
Every
sponding (ii)
of
(v)
with
GIxG 2
, their
Gr
which
is c l o s e d
categorical
G
law
finite
determines
group if
sequence
for
on
an
, the law , the
then
form
that
variety
is
converse
~
of
category
can
x l , x 2 ..... x n . . . . .
every
the
be
It is
groups
homomorphism
G
true,
, the
law
is
variety
described
by
varieties.
(variety)
of
abelian
theorem
a set
qroups.
of
laws.
The
corre-
<
. The
. nilpotent
qroups
of
class
c
[ x l , [ x 2 , [ x 3 .... [ X c , X c + l ] ] ] ] of
all
also.
of
of
in w h i c h
is a w e l l - k n o w n
examples
[ X l , X 2]
variety
clearly
a variety.
the
a series
corresponding V= = S ~
of
element
=C
(iii)
.
= e
(v)
2.1.
(i)
to h o l d
, f[v]
set
laws
Birkhoff
= F ( x I ..... x n .... ) is
and
v = v ( x l , x 2 ..... x n)
[v]
v
~ G
[64]).
. Clearly
law
, x,x'
by
subcategory
-i -i x i ..... X n , X I ..... x n
A
~ A
a Varlet_3 {
letters in
by
(a+xa',xx')
Definition
cartesian
in n - v a r i a b l e s
=
is g i v e n
.
subobjects,
(called
product
groups
1.2.
A
the
(a,x) . ( a ' , x ' )
(1.7)
The
e A
soluble
groups
of
length
~< ~
.
of
(iv)
V = P = = ( c I ..... c n) of
(v)
class
V= = B =q ing
(vi)
A
It
is
n V=
v
in
G
f
the
abelian
law
[xl,x2]xq
is
= A= =bq
said
we
groups
= v(x I
that
that
we
to
Xn)
may
a
: Fco ~ G
be
have
of
of
Let
(v)
group.
We
j
be
is is
. The
correspond-
V.
two
varieties
=. A. b e. q
Bq
is
= A= =bq
. The
again
, the
corresponding
exponent is
a law
and
q ~ O
of
if
exponent
and
zero
only
if
if a n d
only
if
be
the
let
obtained
the
by
image by
a l , a 2 ..... a n
be
elements
element
£ G
.
evaluating
of
[v]
f(xi)
= ai
~ F
v
at
under
( a i , a 2, .... a n )
a homomorphism
, i = I ..... n
, and
f(xj)
arbi-
> n
a set
define
follows.
of
exponent
V=
consider
defined
for
q
-
.....
a
exponent
.
. In p a r t i c u l a r , ~h
of
intersection
of
y
groups
a = v ( a l , a 2 ..... an)
trary
as
that
variety is
qroups
x~
In p a r t i c u l a r
. Then
say
Note
of
a variety.
(2.1) We
variety
obvious
it c o n t a i n s
Let
is
the variety of polynilpotent . . . . . . . . . . . .
~< (c I ..... c n)
, the
law
variety
A_ _b
row
c i ~> 1 '
It
of
the is
laws
defining
verbal
the
V
subqroup
subgroup
of
and VG
G
let
of
G
G
generated
be
an
arbitrary
associated by
all
with
elements
of
VG
de-
the
form
(2.2)
Note
v ( a I ..... a n )
that
n ~
depends
pends
on
and
not
is a n
endomorphism,
on on we
v the
have
, v
6
. We choice
(v)
shall of
, a I ..... a n
see laws
below
~ G
.
that
defining
~
. If
only g
: G ~
G
(2.3)
g ( v ( a l,a 2 ..... a n ))
Hence
the
group
VG
is
invariant,
quotient
group
G/VG
is c l e a r l y
quotient
of
lying
in
G
= v ( g a l , g a 2 ..... ga n)
V
in p a r t i c u l a r in
, more
~
it
~ indeed
it
is n o r m a l . is t h e
The
largest
precisely:
=
Every
group
through
homomorphism
f
: G ~ Q
with
Q
in
V
factors
uniquely
G ~ G/VG f
G
>
Q
(2.4) ~
G/VG We may lows.
rephrase The
this
functor
(2. ~)
is
left
from not We
adjoint
this on
to
chosen
: Gr ~ V
the that set
(ii)
Let
~ = ~c
largest
1.3.
A group
F
in
of
following
every
PG
function
the
Grou~s
= Gab
VG
of
G
P
, and
= Gc+ I that
depends
as
fol-
~ =
=
, and
property:
: S ~ G
there
To
on
V
and
is
the
just
the
abelia-
[G,G]
G/VG
every
exists
on
_ _
is n i l p o t e n t
V-free_
only
examples.
: Gr ~ A b
VG
is a p p a r e n t
.
= G/Gc+ I
is
of class
~ c
in a V a r i e t z : _ t h e _ C o p r o d u c t
is c a l l e d
universal f
functor
It
: ~ ~ ~{
VG
a couple
" Then
~
and h e n c e
with
quotient
Free
E
V
. Then
functor
G/VG
functor
defining
= Ab
nizing
theory
by
laws
Let
= =
of category
of
section
=
language
defined
embedding
this
V
in t h e
= G/VG
property
the
fact
conclude
(i)
the
P
PG
f!
"
set group
a unique
the
in a V a r i e t y
S
¢ F G
if in
it ~
homomorphism
satisfies and
to
f'
: F ~ G
extending
f
f
S
> G i-7 I
(3.1)
N
i I ~
J
f!
F ~
Given to
S
, the
above
isomorphism.
(absolutely) quired there
In
free
tending
order
to
group
universal exists
property
on
f'
F
V-free
= F(S)
set
S
. Then
for
let
f
: S ~
: F ~
G
and
group
let
F = F(S)
F = F/VF G
hence
be
a
on
has
the
function,
a unique
f'
S
up
be
the
re-
then
: F ~
G
ex-
f
f
S
_> G
(3.2)
/
~--~/v{.
Of
construct
the
the
property;
a unique
characterizes
course,
abelian,
G{-free
N
-free
"
f'
groups
are
groups
are
every
group
just
just
free,
free
Ab-free
groups
nilpotent
of
are
class
just
c
free
, etc.
=C
It
is c l e a r
that
tient
of
a V-free
exact
sequence
(3.3)
is
R ~---~ F
cabled
Later
group
on
a V-free
we
PROPOSITION if
F
the
subqroup
shall
3.1.
in
, i.e.
the
A qrou D by
V= G
may
~ F/R
be
represented
. The
as
associated
a quo-
short
---~>G
presentation
need
is q e n e r a t e d U
F
G
qenerated
G
following
F S
of
result
is V - f r e e a n d t for by
T
.
on
on
the
V-free
set
every
finite
is V - f r e e
on
S
groups.
c F
subset T
.
if T
and o__[f
only S
,
PROOF:
If
F
and
the
the
converse
subgroup
function. rained
is y - f r e e
let
To
in a
fT' : U ~ G
G
be
any
f'
: F ~ G
U
Thus
T
group
in
of
f' (x) let
x
clearly
by
f
a
T
= f~(x)
F
and
let
x
~ F
finite gives
. We
is g e n e r a t e d
is V - f r e e
~
by
to
( U.
¢ S
consider
generated
: T ~ G
. Define
well-defined.
, then
generated
subgroup fT
S
U
define
restriction
on
, i = 1,2
f
to
finite
subset
T.
of
S
. To
: S ~ G
subset
be
a is c o n -
of
S
. The
a unique
show
, where
S
prove
x
T
to
that
U.
l
the
T
. Clearly
rise
have
on
by
f'(x)
is
is g e n e r a t e d
by
is g e n e r a t e d
by
1
. Then
x
~ V
where
V
l
TI
o T2
. Since
termined
we
the
extensions
that
f'
Finally,
is
we
to
to
the
extend
S
show
ourselves
=
indeed
since
Finally
, f' T2
, f~luT 2
are
uniquely
de-
have
f' (x) Ti so
f' TI
=
f' (x) T2
well-defined.
generates
that
the
f' (X) TIUT 2
F
the
a variety
case
of
Clearly
V
definition
to
the
uniqueness
always
a coproduct
f'
of
general
has two
is of
a homomorphism. f'
is o b v i o u s .
coproducts.
objects;
case.
Let
it G.
We is
restrict
clear
how
, i = 1,2
be
1
groups
in
product)
V GI
. Their *V G2
(3.4)
GI
where
GI , G2
tions
Ji
show
Thus
let
is g i v e n
*V
: Gi ~ G I
that Q
G2
denotes
Ji
We
coproduct
GI be
*V
: Gi
*V
G2
a group
G2
~ GI
~
called
verbal
or
varietal
* G2/V(GI*G2)
free are
product
given
by
* G2
~ GI
satisfies
the
in
, (also
by
= GI
the
in
~
and
let
of the
*V G2
GI
G2
The
injec-
composition
"
required fi
and
: Gi
universal ~ Q
be
property. homomorphisms.
Then there exists a unique f : GZ *V G2 ~ Q
f' : GI,G 2 ~ Q
and hence a unique
such that the diagram
GI*G2
>
/
GI*vG 2
/
f
f2
is commutative. We finally remark that Jl
Ji
has a left inverse~ the left inverse of
is obtained by
Gi ~
_
>
Gi
(3.5)
G2
where
t
obtained.
is the trivial map. Analogously, It follows in particular that
the left inverse of
Ji ' i = 1,2
J2
is injective.
is
CHAPTER
Our
objective
of groups.
ELEMENTS
OF H O M O L O G Y
THEORY
chapter
is t w o f o l d .
First we
in this
some notation
and
assemble
We present
some basic
these
possible
to H i l t o n - S t a m m b a c h
to s t a t e
some r e s u l t s
but
II
about
facts
facts without [43],
for s o m e
reason
are n o t c o v e r e d
introductory
texts.
Of course,
in
VI.
for t h o s e
theory
whenever
second
objective
Our
that
in m o s t
results
introduce
refering
of groups [43] o r
to
the h o m o l o g y
proofs,
Chapter
the homology
about
want
we
is
are classical, of
give
the o t h e r complete
proofs.
II.l.
Let
G
be
inteqral ideal
in
by
A
of
_(Co_)Hom_olo~l{
multiplicatively.
auqmentation
We denote
e : zG ~ Z
by
zG
its
and auqmentation
e .
MOdG~
the c a t e g o r y
G-modules.
for e v e r y
(I.I)
and
rinq with
IG = k e r
of right Then,
a group written
qroup
We d e n o t e
Definition
Let
n I> O
is d e f i n e d
A
of
left
be a left
, the n - t h
and b y
and
let
cohomoloqy
M__od_~ the B
category
be a r i g h t
qroup with
G-module.
coefficients
by
Hn(G,A)
the n - t h h o m o l o q y
n = EXtzG(Z,A )
qroup of
G
,
with
coefficients
in
B
is d e f i n e d
by
(1.2)
In b o t h garded
H
n
formulas,
(G,B)
z
as a t r i v i a l
= TorZG(B,z). n
, the
additive
heft G-module.
group
of the
integers
is to b e r e -
±2
It
is c l e a r
from
(I.i)
G-modules
gives
We
(correcting
remark
rise
H n ( - , -)
may
G*
objects
The
G-module.
be
The
of
G*
G*
via
f
of
of
. The
by
induced
the
8 = f*
short
sequence
of misprints
in in
(contravariant) pairs
(G,A)
exact
sequence
(co)homology.
[43],
p.190)
functor
with
of
G
on
that
the
a group
category and
A
a
~
(G',A'
homomorphism Here
A'
f
: G ~ G'
is to b e
and
regarded
a homomorphism
as a G - m o d u l e
homomorphism
: H n ( G ' , A ,) ~ H n (G,A)
composition
is
Analogously
we may
regarded
a
as
: (G,A)
G-modules.
Hn(G,
where
are
a group
(f,~)*
is g i v e n
a
any
exact
a series as
that
morphisms
consist
: A t ~ A
(1.2)
to a l o n g
regarded
(f,~)
in
and
the
A ,)
8=f*>
obvious
define
(covariant)
H n ( G , A ,)
~* > H n (G ,A)
"change-of-rings"
a category
2,
functor
2,
on
map.
such . We
that leave
Hn(-,-) the
may
details
be
to
the reader.
II.2.
A
function
d
: Q ~ A
where
Derivations
A
is a Q - m o d u l e
is c a l l e d
a derivation
if
(2.1)
The
d(x.y)
set of
abelian
all
group
= dx+xdy
derivations structure.
from Denoting
x,y
,
Q
to this
~ Q
A
.
obviously
group
by
has
Der(Q,A)
a natural we may
i3 define
a functor
D e r (Q,-)
By
[43] ~ T h e o r e m
The
functor
natural
equivalence
9A
given by For
VI.5.1
Der(Q,-)
(2.2)
A
~{/Q
the m o d u l e
denotes A
from C o r o l l a r y more
: dx
given
a functor
in [43]
is a
,
~ D e r (Q,A) in t h e o b v i o u s
of groups
: Der(G,A)
over
as a G - m o d u l e
that
is a n a t u r a l
way
this
functor
G Q
. For
via
equivalence
~ Hom~{/Q
Q
g
g
: G ~ Q
It f o l l o w s
is c o r e p r e s e n t a b l e ~ N
of
functors
with
A ~QQ 1
by
(gG(d)) (y) =
A~Q
denotes
the
PROPOSITION
2.1.
is a free
Let
F-module
(dy,gy)
semi-direct
I.i) . As a n a p p l i c a t i o n
IF
there
with
, d
the category
there
precisely
: G{/Q ~ ~
VI.5.4
NG
(2.4)
Here
, x ~ Q
define
more
~ HomQ(IQ,A)
is to be r e g a r d e d
precisely,
(2.3)
functors
: Der(Q,A)
we may
~b.__
we have:
of
Der(-,A)
where
~
is r e p r e s e n t a b l e ~ ~
9A(d) (x-l)
fixed
: ~
, y
~ G
product
, d
of
£ Der(G,A)
Q
by
A
(see S e c t i o n
we reprove
F
be a free on
S-i
=
qroup
(xi-i)
on
S =
, i ~ I .
(x i)
, i ~ I . Then
±4
Given
PROOF:
show
that
h
F-module
there
extending that
an
f
exists
. We
uniqueness
: S ~ A
~ F
fact
Clearly, on
F
(2.4)
The
that
F
h'
, so
derivation ~
=
is
that
may to
d(s)
=
In
~
the
calculations
vatives as
is
or
follows.
in
prove
existence
(S-I)
~ A
: IF ~ A
generates
IF
of
as
consider
we
the
must
F-modules
F-module, function
,
the
be
s
s
a group
homomorphism
projection
regarded
as
a derivation
turn
~
,
s
d
A
h'
~ F ~ F
a morphism
is
in
: F ~ A
with
by
(2.2)
to
s
.
: F ~ A ~ F the
identity
Gr__/F . B y
(2.3) ,
e S
corresponds
a module
homomor-
= d(s)
required in
the
= f(s-l)
extension
F
be
of
(co)homology
derivatives free
on
,
of
f
~ S
.
groups
are
often
the
set
the
useful S =
(x i)
so
called
Fox
(see
[29]).
They
, i
e I
. Then
deriarise by
(2.2)
By
Proposition
Fox-derivative sponding
to
the
so
with
partial Let
: f
S-I
f(s-l)
%0(s-i)
Thus
f
that
yields
with
rise
: IF ~ A
function
homomorphism
(f(s-i),s)
free
h'
d
To
a
by
composed
it g i v e s
phism
note
is c l e a r . given
and
a unique
first
h(s)
The
A
NZF
: Der(F,ZF)
2
2.1
the
IF
module
is d e f i n e d i-th
to be
projection
HomF(IF,ZF)
the
is F - f r e e derivation
on
(S-I) ~. !
. The
: F ~
zF
i-th corre-
15
IF ~ i?I We c o n c l u d e
that
(2.5)
~i> ZF
(ZF) i
if
(x-l)
=
~ ~i(xi-l) i~I
,
x ~ F
then
(2.6)
~ (x) = ~. l 1
,
We m a y e x p r e s s
this
in the
PROPOSITION
Note
also,
2.2.
result
(x-i)
=
c L i£I
Oi(xj)
be a normal
summarize
~i(x) (xi-l)
= 8ije
II.3.
N
following
way.
,
x ~ F .
that
(2.7)
Let
i ~ I .
this
(3.1)
The
subgroup
information
in
i,j
~ I .
5-Term
G
Sequences
with
quotient
in the short exact
E : N ~ h> G
denote
,
g~Q
a left Q - m o d u l e
group
sequence
Q . We of groups
.
Let
A
Then
the
following
(3.2)
0
Der(Q,A)
(3.3)
0 ~ HI(Q,A)
(3.4)
H2(G,B) g ~ H 2 ( Q , B )
6*)B ®Q N a b ~ - ~ B
(3.5)
H2(G,B)
E 8*}B ®Q Nab h * ) H i ( G ,B) g*~HI(Q,B)
sequences g*> Der(G,A)
g*>HI(G,A)
and
let
are e x a c t
B
denote
(see
[43],
h* ~HOmQ(Nab,A)
h*> HOmQ (Nab ,A)
a right Q-module. p.202):
~6* H2(Q,A)
g*>H2(G,A)
6* E.~H2(Q,A) g* H2(G,A)
E
g*)H2(Q,B)
®GIG
g*)B ®Q IQ ~ 0 , ~ 0
,
,
i6 N a b = N / [ N N]
Here
is to be r e g a r d e d
a left Q - m o d u l e
via conjugation,
i.e.
(3.6)
y(u[N,N])
where
u ~ N
We n o t e t h e In o r d e r
and
x
e G
important
to m a k e
E I
= xux-l[N,N]
represents
this
statement
:
N t )
:
N ~
be a m a p of e x t e n s i o n s . morphism.
0
(3.7)
0
Hi (Q,A)
~
H I ( Q , A ,) ~
more precise
> G t
f3 ~
> G
~ Q
Moreover
let
let
~ : A ~ At
diagram
Hi (G,A)
are n a t u r a l .
~ Qt
f2 ~
T h e n the f o l l o w i n g
-"
.
fact t h a t all of t h e s e s e q u e n c e s
fl~ E
y ~ Q
be a Q - m o d u l e
homo-
is c o m m u t a t i v e .
~
H o m Q (Nab, A)
H I ( G , A ,) --
HOmQ(Nab,A')
H 2(Q,A)
t~
t ~'
~ H 2(G,A)
H2(Q,A')~
H 2 ( G , A ')
f~ 0 -~ H I ( Q ',A')
A similar
statement
(3.4) , (3.5) ~ we In these Notes
the f o l l o w i n g
,(3.8)
is true
leave
-~ HomQ, (N~b,A')
for the o t h e r
the d e t a i l s
integer w i l l simplified
H G = H (G,z) n n
sequences
(3.2) ,
to the reader.
play a central notation
,
6~, H 2 ( Q , , A , ) ~ H 2 ( G , . A , )
5-term
the h o m o l o g y w i t h c o e f f i c i e n t s
any non-negative use
-~ H I ( G ' ,A')
f~
n = O,i ....
in
B = Z/qZ
role. We
shall
with
q
therefore
±7 t'T
(3.9)
HnG
Using
the n o t a t i o n
= Hn(G,z/qz)
introduced
(3.10)
Z/qz
(3.11)
HqG
The
latter
may
With with
B
Z/qZ
, q
O , 1 , 2 .... E 6,) N / G #
When
q : O
[G,N] We
, the
for
finally
sentation
recall of
Q
#q G
.
to
the
exact
sequence
(3.2)
(or
3.3))
reads
N
q
superscript
G #o N
we h a v e
"
conventions
=
I.i
q = O,i
,
= G/G
abbreviated
= Gq ab
notational
=
in S e c t i o n
= z / q z ® Gab
H~G
these
n = O,I . . . . .
~0^ N a b = N / G # q N
e v e n be
(3.12)
,
q
h,
Gq ab
~
g*>
Qq ab
m a y be d r o p p e d ,
> O
"
and we m a y w r i t e
. Hopf's
, then
formula.
sequence
O ~ H~Q ~ R/F
Let
R >---~ F
(3.13)
--->~Q
be a free p r e -
reads
#q R ~ Fqab ~ Qabq ~ O
so that we o b t a i n
(3.14)
For
H~Qz i F ~q F
q = O
this
formula
II.4.
Let
N 2--> G
is g i v e n short
} Q
a Q-module
exact
sequence
be
n R/F
is due
structure
o
to H o p f
Extensions
a short
#q R
with
exact by
[45].
Abelian
Kernel
sequence
(3.6).
Let
of groups. A
be
Then
a Q-module.
Nab The
18
(4.1)
E
:
A ~>
is c a l l e d
an e x t e n s i o n
structure
of
in
A
Et :
. The A>
A
G
of
defined
extension ) G'
g?) Q
Q
by
by
E
the Q - m o d u l e
(3.6)
agrees
is c a l l e d
~ Q
if t h e r e
A >--9 G
---~) Q
A
with
the o n e
equivalent
exists
if the Q - m o d u l e
to the
f : G ~ G'
already
given
extension
such that
the d i a -
gram
(4.2)
II
fl
A)
rP
) G' --9>.Q
is c o m m u t a t i v e . It
is w e l l - k n o w n
i.e.
the
correspondence
establishing
[43], the
this
p.207) . In the
extension
(4.1)
(4.3)
Then
6~
:
the c o h o m o l o g y
equivalence
class
(4.4)
We
shall
H2(Q,A)
set of e q u i v a l e n c e
one-to-one A
that
AlE]
We p r o c e e d : A ~ A'
the
5-term and
[E]
may be
A
described
we h a v e
of
Q
Q
by
by
H2(Q,A)
in c o h o m o l o g y
the Q - m o d u l e
class
of
set underlying
sequence
HomQ(A,A)
extensions
of extensions
correspondence
as
A
A
,
is in
. The map
follows
associated
(see
with
the h o m o m o r p h i s m
~ H2(Q,A)
AlE]
= { £ H2(Q,A)
of the e x t e n s i o n
E
associated
with
the
is g i v e n b y
= [ = 6~(1A)
map defined
with
classes
with
s h o w at the end
the c l a s s i c a l
classifies
a number
of this
section
by means
o2 f a c t o r
of assertions
be a homomorphism
that
about
of Q-modules,
the m a p
A
agrees
sets. naturality. and
let
Let
with
19
E'
be
an e x t e n s i o n
PROPOSITION
:
A' >
with
4.1.
n[E']
There
E
:
:
if a n d
PROOF.
first
Suppose
commutative
f'
f'
~,(~)
(4.5)
~'
diaqram
~ H 2 ( Q , A ')
is c o m m u t a t i v e .
6: > H2(Q,A )
HomQ(A,A')
6~ > H 2 ( Q , A , )
now
that
: G ~ G'
. To
Then
(3.7)
yields
II
~.(~) ~,(~) do
=
so w e
H 2 ( Q , A ')
= ~' ~'
. We
have
to
show
the
an extension
construct
existence
of
Q
by
of a map the
Q-mo-
A '
set
that
G T
bedding then
=
H O m Q (A ,A)
E
We
the
diagram
immediately
du le
that
>> Q
onlyif
H o m Q ( A ' ,A') 6 ~ ' >
Suppose
have
II
~* ~
whence
such
we
))Q
> G'
that
. Then
: G ~ G'
f' I A' ~
is c o m m u t a t i v e
~ H 2 (Q,A')
~'
> G
~ E'
=
>> Q
exists
A>
(4.5)
the
> G v
easy
:
A' ~
= A' ~ G / T is a n o r m a l A' to
where
and
that
g'
> G
subgroup.
~ A' ~ G check
h'
%'
T = The is
>> Q
.
{(~(-a) , h ( a ) ) } map
h'
induced
is by
. It
induced A'
is e a s y by
~ G ~ G
the g> Q
to
see
emIt
is
20
f
induced
by
the
: G ~ G
embedding
E
:
G ~ A'
A> c~l
E
By
the
first
:
part
that
A[E
E'
, so
that
] =
~'
we
Notice be
many
sitions We
we
maps 4.3
finally
f'
obtain
claim
with
the
that
~
of Let
)
obvious
course
Proposition ~ : Q ~ Q
.
AlE
=
we
have
]
follows
that
E
is e q u i v a l e n t
to
a map
(x
~ G'
that
>
is u n i q u e ; properties,
in
fact
there
general
will
(see
Propo-
G
c~
G
property.
universal
property
4.1. be
in
square
universal
this
f~
required
the
f A'
Of
diagram
.
>
~
reader.
a commutative
£I
not
remark
an
yields
4.1.
. It
> G
V.6.1.)
A
satisfies
)>Q
G
properties.
do
and
# G
: G
required
that
"
= A[E']
f
the
f cl I
= ~,(A[E])
indeed
f'
with
---9> Q
Proposition
~.(~)
so
) G
A')
of
~
a homomorphism
and
let
Details could
are be
left
used
in
to t h e the
proof
21 be
an e x t e n s i o n
PROPOSITION
with
4.2.
There
E
is c o m m u t a t i v e
PROOF:
First
AlE]
:
> G
only
suppose
commutative
exists
A )
if and
= [ ~ H2(Q,A)
that
f
: G ~ G
. Then
we
have
such
that
the
diaqram
-->> Q
if
[* (~)
(4.6)
= [
is c o m m u t a t i v e .
Then
(3.7)
yields
a
diagram
HomQ(A,A)
6~
> H2(Q,A)
Horn@ (A ,m whence
immediately
Suppose
now
: @ ~ G
Here y
G~
~ Q
and f~
g :
~* (~)
= [ .
that
~*(~)
= [ . We have
. To do
so w e
construct
is t h e
subgroup
with
g(x)
is
induced
G[ ~ G
=
of
£(y) by
the
consisting
maps
projection
induced
by
E e :
A)
> G~
>>
E
A~
>G
>)Q
h
is G×Q
the projection
of
By
the
first
part
of P r o p o s i t i o n
existence
4.2
~ Q
GxQ
we
all
induced . It
~ G
diagram
:
the
of
a map
an e x t e n s i o n
G×Q
. The
to s h o w
have
by
(x,y)
~ G
,
A ~ G ~ G×Q
is e a s y
yields
, x
to s e e
that
a commutative
22
£*(~) sO t h a t
= £*(A[E])
~[E e] = ~ = ~[E]
so t h a t w e
indeed
obtain
: ~ G
with
the required
Again 4.3
will
and only
There
E
e
>~
determined,
is n o t h i n g
G -->) Q
in g e n e r a l
else but
. Of course
(see P r o p o s i t i o n
the pull-back
the universal
of Proposition
o f the
property
of
4.2.
Let
:
A>
>~
>>~
:
A'>
> G t
>> Q
exists
f : G ~ G'
c~.(~[~;]) if
(4.7)
correspondence
PROOF:
If
we use
the p r o c e d u r e s
f
=
holds~
to-one
construct
to
makinq
the d i a q r a m
commutative
if
if
(4.7)
Moreover,
f
in t h e p r o o f
E t
be qiven.
G~
and
4.3.
is e q u i v a l e n t
properties.
be used
PROPOSITION
that
a map
~
that
~ : Q ~ Q could
It f o l l o w s
not be uniquely
). W e r e m a r k
maps G£
f
= AlE e]
exists
the diagram
with
~,*(A[E']) f : G ~ G'
the h o m o m o r p h i s m s the d e r i v a t i o n s
then clearly
(4.7)
o f the p r o o f s
d
holds.
are
in o n e -
: Q ~ A'
To p r o v e
of Propositions
4.1
the converse and
4.2
to
,
23
lJ
Now
we
have,
E '£
:
A'~
E'
:
A' >----> G'
by
> G,~--9>
--->> Q
(4.7),
A[~
so t h a t
IJ
] = ~,(A[E])
is e q u i v a l e n t
to
=
~*(a[E'])
E 'e
. Then
= A [ E '~ ] ,
f
may
be
defined
as c o m p o -
sition
f
If
d
fl(x)
: Q ~ A'
: G ~ G c~ ~ G '6 ~ G'
is a d e r i v a t i o n ,
= d' (x)-f(x)
d'
is a g a i n x,y
~ G
(4.8)
a group
are
fi(x-y)
is c l e a r
where
~ A '
homomorphism
d'
: ~ ~ G'
is d e f i n e d
defined
by
by
~G' inducing
= d' ( x - y ) - f ( x . y )
~
and
~
. To
prove
this,
let
that
that
proof
d' (x)
= d' ( x ) - I f ( x ) - d '
(y)" ( f ( x ) ) - l ] - f ( x ) - f ( y )
fl(x)"fl(Y)
fl
two homomorphisms
shows The
:
~ G
fl
. Then
=
It
, x
then
induces inducing
~
and ~
= f l ( x ) " (f(x)) -I
of. P r o p o s i t i o n
4.3
and
~
. Conversely, e
induces
is t h u s
then
the
f,fi:G
calculation
a derivation
complete.
if
d
: Q
~ G' (4.8) A'
24
Finally
we recall
by means
of
as defined
factor by
E
abelian
with
gs
sets
(4.4).
(4.9)
with
the description
: A>-~h> G
of representatives
The
function in
show
that
where
cocycles, by
4.4.
Consider
(4.10)
[e]
(see
with
then construct exact
the e x a c t
(sx)
a function
,
set.
x,y
~ Q
~ Q
is j u s t a set
~
: Q×Q ~ A
interpreted
the normalized
(see
[43],
that
different
p.216-217).
the c o h o m o l o g y [57],
a function
by
.
It m a y b e
denotes
and
, x
i.e.
sections
class
[~]
as an
standard It is e a s y s
yield
~ H 2 (Q,A)
p.lll).
sequence
the e x t e n s i o n homomorphisms
u>~IQ (4.9) ~I
,[,~oI
(see
' ~2
rows
~%o2
-1
B'
02
(4.11)
A
= A[E]
A ) ~ > ZQ ® G I G
associated
. Define
form
so t h a t
a section,
that
a factor
is a c o c y c l e
defined
PROPOSITION
G
= sxsy(s(xy))
is c a l l e d
~
our map
extension
be
. Note
in
in i n h o m o g e n e o u s
cohomologous is w e l l
~
the
with
kernel)
g~Q
= e
Q
HomQ(B~,A)
resolution
PROOF:
of
(with a b e l i a n
this description
s : Q ~ G
s(e)
e(x,y)
element
to
Let
and
(4.10)
and r e l a t e
We c o n s i d e r
kernel.
= IQ
of e x t e n s i o n s
II
[43],
such
Theorem
that
VI.6.3) . We may
the d i a g r a m
with
25 is c o m m u t a t i v e .
Note
that
the c o m p a r i s o n
theorem.
(4.12)
~l[X]
(4.13)
~2[xly]
To prove
the e x i s t e n c e
of
91
T h e y m a y be c h o s e n
= I ® (sx-1)
,
by
to be
x ( Q
,
= sxsy(s(xy)) -1 : 9(x,y)
commutativity,
is a s s e r t e d
' 92
,
x,y
~ Q
consider
= 1 @ [sx(sy-1)-(s ( x y ) - l ) + ( s x - 1 ) - ( s x s y ( s ( x y ) ) - l - 1 )
(9102-K92) [xlY]
]
= 1 ® [ (sxsy(s (xy)) -I-I) s (xy) - (sxsy(s (xy)) -I-I) ] = I ® [(sxsy(s(xy))-l-l)(s(xy)-l)
But this hand
is zero,
side,
(4.11)
where
since
(sxsy~(xy)]-l-1)
it o p e r a t e s
as zero.
] .
m a y be m o v e d
Applying
to the
HomQ(-,A)
left
to d i a g r a m
we o b t a i n
... ~ H o m Q ( Z Q @ G I G , A )
h*
/
HomQ(A,A)
H2(Q,A)
•
°
(4.14) ~2 >
... ~ HomQ(B~,A)
where
the upper
(4.14)
sequence
it is then c l e a r
A[E]
thus p r o v i n g
II.5.
Let dule.
(5.1)
C
Universal
be an a b e l i a n Then
there
is part of the 5-term
sequence
(3.2).
~ O
From
that
= 8~(I A) = [9~(1A) ] = [9]
Proposition
[ ]> H2(Q,A)
HomQ(B~,A)
~ H2(Q, A)
,
4.4.
Coefficientsi
group,
are n a t u r a l
regarded exact
0 ~ Ext(Hn_IQ,C)
the K ~ n n e t h
as trivial
sequences
~ Hn(Q,C)
Theorem
left or r i g h t
([43],
p.222)
~ Hom(HnQ,C)
~ O
Q-mo-
,
26
(5.2)
0
These
sequences
Both our
of t h e m
HnQ
are
(5.2)
but
we
5.1.
Z/qZ-module.
splitting
need
a generalization
q
the
be a n o n - n e q a t i v e
following
sequences
0-" Extlz/qz(Qqb,C)E
H2(Q,C)~
(5.4)
"'" ~ T°r2z/qz,~ aqb
~ Q aqb ® C -
S 9
hV
We ) F
only prove g
i
~ Q
o
(see of
(3.13)).
,c)
a free
in g e n e r a l .
of
the
sequences
inteqer
and
let
H]Q
6. k
homology
Set
S/F~q
of
K = im h~
(5.4) of
s
5-term
are e x a c t
For (5.1),
C
be
a
and n a t u r a l .
z,~ T o r
>H2(Q,C)
presentation
Q
sequence
being and
g:>
ab
with
, so that we h a v e
0 -" H~Q -~ S / F ~
(5.6)
0 -~ K -~ F qab -~ Qqba -~ 0
We a p p l y
Homz/qz(-,C)__
= Homz/qZ
•
dual.
Let
let
Q aqb
-o
coefficients short
exact
z/qz sequences
0
Hom(Qqb,C)
i
(5.8)
0 ~ Hom(K,C)
where
we have
~
S ~ K -~ 0
to t h e s e
, Ext 1 = EXtz/qz
(5.7)
q
we
Hom(Faqb,C)
~ Hom(S/F#qS,C)
a zero
at the
A
sequences.
right
Using
the a b b r e v i a t i o n s
obtain
~
Hom(K,C)
~ Exti(
~ Hom(HqQ,C)
hand
end
of
"
/qz(Q]b,c)
Z/qZ-modules
(5.5)
Hom
sequences.
Hom(H2qQ,C ) -,. Ext2"z/qz(Qq~'C)aD ~
(5.3) , the p r o o f
be
b e the a s s o c i a t e d
exact
is n o n - n a t u r a l ,
(5.3)
PROOF:
coefficient
~ O .
n = 2
Let
Then
Z' > Tor (Hn_IQ,C)
Hn(Q,C )
the u n i v e r s a l
the
shall
in d i m e n s i o n
PROPOSITION
~'>
called
split,
applications
® C
q ,C) ~ O Qab
~ Extl(K,C)
(5.7)
since
~
Fq ab
' ...
is
'
"
o
27 free over
Also, more
Z/qZ
. Using
(5.1)
H o m ( F q b , C ) = Homz(Fab,C)
= HI(F,C).
it follows
following
we have
= HI(Q,C)
it is e a s y
Fq ab
n = i
H o m ( Q q b , C ) = HOmz(Qab,C)
to see that
Hom(S/F~qS,C)
from the long e x a c t
ExtI(K,C)
since
for
is
Extz/qz-Sequence
~ Ext2(Q~b,C)
Z/qz-free.
= HOmQ(Sab,C)
. Further-
that
,
We m a y c o m p i l e
the above
information
in the
diagram
0
0 -" Hom (Q~b'C)
-," Hom (F~h ,C) ~ Hom(K,C) II
II O --
O
~ Extl(Qab,C)
H I(Q,C)
--
H I(F,C
HOmQ(Sab,C)
6* H 2 (Q,C) -->
Hom (H2qQ, C)
=
Hom (HqQ, C )
~
Ext
~ O
(5.9)
Ext
where
the second
It is t r i v i a l is exact.
line
that
E
The p r o o f of
We are of course
is the 5-term
interested
(K,C)
cohomology
is m o n o m o r p h i c (5.3)
i
q (Q b ~C)
sequence
(see
(3.3)).
and that the right m o s t c o l u m n
is thus complete. in the c a s e where"
q ,C) Ext 2Z/qz (Qab
2
Extlz/qZ(K,C)
= 0
28
z/qz,~q
T °r 2
for t h e n
sequences
for e x a m p l e
LEMMA i__ff
5.2.
q Qab
PROOF:
K
q ~ 0
= O
become
exact.
. The
short
However
Z/qZ-module
This
is so,
if,
we h a v e
K
is p r o j e c t i v e
if and
only
is. --
It
jective
module.
that
z/qz every
K
is p r o j e c t i v e ,
sequence
Let
z/qz-projective,
free
q ~ i
and
q Qab
(5.6)
and
z/qZ-module
summand
is i n j e c t i v e
in a free m o d u l e
5.3.
is s e l f - i n j e c t i v e ,
it is a d i r e c t
Thus
summand
COROLLARY
that
follows
is p r o j e c t i v e
direct
(5.4)
Tor~/qZ(K,C)
Z/qV-projective.
It is w e l l - k n o w n
K
Q aq b
(5.3),
is
Let
noetherian. if
~Uab ,C)
let
in a free,
C
K
and h e n c e
(5.6)
is p r o j e c t i v e . and
(of course)
is i n j e c t i v e .
sequence
splits,
and
Now in-
splits.
As
Conversely,
if
is p r o j e c t i v e ,
be a z / q z - m o d u l e .
If
Q aqb
also.
-is -
then
H2(Q,C)
~ Hom(H~Q,C)
H2(Q,C)
~ H~Q ® C
,
(5 .io)
The
following
homomorphism with
N
~
(3.13)
(5.3).
and of
N E
.
yields Let
some E
additional
: N >---) G
a Z/qZ-module.
information
~ Q
Then
on the
be an e x t e n s i o n
the h o m o l o g y
5-term
reads
H ~ G ~ H~Q 6 ~ > N
(5.11)
~ Gqab ~ Qqab ~ O
.
then have
PROPOSITION
PROOF: and
in
central
sequence
We
proposition
We
5.4.
consider
construct
H(A[E])
a free
the d i a g r a m
= 5 E.
:
HqQ
~
presentation
N
.
E'
: S >
h'
> F
g'
;~Q
of
a
Q
29
E'
:
(5.12)
) F
sl E
By
S>
(3.7)
:
it g i v e s
))Q
I
11
N )----> G ---97 Q
rise to a c o m m u t a t i v e
0
of 5 - t e r m
sequences
~ H~Q ~ S / F ~ q S h~)Fabq ~ Qqab ~ 0
.t
,;
s' I,
H q G ~ HqQ bE---->N
It f o l l o w s
diagram
~ ~
,
G q ~ Qq ~ 0 ab ab
that E b, = s ' I k e r ( h ~ : S / F # S ~ Fq~ ) q aD
We m a y
then read off from d i a g r a m
(5.11)
~6~,(s,)
It r e m a i n s
of 5 - t e r m
arising
from
E
6*E t (s')
sequences
b{
HOmQ(Sab,N)
with coefficients
,
N
A[E]
Thus
is c o m p l e t e .
the p r o o f
> H2(Q,N )
~ Hom(S/F#qS,N)
(5.12)
= b~(l N)
s t a t e the K N n n e t h be
the
[I 6* E' > H2(Q,N)
HomQ(Sab,N )
G = GI×G 2
To do so w e c o n s i d e r
(5.10)
s*~
N e x t we
= ~[E]
in e o h o m o l o g y
HomQ(N,N)
But n o w
that
= 6.
to s h o w that
diagram
(5.9)
their d i r e c t
, so that we
= b~,s*(iN)
Theorem. product.
Let Then
indeed o b t a i n
= 0~, (s')
GI,G 2 there
b e two g r o u p s
and
is a n a t u r a l
exact
let
30 sequence
(5.13)
(K~nneth-sequence~
O ~
[43],
sequence
splits,
The M a y e r - V i e t o r i s
with amalgamated
to o b t a i n
the c o p r o d u c t Let
GI,G 2
Sequence
G
the
We w i l l
the c o p r o d u c t
theorem
be
Denote by group
subgroup.
and
let
the free p r o d u c t
U . It is w e l l - k n o w n
(co)homology
then
of
U
Theorem
of a free p r o d u c t
specialize
index
to free p r o d u c t s
Also, we w i l l
generalize
sets.
be a s u b g r o u p of
GI
~ O
in g e n e r a l .
and the C o p r o d u c t
theorem.
to a r b i t r a r y
two g r o u p s
T o r ( H i G 1 , R2 ) ~G
but n o n - n a t u r a l l y ,
In this s e c t i o n we f i r s t c o n s i d e r
in o r d e r
p.223)
® HiGI~H~G2~ ~ HnG ~ ~ i+k=n i+k=n-I
The K ~ n n e t h
II.6.
see
and
G2
GI
and
with amalgamated
G2 sub-
that
h1 u
~
>
c~
h2Y
(6.1)
g~[ g2
02 is a p u s h - o u t gi
: Gi ~ ~
PROPOSITION G-module.
>
diagram
in
and h e n c e
6.1.
Let
A
Gr
. Also,
gihi
: U ~ G
be a left
T h e n t h e r e are e x a c t
0 ~ H °(G,A)
it is w e l l - k n o w n are
injective.
G- module
sequences
g*) H ° ( G I ,A) @9 H ° ( G 2,A)
that the m a p s
and
let
B
be a r i q h t
(Mayer-Vietoris)
h* > H o (u,A)
~ H i (~,A)
~
. . .
(6.2) ... , Hn(~,A)
... ~ H n ( U , B )
g*> Hn(GI,A)
h, ) Hn(G1,B)
@9 Hn(G2,A)
@) Hn(G2,B)
h * > Hn(U,A)
~ Hn+l(a,A),...
g, ~ H n ( ~ , B ) ~ H n _ I ( U , B ) . . . .
(6.3) ... ~ H I ( ~ , B ) ~ H o ( U , B )
h , > Ho(Gz,B)
(~ Ho(G2,B)
g * > Ho(G,B)
~ O
31
where h.
g*
=
=
{g[,g~}
, h*
=
; and
g.
=
,
{hl. ,-h2. } .
PROOF:
We
first
show
that
zB %
(6.4)
is b o t h
monomorphic.
and
property
>
a push-out
IG
square
it is e n o u g h
in
to s h o w
Modthat
. In o r d e r hi,
in
to p r o v e
6.4)
is
But we h a v e
IU = zG ® G i ( Z G i
is a s u b g r o u p
since
g2*
@G2 IG 2
ZG ~
U
®G1 IGI
z~
gl. ~
a pull-back
the p u l l - b a c k
phic;
~
h2.l Z~
Since
hl*
iu
G.
of
Gi
the
is a s u b g r o u p
IU)
@U
map
of
G
IU ~ IG 1•
zG i %
, tensoring
with
zG
is m o n o m o r over
G.
1
shows
that
In order versal
hi.
is
to p r o v e
property.
~.1 : ZG ®Gi maps
l
~i,~2
(6.5)
IG.l -- M yield
(6.4) let
M
a
is a p u s h - o u t be
~lhi.
with
(unique)
also.
the d e r i v a t i o n s
fi
: G i ~ M ~ G. l
with
= ~2h2 .
pair
and
we
verify
the u n i -
let
be given.
of d e r i v a t i o n s
By
(II.2.2)
di : Gi ~ M
the with
.
dl,d 2
correspond
G. ~ M ~ G I
square,
any G-module
dlh I = d2h 2 : U ~ M
(II.2.3)
square
that Thus
By
the
monomorphic,
~ G. i
l
the
to g r o u p identity.
homomorphisms Of c o u r s e ,
32
hi --~
U
fl G1
)
M ~ GI
h21
f2 ~ M 3 G2
is commutative. map
Since
(6.1)
f : G ~ M ~ G . Also,
push-out
property of
Applying
(II.2.3)
: IG ~ M
(6.1)
and
M ~ G
is a push-out it follows that
(II 2.2)
square we obtain a (unique)
from the uniqueness
G ~ M ~ G ~ G
part of the
is the identity.
again we obtain a
(unique)
map
satisfying
~'1 Thus
~
(6.4)
=
~gi*
: zG ®Gi
is both a pull-back
IG
1
~ M
,
and a push-out
i
=1,2
.
square. Now consider
the
diagram
(6.6)
0 ~ ZG® IU
{hi* '-h2* ~ -~ zG®
{hi* '-h2* } -> ZG@GIZG i ® ZG®G2ZG 2
O ~ ZGQuZU
{hi.,-h2. }
I
o ~ z~%z
are exact.
the top row is exact.
isomorphic
Since
> IG ~ O
-> ZG ~ O
I
® zS®G2z
> z
(6.4)
and a
is a pull-back
The second row is exact,
~ o
since it is
to
o ~ z~
(i,-i}> zB @ z~
It follows
that the bottom row is exact,
Hom~(-,A)
and
subgroup
IS 2
~
> zB®~Iz
Obviously all columns push-out
IG I @9 zG®
V
of
B ~ G
--
<±,I>> z8 ~ o .
also. Now apply the functors
to the bottom row of
we have
(6.6). Using that for any
33
~ V z,A)
(6.7)
ExtG(ZG
= EXtv(Z,A)
= Hn(V,A)
,
(6.8)
TornG(B P ZG ~V Z) = T o r Vn ( B , Z )
= Hn(V,B)
• '
we conclude and
t h a t the r e s u l t i n g
sequences
are
just
(6.2)
(6.3).
COROLLARY let
A
6.2.
Let
G = Gi , G2
be a left G - m o d u l e ,
coproduct
injections
(6.9)
g*
(6.10)
g,
Moreover
gi
and
be let
: Gi ~ ~
: Hn(G,A)
the free p r o d u c t B
GI
be a r i q h t G - m o d u l e .
' i = 1,2
-~ Hn(GI,A)
Hn(GI,B)
of
@9 Hn(G2,B)
induce
and
G2
T h e n the
isomorphisms
@~ Hn(G2,A)
,
n >I 2 ;
-~ Hn(G,B)
,
n >i 2 .
the s e q u e n c e s
0
H ° (G, A)
g* > H ° ( G i,A)
Hi(G,A)
0 -- HI(GI,B)
@9 H ° (G 2 ,n) ~ A
g * > HI(GI,A)
@9 HI(G2,B)
B ~ Ho(GI,B)
are
long e x a c t
@9 HI(G2,A)
~ 0
g*) HI(G,B)
• Ho(G2,B)
g*) Ho(G,B)
~ O
exact.
PROOF:
Use
(6.2) , (6.3)
Hn(e,A) A
for
=
0
=
U = e
(e,B)
H
for
that
n >i i .
n
We r e m a r k
that
if
Corollary
6.2
is true for
and
B
are t r i v i a l n = i , also.
of this
fact to the reader.
We
zations
of b o t h P r o p o s i t i o n
6.1
sets.
and n o t e
We w i l l be c o n t e n t
G-modules,
the a s s e r t i o n
We leave t h e o b v i o u s
finally note
that there
and C o r o l l a r y
6.2
proof
are g e n e r a l i -
to a r b i t r a r y
to state and p r o v e e x p l i c i t l y
of
the
index
34
generalization
PROPOSITION ~et
of C o r o l l a r y
• G. be the free p r o d u c t of (G.) , i ~ L l i i~I be a left G-module~ and let B be a r i q h t G-module. T h e n the
A
coproduct
6.3.
Let
G =
injections
gi
: G
(6.11)
g*
: Hn(G,A)
(6.12)
g,
:
In order
repeat
in the proof b e l o w
Proposition We first
~ G
i -~
i £ I
induce
n Hn(Gi,A) ieI
~ Hn(Gi,B) i£I
PROOF:
n >~ 2
2 Hn(G,B )
to keep our p r o o f s
,
n I> 2
as t r a n s p a r e n t
some a r g u m e n t s
isomorphisms
o
as possible,
a l r e a d y used
we will
in the p r o o f
of
6.i.
show that
the c o p r o d u c t
(6.13)
Given
6.2.
injections
g*
(6.13)
it is enough induce
: Der(G,M)
we obtain,
-7
using
Hom~ (IG ,M) ~
to p r o v e
that
for any G - m o d u l e
M
an i s o m o r p h i s m
H Der(Gi,M) i£I (II.2.2)
H HornG IGi ,M) i~l i i£1H H o m ~ ( Z G
@Gi
IG i,M)
= H o m e ( i ? I zG ®Gi IGi,M) Hence
we m a y c o n c l u d e
that the c o p r o d u c t
injections
induce
an isomor-
phism
(6.±4)
g,
Applying n i> 2
:
@9 zG ®Gi i~I
ExtG-I (- ,A)
yields
(6.12).
, n i> 2
IG i ~ IG yields
.
(6.11) ; a p p l y i n g
Tor~_i(B,-)
,
35
We
now
prove
derivation In
order
(6.13). d
to
d I. : G I. ~ M
The
: G ~ M
with
construct , i e I
homomorphism
an be
given.
homomorphism
G~ ~ M 1
Using
the
property
the
homomorphism
-- M ~
G
with
Gi ~ M G ~ M
obtain
a derivation
yields
an
is
inverse
complete.
d of
coproduct
inverse
group
universal
the
of By
~ G. 1 of
the
: G ~ M
. It
(6.13).
the
each
G. ~ M 1
free
~ G
the
in
let
(II.2.3)
G
g*
is g i v e n
by
injections
g*
with
~ Gi ~ M
~ G ~
g*
produ~zt there
identity. is e a s y Thus
gi
the
: Gi ~ ~
1 we
corresponds
~ G
the
1
"
is a u n i q u e
see
that
proof
of
to
a
identity.
conclude
may
Applying to
a
derivations d.l
~ G
composing
that
homomorphism
(II.2.3) this
we
procedure
Proposition
6.3
CHAPTER
EXTENSIONS
In this
chapter
variety
~
we define
and d e d u c e
The definition duce
exact
ordinary these
of
stantiated
that
in the
direct
product
in
in e x a c t l y
Gr
V
~Q~Q
the
V(Q,-)
VMOdQ
A ~ Q
and examples
remaining
in g r e a t e r properties. sequences ordinary
are
V
(co)homology. the c o p r o d u c t
deduce
a change universal
with
In S e c t i o n
5-term
a
as
This
It
point
is s h o w n
H2(Q,A) clear
) is n o t
Q-modules
of view
that
dein
from
at l e a s t
formally,
is s u b -
if the
classifies
semi-
extensions
extensions
the natural
M~{Q
A
apparent
classifies
that
2 we
sequences
correspond,
V(Q,A)
. In S e c t i o n
domain
o f the
but
the
full
for w h i c h
the
semi-direct
4 various
in
subcategory
characterizations
of
given. of
this
5 we
that correspond
of
3.
chapter
In an e l e m e n t a r y
In S e c t i o n
analog
prove
in
sections
detail.
V(Q,-)
, V
then
it b e c o m e s
i.
to t h e
functor.
V
same w a y
of t h o s e
is
V
Section
is in
(and
consisting
product
The
A 3 Q
associated
II.3) . It b e c o m e s
functors
important
. As a c o n s e q u e n c e
functors
analogous
group
, V
in S e c t i o n
(see S e c t i o n
(co)homology
V
KERNEL
properties.
is g i v e n
the
ABELIAN
functors
their basic
that are
(co)homology
second
IN V W I T H
the
, V
sequences
sequences
to the
V
III
prove
to the
theorem
of v a r i e t y coefficient
manner
with
exact
6 we
sequences
and for
functors
V
their
(co)homology
relevant
for
~
sequences , V
the
II.6.2) . In S e c t i o n
finally, V
, V
, V
of c o e f f i c i e n t
establish
(Corollary
sequence,
the
we e s t a b l i s h
the e x i s t e n c e long
In S e c t i o n
deal
in S e c t i o n .
7 we
8 we
in
38
Most
of the
ticular, 2,
results
we w a n t
to m e n t i o n
5) , B a r r - B e c k
Leedham-Green up to our
presented
[9]
[53],
~
the
(Sections [54],
elementary
The
be an a r b i t r a r y
following 2,
[55]
approach
III.l.
Let
in this c h a p t e r
are
a left Q - m o d u l e
intention
is to d e f i n e
to be
and
and b y
B
~ Ab= ,
(1.2)
V(Q,-)
: MOdrQ
~ Ab
=
on m o d u l e s
presentation
let
A,B
of
Q
are g i v e n
= ker(f*:H2(Q,A)
(1.4)
V(Q,B)
= coker(f,:H2(F,B)
We h a v e
to s h o w
tation
f : F
--9>Q . We do this
V(Q,B)
being
dual.
: F'
)~ Q
exist
f'
maps
h,h'
be
(Sections
8).
Ideas
in S t a m m b a c h
3, 4),
leading
[78],
[79].
Q
be a g r o u p a right
in
~
. By
Q-module.
A
Our
these
groups
another
making
as
follows.
Let
f : F ~ Q
be
, then
V(Q,A)
Let
[50]
(Sections
.
(1.3)
that
found
[I]
In p a r -
functors
: MOd6Q
a V-free
Andr~
5, 7,
we denote
V(Q,-)
values
2, 4,
known.
G[~_~i~x~Lx_~IQa~i
(I.I)
Their
papers:
Knopfmacher
(Sections
variety,
we denote
5),
are w e l l
~ H2(F,A))
~ H2(Q,B))
do n o t d e p e n d
for
V-free
the t r i a n g l e
,
V(Q,A)
on
only,
presentation
the c h o s e n the p r o o f
of
Q
presenfor
. Then
there
39
F
<
h' h
F'
(1.5) Q
commutative.
It f o l l o w s
that
H 2 (Q ,A)
(1.6)
/ H 2 (F ,A)
h'*
<
>
H 2 (F ' ,A)
h* is c o m m u t a t i v e . ker
f'* ! k e r
The
effect
: A ~ A' define
~.
Hence f*
of
ker
, thus
V(Q,-)
= ker
proving
~.
h*f'*
ker
on homomorphisms
be homomorphisms.
Then
.
0 ~ V(Q,A)
~ H 2(Q,A)
~ H 2(F,A)
(1.7)
o - Q(Q,A')-
H2(F,B)
H2(Q,A') ~
~ H2(Q,B)
H2(F,A ')
~ v(Q,B)
~ O
i
(1.8)
9.1
9.1
H2(F,B')~
We
remark
that
for
~*~v
H2(Q,B')~
V = Gr
we
(1.9)
V(Q,A)
= H2(Q,A)
(i. IO)
V(Q,B)
= H2(Q,B)
f'*
and
conversely
equality.
, V(Q,-)
, p : B ~ B' and
f*
have
,
V(Q,B')~
0
is as the
follows.
following
Let
diagrams
4O
so t h a t the
V
second
PROPOSITION an__~d B
generalizes homology
1.1.
group
For
functor.
all V-free
V(F,A)
Next we
study
We a l s o
qroups
group
note
F
functor
and
the o b v i o u s
and
V
result
for a l l F - m o d u l e s
A
: 0 = V(F,B)
the behavior
first variable.
the d u a l i z a t i o n
in
and
let
A
to
be
the
V
, V
with
. Thus
let
a left Q'-module.
property
respect
our discussion V
f : F --9~ Q
The universal
of
We r e s t r i c t
reader
that
cohomology
we have
(1.11)
V
the s e c o n d
, f'
: F'
of ~-free
to
g
V
to m a p s leaving
: Q - Q'
Choose
be
in t h e to the
a homomorphism
V-free
presentations
a map
h
--9)Q'
groups
yields
: F ~ F'
such
square f
F
>> Q
hl
~g f,
F'
is c o m m u t a t i v e .
)) Q
Hence we
obtain
0 ~ V(Q',A)
the
commutative
diagram
~ H 2 ( Q ' ,A) ~ H 2 ( F ',A)
l
(1.12}
g* I O ~ V(Q,A)
with It
exact
rows
defining
is n o w o b v i o u s
full
subcategory
and
A
reader
to
that VG*
a Q-module formulate
g*l
h*~
~ H 2(Q,A)
~ H 2(F,A)
g* V(-,-)
of
G*
: V(Q',A)
~ V(Q,A)
may be regarded consisting
(see S e c t i o n
the a n a l o g o u s
as a f u n c t o r
of pairs
II.l) . A g a i n w e statement
for
(Q,A) leave
V(Q,B)
on the
with
Q
it to the
in
41
We
finally
trivial
remark
module
the obvious,
but
the d e f i n i t i o n
of
important g*
in
fact that
(1.12)
for
makes
A
a
V(-,A)
into a
functor
(1.13)
V(-,A)
Similarly,
if
B
(1.14)
: V
is a trivial
V(-,B)
to
section
module,
we h a v e
The
5-Term
Sequ_ences
we e s t a b l i s h
5-term
sequences
for
V
and
V
analogous
(II.3.2) . . . . . (II.3.5) .
THEOREM in
2.1.
V . Let
Then
there
Let A
E : N )h>
G g-~)Q
be a left Q - m o d u l e
are e x a c t
be an e x t e n s i o n and let
B
0 ~ Der(Q,A) g*> Der(G,A)
(2.2)
0 ~ HZ(Q,A) g*> HI(G,A)
(2.3)
V(G,B)
g*>V(Q,B)
6 E* ) B ® Q N a b
(2.4)
V(G,B)
g*)V(Q,B)
6E * ) B @H~ IN a(b G h*> ' B ) u
PROOF:
Let
: F---~>Q
(II.3.2)
f : F --~)G
and columns.
be a riqht
with
G
Q-module.
the
h*> H O m Q ( N a b , A ) 6 ~ > ~ ( Q , A ) g*> V(G,A)
h* > H O m Q ( N a b , A )
be a V-free
is a ~ - f r e e
we o b t a i n
of qroups
sequences
(2.1)
gf
a functor
Ab
: V ~
IIi.2.
In this
Ab
~
h,> B@GI G g * > B @ Q I Q
of
Q
commutative
g*bV(G,A)
~ 0
g~Hi(Q'B)
presentation
presentation
following
6* ~V(Q,A)
of
. Using diagram
,
,
,
~ 0
P
G . Then the
5-term
with
exact
sequence rows
42
0
O
V(Q,A)
g*>V(G,A)
zn i /
(2.5)
0 ~ Der(Q,A)
~ Der(G,A)
/
I
1
W
HomQ(Nab,A) ~H2(Q,A) g% H2(G,A)
~
(gf) "I
~f*
H 2(F,A)
It is t h e n o b v i o u s V(Q,A)
6~ _ : HOmQ(Nab,A)
and t h a t s e q u e n c e
We leave lity
that
it to the r e a d e r
(see
(II.3.7))
L a t e r on we when
B
=
.
Using
H 2(F,A)
factors
through
is exact.
to f o r m u l a t e
and to give
shall n e e d
Z/qZ
(2.1)
~ H2(Q,A)
=
the e x a c t
statement
of n a t u r a -
its proof.
sequence
(2.3)
the n o t a t i o n
(or
(2.4))
introduced
in the s p e c i a l
in S e c t i o n
II.3
case and
writing
(2.6)
V q G = V(G,Z/qZ)
we have the e x a c t
(2.7)
sequence
vqG
g*> v q Q
6E * > N/G
Let
R >h_h> F
g)> Q
e
q
N
h.) Gq ab
g*)
q ~ 0 Qab "
Next we note
COROLLARY
2.2.
be a v - f r e e
presentation.
(2.8)
V(Q,A)
= c o k e r (h* :Der (F ,A) ~ H O m Q ( R a b , A ) )
(2.9)
V(Q,B)
= ker(h,:B
®Q Rab ~ B ~F IF)
In p a r t i c u l a r ,
(2.10)
V q Q = (R n F #
q
F)/F #
q
R .
,
Then
43
PROOF: fact
This that
immediately
that
We
conclude
C
be
V(F,A)
with
an
a trivial
tation
follows = 0
R ~---> F
--~> Q
0 ~
(2.12)
O ~ V(Q,C)
It
is
apparent
Q-module!)
~
Q
we
this
that
under
(2.14)
V(Q,C)
= Tor~(C,Q)
scribe
a group
those
E
G
E
in
is
in
V
V
. It to
as
group
Let by
~[E]
=
(II.4.4). ~ H2(F,A)
The
GrouR
in
and
let
A
Q
by
A
V=
E
Q
be
free
in
V= . L e t
abelian
presen-
sequences
-- H o m ( R , C )
~ ~(Q~C)
given
0
~ O
,
.
hypotheses
(C
a trivial
by
is
is c l e a r is is
f
: F
the
that
in in
of
Extensions
be
a Q-module.
We
want
to
de-
g~Q
this
~ ~ H2(Q,A) Let
of
: A ~h>G
. If
equivalent abelian
the
,
III.3.
extensions
(3.1)
with
let
i.e.
exact
the
using
have
= Extlz(Q,C)
be
a V-free have
(2.3)
I.II).
~ C ® R ~ C ® F ~ C ® Q ~
~7(Q,C)
Q
(2.1),
= A_b___ a n d
~ Hom(F,C)
(2.±3)
Let
(see
Choosing
Hom(Q,C)
from
we
Let
of
(2.11)
sequences
= V(F,B)
example.
Q-module.
from
V ~
case if
, also.
E
we
shall
is
Note
in
say V
that
if
that
, then E
is
the
extension
every in
extension ~
, then
A
, also. be
the
---97 Q
2-cohomology be
a ~-free
class
associated
presentation
of
with Q
E
. Define
44
=
where
: H 2 (Q,A)
f*
PROPOSITION
3.1.
and
is
A
~ F
PROOF:
f*(~)
The
in
Consider
f
A>
:
Gf
that
in
V
universal t
Gf = A ~ suppose
F
image
, so
that
that
Gf
is
that
~ = is
in
3.2.
Let
PROOF:
Proposition
morphic
-->>Q image
the of
~ =
V=
COROLLARY
: F
only
if
~ = 0
in
V=
E
of
G×F
the
= 0 A
then,
is
V-free
extension
and
Ef
and
A
~ F
is
the
group
>> Q
be
in
V=
group
and F
then
. It
follows
~ F
is
in
~
G
hence
in
Gf
is
yields
a
that V
. Then
. Conversely Gf = A
, being
an
; then
A
~ F
,
epimorphic
also
: A ~
3.1
> G
we
A 3 F
} .
, then
f*(~)
. But
group A 3
the
= O
V
Q
£ G×Flgx=fy
in
of
f*(~)
is
By
if and
V
by
property
Gf
Gf
f
of
G
: F ~
in
~f g>>
{(x,y)
suppose
splitting
~
is d e f i n e d
Now
. The
is
state
))F
A>h>G
Gf =
so
Gf
~
Pi
that
E
then
diagram
(3.2)
Recall
may
.
:
E
. We
extension
V
the
E
H 2 (F,A)
-
, is
conclude
F
is in
in V
in
that V
V
for
. Thus
also.
a y-free A
~ Q
~
Q
is
i n V.
presentation
, being
an
epi-
45
THEOREM (i)
3.3.
If
Let
A ~
Q
Q
is n o t
is o u t s i d e
(ii)
(ii)
.
is
in
A ~ Q
are
classified
Assertion
we
Since A
as
is
V
by
= O
definition
V
that
a Q-module
in
V
be
only
interest
cateqory
sider
f
A of
with
: G ~ Q the
be
f
in
~
then that
The
E
Corollary
3.2.
: F --~Q
be
it
follows
conclude the
that
extensions
E
: A >---> G --9>Q
: A~---> G ---~Q
in
V
For
the
a V-free
statement presentation.
by
Proposition
E
is
in
V
~
are
classified
by
seen
in S e c t i o n
3
in
3.1 if a n d
only
~ H 2 (F,A))
Cateqory
let be
is
Q
the in
of A b e l i a n
be
in
kernel
V of
V
. There
subcategory
VMOdQ
A ~ Q
abelian
.extension
V(G,A)
can
full
a Q-module.
extensions
from
: H 2 (G,A)
and
A ~ Q
in t h e
Q-modules
Let
if
A
be
every
the
Let is
. We
is
a variety
A
V(Q,A)
, so
III.4.
Let
, then
Q V
let
, then
is c l e a r
ker(f*
which
and
V
follows.
in
~ = f*(A[E])
V
in
) A ~ Q
~ F
in
by
(i)
proceed A >
that if
V
I ff
PROOF:
be
in
~
. The
Kernels
. We have an
extension
is t h e r e f o r e of
M~Q
category
E
: A>
~ G --~Q
considerable
consisting VMOdQ___~
of
is c a l l e d
those the
kernels. a homomorphism
diagram
G
(4.1)
If A ) h>
A ~ Q
g)> Q
(not n e c e s s a r i l y
surjective)
. Con-
46
By
(II.2.3)
(see
correspondence vations
d
: G ~ A
if
A
~ Q
vanish
is
on
derivation
d
PROPOSITION
4.1.
every VG
Now
consider
: G ~ A
A
[v]
f
~ F
then
V(A
~ Q)
PROPOSITION in
every
Using 4.2
~ Q
is
~ Q
is
with
gf'
given
conclude
on
VG
f'
from
and
: G ~ A
(4.2)
. We
may
thus
, then
for
every
, every
= f
deri-
by
homomorphism
V
a one-to-one
.
may
in
there
being
~ F
every
we
: G ~ Q
case
is
: F
, v
derivation
~
that
special
~ Q
every
every
v
f
the
If
for
every
be
, x
vanishes
I__ff A
: G ~ A
correspondence
, then
xl,x2,...
(v)
that
~
VI.5.4)
derivation
d
that
~ Q every
state
qroup
G
: G ~ A
and
vanishes
.
generators ~
f'
(dx,fx)
, so
homomorphism
on
=
in
VG
Corollary
maps
, the
f' (x)
must
v
[43],
between
(4.2)
Now
also
e
d
V
. Then
f
: F
Let the
~ Q
the
in
V
(v)
= e , so
4.2.
Let
~ Q
: F
where
variety
be
every
it
be
Q-module
if
d for
on
all
is
in
V
defined
by
the
laws
is
derivation
in d
by
every
f
VMOdQ ~ A
if
: F , v
have
thus
and
(v) only
vanishes
4.1
vanishes
~ F
~
on
laws
Proposition ~ A
v
group
the
: F
. We
=
: F
by
Iv]
I Q
A
Gr-free
defined
derivation
vanishes A
, the
follows
. Conversely,
that
every
= F
, then
~ A
V
G
on
~ Q ~
(v)
,
proved
and
let
if
for
on
[v]
Q
£ F
(v) Fox-derivatives as
follows.
(see
Section
II.2)
we
may
rephrase
Proposition
,
47
COROLLARY in f
V
4.3.
. Then
the
: Fco ~ Q
as
zero
Let
the
on
A
PROOF:
Suppose
phism
II.2.2)
V
be
defined
Q-module
A
elements
by
is
the
in
f0i[v]
laws
~M~{Q
e ZQ
v
if
, v
~
~
and
(v)
(v)
and
only
if
let for
, i = 1,2 ....
Q
be
every act
.
f
(4.3)
: F
HA
~ Q
is
: Der(Fco,A)
given.
Then
-T H o m F
we
have
a natural
isomor-
(IFco,A) CO
associating : IF
~
with A
the
derivation
defined
~(x-i)
= dx
By
II.2.2
we
(4.5)
dx
where f
we
have
: Fco ~ Q
that
dx
only v
if
e
all
by
(4.7)
~Q
the
ZvQ
the
is
homomorphism
Co
the
free
on
derivations zero we
zero
d
on
thus
A
A
ZvO : z Q / i 0
.
ZFco ~
~ fOi(x)~(xi-i) i=i ZQ
induced
by
(xi-l)
, i = 1,2,...
it
follows
: Fco ~
A
if
f0i(x)
that
elements
if
and
only
A
is
f0i[v ]
in
_VMo%
of
zQ
v
generated
£
(v)
, f
if a n d
, i = 1,2, . . . .
.
ideal
ring
=
.
, i = 1,2 .....
quotient
map
conclude
the
on
2-sided
f0i[v]
~ F
denote
: Fco ~ Q
as
x
= q;( ~ O i ( x ) ( x i - i ) ) i=l
to
as
4.2 f
operate
(4.6)
by
all
the
have
IFco
operate
for
(v)
Denote
and
f
Proposition
,
~(x-i)
used
for
i = 1,2,... Using
=
. Since
= O
: Fco -- A
by
(4.4)
Proposition
d
by
: F
all
~ Q
,
48
We
may
then
COROLLARY
rephrase
4.4.
The
our
result
Q-module
A
as
follows.
is
in
if
VMOdQ
and
only
if
A
is
a
so
that
?~Q-module.
Next
we
give
(i) (ii)
y
some
examples:
= G{
. Of
course,
= Ab
. Obviously
a module
in
VMOdQ
versely,
if
A
abelian,
so
consists
precisely
= Bq
(iii)
sider
the
(4.8)
is e a s y
extension
a trivial A
is of
in
the
a by
f
~ Q all
( ZF
~M~dQ trivial ~q
(iv)
V
= N
is c e n t r a l , Q-module.
then
. It
A
follows
~ Q
Con-
= AxQ
that
is
VMOdQ
Q-modules.
is
x~
, so
that
we
have
to
con-
, i = 1 , 2 ....
that
+ 2 +xq-i i xi+xi+ . . . .
=
: F
~ Q
is
i = I
. Thus
we
see
elements
of
. We
i ~
,
allowed that the
l+a+a2+...+a
(4.zo)
~
a trivial
0
Now
= M~d=Q
Q-module,
in
defining
to v e r i f y
0i[xq]l
(4.9)
~M~dQ
elements
Oi[X~]
It
case,
is n e c e s s a r i l y
is
law
this
every
that
. The
in
to ~Q
send is
xI
the
I
to
an
ideal
of
.
arbitrary ZQ
element
generated
form
q-I
,
a
~ Q
.
claim
=C
PROPOSITION
PROOF:
We
4.5.
proceed
Let
by
V
= N
, then
induction
on
ZvQ
c
= ZQ/(IQ) c
For
c
= I
, example
(ii)
49
estab].ishes show
the
that
(4.11)
IvQ ~
v =
be
the
result,
law
IQ c
y
Then
we
=
V
= N
for
any
where
we
have
derivation
Now
let
f
: F
fdy
~ IQ c - I
that
indeed
In
, and
particular
the
~ Q
. We
first
.
: F
~
zF
-1 -I -1 - xlyx I dXl-Xlyxl y dy
fact
be
by
fd[xl,Y]
(4.i4)
d
(i-xiYxli)dxi
used
~ 2
. Set
d[Xl,Y ] = dXl+XldY =
c
=C
[ x 2 , [ x 3 .... [ X e , X e + l ] . . . ] ]
have
(4.~3)
= ~-- " L e t
3 ..... [ X c , X c + l ] . . . ] ] ]
=
(4.12)
~1
. Let
[Xl,[X2,[x
defining
for
any
Lemma
+ xi(l-yx~iy-l)dy
that
d ( z -i)
= -z-ldz
homomorphism. 4.6
below
By
, z
~ F
induction
f(l-xlYx~
i)
£ IQ c
, so
£ IQ c
we
may
set
d
=
~.
, i = 1,2,...
proving
1
I Q ~ V
(4.15)
Next
we
show
(4. i6)
[v]
IQ c
that
=
a homomorphism
IQ c ~
~Q
. To
do
[ x i , [ x 2 ..... [ X c , X c + l ] ] ]
f
: F
: F
~A
~ Q
with
so w e
~ F
f ( X e + i)
consider
,
= e
£ Q
, and
the
derivation
(4.17)
aO
where using
~
c+1
: ZF ~ A
(4.13),
is g i v e n
by
~(I)
= a
6 A
. We
then
have,
5O
(4.i8)
~c+l[V]
= f ( l - X l Y X l I) O c + i X l + f ( x l ( l - y x - i y -i)) ~ c + i Y = O + f(xi-i)~c+iY
since
0c+iX i = O
(4.19)
~c+l[V]
so
Since
. By
induction
we
obtain
IQ c ~
Formulas
to prove
LEMMA
If
)
~ Q
may
be
chosen
arbitrarily
it
~ F
proceed
by
c 9 2
y : [x,z]
and
(4.2i)
c
, then
prove
(l-y)
induction . We may
, x
l-y
.
the
proposition.
the
y
Let
IvQ
(4.i5)
It r e m a i n s
We
c
.
that
(4.21)
4.6.
= f ( ( x l - l ) (x2-i) ... ( X c - l ) ) a
f ( x I) ..... f ( x
follows
form
= e
= f ( ( X l - l ) (x2-1) ... (Xc-l)) ~ c + l ( X c + l )
~Oc+l[V]
trivial.
f(y)
that
(4.20)
PROOF:
and
~ F
, z
= l-xzx
-i
on
~ IF c
c
obviously £ Fc_ i
z
~ For
c = i
concentrate
the on
assertion elements
is of
the
. Then
-i -i
-I
=
(zx-xz)x
z
=
((i-z)(i-x)-(i-x)(i-z))x
-i -I
Since l-y
We
we
may
assume
inductively
that
i-z
z
£ IF c - i
, it f o l l o w s
~ IF c
conclude
this
section
with
the
following
general
result.
that
5i
P R O P O S I T I O N 4.7. The c a t e q o r y of trivial Q - m o d u l e s p r e c i s e l y of the qroups
PROOF:
If
is in
~ n ~
versely, then in
A
if
is in
in
VMOdQ
~ n ~b
in
=VM__~Q consists
•
, then c l e a r l y
A
r e g a r d e d as abelian group
for it is an abelian subgroup of a g r o u p in A
is in
A S Q = A×Q
V
n
Ab
and the Q-module
, w h i c h c l e a r l y lies
in
~ . Con-
structure is trivial,
~ . It follows that
A
is
VMOdQ
Note that
it follows from P r o p o s i t i o n 4.7
if and only if proved using
ZvQ
is a z/qz-algebra.
formula
that
~
This may,
is of e x p o n e n t of course,
q
also be
(4.9).
III.5. The C o e f f i c i e n t Exact S e q u e n c e s
In this section we shall deduce exact c o r r e s p o n d to the long exact
sequences
for
(co)homology sequences
~
and
V
that
in o r d i n a r y
(co)-
homology. Let
V
be any v a r i e t y and let
P R O P O S I T I O N 5.1. The functor
Q
be
in
Der(Q,-)
In fact~ there
is a natural e q u i v a l e n c e
(5.1)
~A : Der(Q,A)
PROOF: Using
(II.2.2)
V .
: VMOdQ ~ ~ ~
~ HOmzvQ(~Q
and the fact that
F
be V-free.
Then
with
®Q IQ,A)
A
HA ~A : Der(Q,A) ~ HomQ(IQ,A)
C O R O L L A R Y 5.2. Let
is representable.
is in
-7 Hom
ZvF ~
VMo~
Q(~Q
IF
we have
~Q IQ,A)
is a free 7vF-module.
PROOF: The proof is the same as the proof of P r o p o s i t i o n II.2.1. Of course, we have to use
(5.!) and the fact that
A ~ F
is in
V . We
52
will
omit
THEOREM
the
5.3.
VMod
and
Then
there
details.
Let
let
> A
A'
~ A"
) B
B' >
are e x a c t
~ " 9 B"
be a s h o r t he
a short
exact
exact
sequence
sequence
in in
~M~.
sequences
O ~ Der(Q,A')
~>Der
(Q ,A)
~*>Der(Q,A")
~-~
(5.2) (l !
~g(Q,A')
V(Q,B')
/I~V(Q,A)
~)V(Q,B)
~*~g(Q,A")
~*)V(Q,B")
°9
(5.3) --~> B' %
PROOF:
Let
f : F--9> Q
IQ ~
B ®Q IQ ~ B "
be a V - f r e e
®Q IQ - O
presentation
of
Q
and
consider
the d i a g r a m
V(Q,A')
~
V(Q,A)
~
V(Q,A")
/
-- D e r ( Q , A>" )
... -- Der(Q,A)
(5.4)
f*i
first
f*
: Der(Q,M)
vertical and show
note
maps
that
~ Der(F,A")
wF = O
of
that
Den(F,-)
f*~
~ H 2 (F,A)
~ H 2 (F,A")
-WF - ) H 2 (F,A')
M
5.2
- _V_MO_dQ ~ Ab
@F
ZvF @F
is an e x a c t
the
two
to p r o v e
sequence
5.1
~ HOmzvQ(ZvF
the m o d u l e
so that
In o r d e r
the r e s u l t i n g
- H 2 ( Q , A '')
the m a p
is m o n o m o r p h i c ,
. By P r o p o s i t i o n
Den(F,-)
and b y C o r o l l a r y
f*l
are m o n o m o r p h i c .
the e x a c t n e s s
that
f*l
for a n y Q - m o d u l e
~ Der(F,M) f*
~ H2 (Q,A)
f*l
... ~ Der(F,A)
We
w Q> H2 (Q,A')
(5.2)
left m o s t
the e x i s t e n c e it is e n o u g h
to
we have
IF,-)
IF
: V=MOdQ ~ A b
is Z v F - f r e e .
functor~
this
It f o l l o w s
implies
that
of
53 wF = 0 . The proof
for the s e q u e n c e
Note
the a n a l o g o u s
that
sequence
in
that d e s p i t e
~Q
(5.3)
is dual;
sequences
(not
VMOdQ)
the e x i s t e n c e
it m a y t h e r e f o r e
associated are not,
with
a short e x a c t
in g e n e r a l ,
of the s e q u e n c e s
be omitted.
exact.
Note
also
(5.2) , (5.3) , the f u n c t o r s
v (Q,-) : ~ are not,
in general,
derived
III.6.
Let
G.l ' i = 1,2
tal p r o d u c t
THEOREM
6.1.
the c o p r o d u c t
functors
The C o p r o d u c t
be groups
in
(see 1.3.4) , i.e.
Let
A
be in
injections
in the m o d u l e
Theorem
~_
and
induce
V(GI
(6.2)
V ( G L , B ) ~ V ( G 2 , B ) -T V ( G I
PROOF: fi
We o n l y prove
: Fi ~ Gi
' i = 1,2
V(FI (6.3)
T-free
~ V(GI'A)
the p r o o f
, be y - f r e e
* F2)~
the v e r t i c a l presentation
maps of
be their
in the category'
B
be in
varieV •
V M o d ~ V . Then
,
*V G2'B) (6.2)
being
presentations.
are induced
h
dual.
Consider
Let the d i a g r a m
>> Fi *V F2 fv ~
> Gi * G2
GI *V G2
let
of
f~
* G2)~
GI *V G2
e V(G2,A)
Fi * F2
f'~ V(GI
where
(6.1),
.
isomorphisms
(6.1)
*V G2'A)
let
and
[M~Q
for V and V
their c o p r o d u c t
VMod V
category
by
. Set,
g>> Gi *V G2 fl,f2
. Note
for short,
that
f V
is a
54 G = G I . G2
' GV
= GI
*V
G2
'
F = F1 , F2
' FV
= FI
*V
F2
"
(6.4)
Diagram
(6.3) gives rise to a diagram of 5-term sequences
0 ~ Der(G v,A) g ~
Der(G,A)
~ HOmGv((VG) ab,A)
6% H2(Gv,A)
f~l
f*l
f'*~
f~l
(6.5)
O ~ Der(F V,A) h*> Der (F ,A) ~ HOmFv((VF) ab,A) Since
A
is in
derivation tion
d' : G
V
vanishes on
VG , thus giving rise to a deriva-
since
A
is in
~ Der(G,A)
VMod__Fv , we have
h* : Der(F V,A)
-T Der(F,A)
This implies that both homomorphisms Since
f'
Corollary
that every
~ A . Hence we get an isomorphism g* : D e r ( ~ , A )
Similarly,
f*l
6*> 1{2(FV,A) h*) H ~ , A )
V M o d ~ 7 , it follows by Proposition 4.1
d : G ~ A
g*) H~G,~
in (6.3) II.6.2
is epimorphic,
f'*
6*
in (6.5) are monomorphic. in (6.5)
is monomorphic.
By
we have
I{2(G,A)
~ H 2(G I,A) ~ H 2(G 2,A)
,
H 2(F,A) -T H 2(F I,A) @9 H 2(F 2,A)
so that (6.6)
Since
ker f* = ker f [ e
Ji : Gi
GV
as well as
1.3.5) we conclude that
ker f~ = V(GI,A) Ji' : Fi ~ FV
@9 V(G2,A)
have left inverses
(see
55
both
split.
g*
: H2(~,A)
~ H2(G,A)
h*
: H 2 (Fv,A)
~ H2(F,A)
Since
f'*
V(Gv,A)
is m o n o m o r p h i c ,
= ker(f~
coproduct
(6.6).
injections,
III.7.
In this
s e c t i o n we
considering
the
with
. Thus
V
S i n c e the
c W
~ H 2 (F,A))
e V ( G 2,A)
isomorphism
this c o m p l e t e s
,
is c l e a r l y
Exact
shall d e d u c e
sequence
let
an e x a c t
V(-,-) V
¢ W
and and
induced by the
the proof.
The C h a n q e of V a r i e t z
functors
that
~ H2(Fv,A))
: H 2 (G,A)
= V ( G I,A)
latter b y
it f o l l o w s
: H2(Gv~A)
ker(f*
the
,
W(-,-)
let
Q
Sequence
that a r i s e s
from
for two v a r i e t i e s
be in
W
Suppose
V,W B
is
a Q/VQ-module.
PROPOSITION
7.1.
There
TQ,B Moreover, V
TQ, B
is a n a t u r a l
: W(Q,B)
is s u r j e c t i v e
transformation
-~ V ( Q / V Q , B ) .
if
B
is in
VMOdQ/vQ
or if
Q
is in
°
PROOF:
Let
is a W - f r e e Consider
F --9> Q
be an ~ { - f r e e
presentation
the d i a g r a m
and
presentation
F/VF--9> Q / V Q
of
Q . Then
is a V - f r e e
F / W F --~ Q
presentation.
56
H 2 (F/WF ,B)
~
H 2 (Q ,B)
~
0
W ( Q ,B) I
6Q,B H 2 (F/rE ,B)
~
H 2 (Q/VQ ,B)
~
V (Q/VQ ,B)
O
(x
(7.1)
B ~F
(VF/WF)-->
B ®~
ab
(VQ)
B ®F I(F/WF)
B ®Q IQ
~
¥~
B
%
I(F/VF)
B
%
0 Plainly
TQ, B
then by L e m m a surjective in
~
since
yields 7.2
it e a s i l y
, then
as w e l l follows
VQ = e
, and
it is i n d u c e d by the
It r e m a i n s
to p r o v e
L E M M A 7.2.
Let
B
the
be
I(Q/VQ)
0
a natura[
~
ab
transformation. as
¥
that
are
TQ, B
it is o b v i o u s i d e n t i t y of
If
B
is in
isomorphisms.
Since
~
is s u r j e c t i v e ,
also.
If
that
~Q,B
r
~MOdQ/vQ
H2(Q,B)
. Then
B ~Q IQ J B QQ I(Q/VQ) .
PROOF:
For any
A
in
Der(Q,A)
natural
in
A . Using
~ / V Q
~ Der(Q/VQ,A)
(II.2.2)
HomQ(IQ,A)
Since
A
is in
Proposition
,
we o b t a i n
~ H o m Q ( I ( Q / V Q ) ,A)
VM__~O_dQ/vQ w e get
from this
4.1
is Q
is s u r j e c t i v e ,
following
in
~Q/VQ
yields
is
57
HomQ(Zv(Q/VQ)~QIQ,A)
again natural
in
for anY
B
@Q IQ ~ Zv(Q/VQ)
r V. M. o. d. Q /"VQ
in
QQ IQ
B @Q Zv(Q/VQ)
COROLLARY
Suppose
7.3. B
t h e n there
the p r o o f of L e m m a
(Leedham-Green
is in
VMOdQ
is an e x a c t
@Q I(Q/VQ)
we h a v e
B @Q IQ = B ®Q Zv(Q/VQ)
thus c o m p l e t i n g
,A)
A . We m a y thus c o n c l u d e
Zv(Q/VQ)
Hence
~ HomQ(zv(Q/VQ)@QI(Q/VQ)
@Q I(Q/VQ)
J B ®Q I(Q/VQ)
,
7.2.
[55]).
Let
V
. I__ff F / V F --+>Q
¢ W
and
let
is a V - f r e e
Q
be
in
V .
presentation,
sequence T
W(F/VF,B)
(7.2)
-~ W(Q,B)
PROOF:
T h i s can be read o f f
theses
stated
of
the m a p
W(F/VF,B)
~
W(F/VF,B)
An application
of the ker
Let
Q
for
is i s o m o r p h i c
~ B ~F
(7.1). and
7.1
(VF/WF) ab
- coker
be an a r b i t r a r y
Then Proposition
(7.4)
from d i a g r a m
-~ O
For u n d e r
Q = Q/VQ
the h y p o -
. By d e f i n i t i o n
we h a v e
(7.3)
REMARK:
Q'B) v(Q,B)
yields
"
sequence
group
in
then yields W = G{
and
the r e s u l t . let
~ = ~
.
a surjective map
TQ, B : H2(Q,B) --9> T o r ( Q a b , B )
B
in
identified sequence
~Qab
' i.e.
w i t h the m a p (II.5.2)
for
E'
for a t r i v i a l Q - m o d u l e . in the u n i v e r s a l
n = 2 . If
Q
is in
Obviously
coefficient ~
, i.e.
if
it m a y be
exact Q
is
58 abelian
we may
give
an e x a m p l e
W(F/VF,B)
This
obviously
maps
exact
We
that
finally
note
and C o r o l l a r y details
: B ® H2(Fab)
B ® H2(Q)
, the k e r n e l
The
one
obviously
Universal
section
we
gets
the
8.1.
Let
Suppose
C
is in
deduce
exact
V
be
sequences
a variety
V n ~
dual W
in the u n i v e r -
to P r o p o s i t i o n
and
V
. Then
. We
7.±
leave
the
... -- TOr 2
We
only prove
recall
from
~M~d__Q . N e x t we
consider
R>h>
we h a v e
an e x a c t
(8.3)
4.7
F
K
the
sequences
that
if
, V
are
q
that
and
exact
-- V q Q @ C --->V(Q,C)
a y-free
of
(8.1)
C
is in
correspond for
let
Q
n = 2 .
be
in V.
sequences ~ E x t z2/ .q z (Qq. aD ,C)~...
Z'>Tor
being
q z [ Q b,C)
dual.
y n Ab
First
, then
C
, 0
we is in
presentation
g>>Q
sequence
O ~ V q Q ~ R/F
by
V
] Hom(vqQ,C)
(8.2) 5 the p r o o f
Proposition
for
in a V ~ { ! ~ Z
(II.5.1) , (II.5.2)
there
(8.2)
~ab,C)
Sequences
of e x p o n e n t
O ~ Ext I_ (Qq~ ,C) ~ V(Q,C) Z/q Z an
PROOF:
Exact
sequences
(8.1)
exact
results
functors
Coefficient
coefficient
THEOREM
Denote
E'
sequence.
7.2 b y c o n s i d e r i n g
to the u n i v e r s a l
Then
of
to the reader.
III.8.
In this
(7.2) ~ for we h a v e
= H2(Fab,B)
onto
sal c o e f f i c i e n t
for
image
of
#
h,
q
R h, ~> and
Fq g*> q ~ O ab Qab consider
the two
" resulting
short
59
(8.4)
0 ~ V q Q ~ R/F
(8.5)
0 ~ K ~ F q ~ Qq ~ 0 • ab ab
T e n s o r ing w i t h
C
yields
~
q
R ~ K -- 0
the d i a g r a m
0
0
_ z/qz,_q -.-> Tor± t~2ab,C)
V(Q,C) 11 1
... ~ T o r ~ / q Z ( K , C )
(8.6)
VqQ
® C ~ R/F
#
R ® C ~)
q
K ® C
Fq ab @ C
=
F ab q ® C
Qq ab ® C
=
Qq ab ® C
0
Since
Fq ab
is
it f o l l o w s
Z/qz-free
from
the
0
long
exact
Tor-sequence
that
T o r ~ / q Z ( K , C ) ~ Tor _ Z2 / q Z ( Q ~ b , C )
thus We
completing
remark
larly ciated
(8.7)
that
sequence with
the proof. sequence (8.1)
(8.4).
(8.2)
can be c o n t i n u e d
to the right)
We o b t a i n ,
Ext-sequence.
long
left
exact
~ T°rZ/qZ(R/Fn
z/qz,~q
T°rn+i the a n a l o g o u s
the
the
(and
sequence
n ~ i
"'" ~ T°rZ/qZ(VqQ'C)n
_
and
using
to
~ab
,C)
#q R,C)
~ TorZ/qZ(vqQ,C) n-l-
"'"
simiasso-
60 COROLLARY exact
8.2.
If
V
is o f
exponent
0 ~ Ext~(Qab,C)
(8.9)
O - VQ
Again
we
only
prove
enough
to p r o v e
that
sequence
free
For
abe!ian.
obvious varieties
in the
case
Hence
from
of
the
K
we
~ Hom(VQ,C)
~ Tor~(Qab,C)
homology (8.9)
is
diagram
V
~ V(Q,C)
® C ~ V(Q,C)
PROOF:
then
, then
have
split
short
sequences
(8.8)
is
q = 0
free
(8.6)
exponent
part.
splits. abelian
~ 0 ,
~ 0 •
Since
But
Tor~
if
and
q = 0
(8.4)
that
this
yields
q ~ 0
, we
are,
= 0
, it
, then
is Fab
splits.
It
is
a splitting
of
(8.9).
of
course,
interested
where
T ° r 2z/qz (Q ~b' c) ~ Tor~/qz(=,C) = O (8.±0)
~xtz/q~2 (Q~b'C) - Ext~/qZ(K,C) : O. This
is so,
projective
for
if a n d
COROLLARY jective,
if,
8.2. then
example,
only
if
K
is p r o j e c t i v e .
Qabq
is.
If
V
is o f e x p o n e n t
for
C
in
V(Q,C)
J Hom(vqQ,C)
(8.12)
V(Q,C)
~ VqQ
finally
is a l s o an
true:
extension
(8.13)
note
that
Let in
C V
~A[E]
the be
thus
obtain
q > O
, and
by
if
Lemma
Qq ab
II.5.2
is --
,
® C
statement in
V
n Ab
characterized
: VqQ
~ C
analogous
by
and
to P r o p o s i t i o n
let
AlE]
=
E
: C >
~ ~ V(Q,C)
> G
II.5.4 D Q
. Then
K is
z/qz-pro-
V n Ab
(8.11)
We
We
But
be
6±
is t h e
homomorphism
E b.
in the
5-term
sequence
bE
VqG
Since we
the
leave
proof
of
it to t h e
~ VqQ
this
*)C
fact
reader.
~ Gq ~ Qq ~ 0 ab ab is a n a l o g o u s
to
" that
of Proposition
II.5.4
CHAPTER
THE LOWER
The key result
of this C h a p t e r
clude
that
gical
hypotheses
G
b y the
simple them
terms
theorem
takes
tions
up
I, 2 w e
giving
o f the are
theorems theorem
are
to p r o v e
Hall
and p r o v e
with
the n o t i o n
some
interesting
that has given by
[38]
5 we apply
free groups
7 we
groups
and
results
the
n+r generators is
free.
some
on groups famous
and
using
It is a m o s t theorems their
striking
of this
proof
fogy theory
that
do n o t r e q u i r e
of groups
its b e s t :
nature.
the
l.i
by a special
to
known famous
may
variety. to p a r a 6 deals
theorem
of nilpotent
of Magnus
turn
to g e n e r a t e
Section
our b a s i c
which
although
uses
non-homological
8 we
Theorem
[60]
yields
groups.
In
presentation that
a group
also be generated
finite
on t h e e x i s t e n c e
nevertheless at
theorem
theorems
groups
of n o r m a l
to p r o v e p-comple-
i.i.
fact
chapter
given
r relators
In S e c t i o n
our T h e o r e m
Here
in S e c -
of the b e t t e r
results.
the d e f i c i e n c y
the H u p p e r t - T h o m p s o n - T a t e - t h e o r e m ments
known
some of
in a n i l p o t e n t
of our basic
of groups.
about
a theorem
as a c o r o l l a r y
by n elements
extend
of
group
and
of t h i s
o f the k n o w n
in p a r t i c u l a r
groups
K
preparation
of a V-free some
homolo-
of
applications
some
3) m o s t
4 we obtain
splitting
of deficiency
prove
(Section
an extension
The
to c o n -
certain
the q u o t i e n t s
After
in a v a r i e t y ,
about
one
the p r e s e n t a t i o n
for a s u b s e t
In S e c t i o n
on s p l i t t i n g
In S e c t i o n
Section
conditions
series.
chapter.
It e n a b l e s
satisfying
between
numerous;
of this
subgroup.
o f P.
central
surprisingly
able
1.i.
f : K ~ G
isomorphisms
lower
SERIES
is T h e o r e m
the r e s t
sufficient
a V -free
CENTRAL
a group homomorphism induces
IV
the
statements
any homological
homological
of many
terminology,
m a c h i n e r y . Here w e
as a tool,
to p r o v e
o f the
see
theorems
the of a
homo
64 Theorem
i.I
for
[75],
where
is
in
Stammbach
to
Section
chapter groups
some
8.
= Gr
is c o n t a i n e d
applications [76],
Our
stems we
V
[77],
information
from
H.
present
Section
~
be
negative
a variety
integer
(I.I)
of
we
V(G,z/qz)
as
defined
in
Sections
VqG
= H~G
. We
shall
(1.2)
as
G~
defined
in
Much
of
the
[79].
Tate's
paper
[82]
is
related
most
of
5 come
from
Baumslag
Basic
and
of the
groups
used
results
on
[i3],
in
rest
this
parafree
[i4].
Theorem
let the
G
be
in
~
. For
for
V:
q
a non-
groups
= VqG
also
'
also.
consider
III.l
= G
Stammbach
found,
[64];
The
[74],
be
varieties
groups,
shall
Stallings
about
Neumann
in
to
[78],
IV.I.
Let
are
in
, III.2.
consider
Gq
= G
i+l
(I.I.2) . We
shall
Recall
the
~
q
that
lower
central
(q)
= G: =r
we
series
have
of
G
i = 1,2,...
Gq
1
'
set CO
Gq=
(1.3)
The
basic
theorem
chapter
is
THEOREM
I.I.
Let
Suppose
that
f
morphism
(1.4) for
~
w
every
as
f,
l
Gq
G
1
of which
we
=
~
~
i i
shall
G
l
make
°
numerous
applications
in t h i s
follows.
f
: K ~
induces
: VqK
G an
-->} V q G
be
a homomorphism
isomorphism . Then
the
f, map
of : Kq ab
f
qroups
in
V
~ Gq ab
and
an
induces
an
f?~ , ~ / ~ ~ ~/~? i ~> i
, and
a monomorphism
fq W
: K/Kq>---> G / G q
. epi-
isomorphism
65 PROOF: and
We p r o c e e d b y
for
i = 2
the e x a c t
induction.
sequences
~5 ~
phisms. phism.
~4 ~
~3 ~
By the 5 - 1 e m m a
this
we a p p l y the 5 - 1 e m m a
implies
-i)ab
that
~I,~4
and
~3
a5
are
isomor-
is an e p i m o r -
is an i s o m o r p h i s m .
Next
to the d i a g r a m
a31
---9> K/Kq_i
~fq
G iq _ I / G qi >
fq
ab
is an i s o m o r p h i s m ,
K iq- I / K iq >---2 K / K q. z
Hence
we c o n s i d e r
~i~
f : K ~ G . By i n d u c t i o n ~2
~3
i > 2
is t r i v i a l ,
(K/K~ -i )ab q ~ O
~2 ~
-I ) ~ G i - i / G i
By h y p o t h e s i s
(1.7)
For
q ~ -I ) ~ K iq - I / K iq ~ K ab
induced by
By the above
the c o n c l u s i o n
(III.2.7)
VqG ~ Vq(G/G and the m a p
i = i
it is p a r t of the h y p o t h e s e s .
VqK ~ Vq(K/K (1.6)
For
~fq-1
> G / G q ---9>G / G q _ i
is i s o m o r p h i c ~
is i s o m o r p h i c ,
by
and the
induction
fq i-i
is i s o m o r p h i c .
first p a r t of the c o n c l u s i o n
is
1
proved. In o r d e r
to p r o v e
the s t a t e m e n t
about
f
consider
the d i a g r a m
w
fq (1.8)
I
I fq •
K/Kq.
G/a q.
)
l
and s u p p o s e f q ( x K q) l
xK q £ ker
= G q . But
1
1
x £ Kq
Under
the h y p o t h e s e
1
fq . Then,
since
fq
'
that
epimorphie.
1
, i 9 I
for all
i >I I , we h a v e
is an i s o m o r p h i s m
by
(1.4)
it f o l l o w s
l
Hence
x
of T h e o r e m
A counterexample
~ Kq
, and
fq
is m o n o m o r p h i c .
l.i the m a p
fq
is not,
to
is as f o l l o w s
to
09
(see [74]).
in g e n e r a l , Let
66 K
= G
= F(x,y)
f
: K ~
G
given
immediate f,
f(x)
f,
~ H2G(=
residually
Gr-free
by
that
: H2K
are
, the
= x
: Kab
O)
is
group
, f(y)
~ Gab an
nilpotent,
on
two
= y[x,y]
is an
have
K
Since = G
W
But
We
of
course,
remark
f
is n o t
however
that
surjective,
quite
x,y
. Taking
isomorphism.
epimorphism. we
generators
q = 0
, so
free
that
generally,
is
groups
f
W
for
, it
Also,
absolutely
= e
. Consider
= f
.
W
y ~
if
im
f
f
.
is e p i m o r p h i c ,
fq
also
W
is.
It
follows,
epimorphic,
that
we
may
whenever
conclude
the that
map
f
fq
is
: K ~ G an
in T h e o r e m
1.1
is
isomorphism.
W
COROLLARY that
f
1.2. : K ~ G
morphism
f,
PROOF:
By
(q)
there
if
follows
K,G
induces
from
exists
n
Theorem
I.I.
1.3.
Let
G
be
suppose
exists
a V-free
group
for
every
PROOF: in (yj)
f
induces
i >~ i
Let
Gq ab , j
Obviously conclusion
an
isomorphism
(xj)
,
an
form
a basis.
~ J
and
the
the
a
e J
I.l)
with
and
itself
in
Gq ab
nilpotent
f,
~
is
Suppose
~ Gq ab
is
isomorphism.
an
G
. The
VqG
in
a homomorphism
V f
and
is c a l l e d
result
, with
free
(q).
: Kq ab
a group
Gq = e n
a group
monomorphism
be
a
Consider map
hypotheses
follows.
Section
are
then
= 0
n Ab ==
epi-
nilpotent easily
. In c a s e
. Then
: F ~ G
an
there
such t h a t
isomorphism
and
, j
f
that F
that
. Then
~ i
a prime,
map
groups
(see
not
the
be
: VqK-->>VqG
definition
COROLLARY is
Let
f of
set
of
the
: F ~ G Theorem
fqcu : F / F q ~
elements
V-free =
defined I.I
in
group
are
by
G/Gq
)
G F
"
, whose on
f(yj)
satisfied,
the
set
= xj so
images
, j
that
e J the
.
67 COROLLARY G p
1.4.
Let
is a f i n i t e l y , then
G
~
be a nilpotent
generated
Since
Hence
H2G
G
is f i n i t e l y
is f i n i t e l y Since by
that
The conclusion
REMARK.
generated
groups
VG
V(G,A)
= 0
We b e g i n shall
(III.8.8)
we have
. By
i Ext(Gab,A)
, for a l l
we
shall
the
G
Recall
from S e c t i o n
ponent
, then
G
is free
Free
p ~ 0
primes
it is f i n i t e l y
related.
VG
is f i n i t e l y
and
that
V:
I.I.I).
Corollaries
if
VG = 0
1.3 and
in the n i l p o t e n t V(G,A)
must
be
1.2
V(G,A)
= O
.
1.2.
variety for all
free
then
= 0
abelian
shows
by
and
that
G
(III.i.il) .
SqbgKgu R Theorems
.
nilpotent I.l
then
in
~
free,
1.3 a n d
some results
Let
p
satisfies
on the
denote
existence
a prime
property
( Pp)
o f V= - f r e e
or zero.
We
if t h e V= - f r e e
(p) .
that nilpotent
, nilpotent
(see L e m m a
Gab
state
variety
are residually
for
for a l l
~ Hom(VG,A)
of C o r o l l a r i e s
if
groups
that
A
some preliminaries.
say that
G
A
of a group with
by applying
if a n d o n l y
= O
section
hence
is V - f r e e
IV.2.
subgroups
= 0
afortiori,
q : O
Conversely,
In this
and,
is t o r s i o n f r e e ,
. An a p p l i c a t i o n
is free.
nilpotent
to see t h a t the g r o u p
V(G,A) if
vPG
. I__ff
(D T o r (Gab,Z/qZ)
is t h e n o b t a i n e d
of e x p o n e n t
Now
with
q : O
(III.8.9)
Gab
It is e a s y
abelian
V
generated
V q G ~ V G ~9 Z / q Z
it f o l l o w s ,
in
of exponent
i_~s V - f r e e .
PROOF:
generated.
qroup
variety
(p) m e a n s
(0) m e a n s
nilpotent
just nilpotent
a n d of
finite
p
and ex-
68
LEMMA
2.1.
GP = e
A qroup
G
is r e s i d u a l l y
nilpotent
(p)
if a n d o n l y
if
.
w
PROOF: there say.
Suppose exists
It
G
is r e s i d u a l l y
N 4 G
with
is t h e n c l e a r
nilpotent
x ~ N
that
and
GPc+I
c N
(p). L e t
G/N
nilpotent
so that
Gp = e
. Conversely,
let
exists
an
integer
that
if
nilpotent (p) P
(i)
n
G
that
such
that
G
(p). W e
x / Gp n
is r e s i d u a l l y
is r e s i d u a l l y
,
"
e ~ x 6 G
there
continue
. Since
G/G p n
nilpotent
(p) .
p-group,
then
a finite
with
some e x a m p l e s
is n i l p o t e n t
G
(p)
is r e s i d u a l l y
of varieties
satisfying
:
v = Gr
~ the a b s o l u t e l y
residua!ly (ii)
given
c
w
we may conclude Note
. Then
. Then
It f o l l o w s
c + l
Gp = e
w
~ G
(p) o f c l a s s
x ~ Gp
--
that
e / x
V = N :
finite
free g r o u p s
are r e s i d u a l l y
nilpotent,
and
p-groups.
~ the
free n i l p o t e n t
: the
free
groups
are residually
finite
p-groups
=C
[34]. (iii)
~ = ~
residually
(iv)
=
V
state
residually
that
example
the
2.2.
a qroup
in be
n A~) a V-free
V.
: the
contains
general
(i) L e t
V
w i t.h
Gab .
be
subgroup
a direct F
o_~f
nilpotent
and
[34]. free g r o u p s
and
in a n y p o l y n i l p o t e n t
residually
examples
a variety
in
in G
summand. G
are r e s i d u a l l y
(ii)
finite
and
p-groups
(iii).
variety
[34].
We may
now
theorem.
free .
a s e t of e l e m e n t s qenerate
groups
p-groups
nilpotent
(iv)
following
THEOREM
j ~ J
finite
$(cj ,c 2 ..... c k)
are
Note
soluble
.
satisfyinq
V n Ab whose Then
and imaqes (xj)
(Po) VG = O in
Gab
, j ~ J
and
let
. Let
(xj)
freely freely
G
b__ee ,
(in qenerates
69 (ii)
Let
V
be
be a q r o u p ments
in
in
G
V_
freely
PROOF:
In b o t h
prime
of
F
of
G
theses fP: ~ fP
vPG
imaqes
cases
G
(ii))
take
f : F ~ G
for a p r i m e
(xj)
are
in
the
Gp ab
, j e J
linearly
subqroup
enlarge
F
set
We
shall
to be
prove
that
generates
the ~ - f r e e
Since
we o b t a i n
V=
let
G
a set of eleThen
, j e J
in c a s e
(xj),
(i)
p
this
larger
on
f
to a set of the g i v e n
a ~-free
group
f (yj ) = x 3 . C l e a r l y
whence
be
and
.
(xj)
(p = 0
p
independent.
o__~f G
, j e J ) freely F
by
I.i
>G/G~.
factors
GPab
a basis.
(xj)
(Pp)
. Let
V-free
form
. To do so,
F/F~
in
images
call
of T h e o r e m
= 0
we m a y
whose
again
and d e f i n e
satisfyinq
qeneratesa
in c a s e
(which we
with
whose
j ~ J
elements
a variety
set
subgroup
(yj)
, j ~ J
satisfies
the h y p o -
a monomorphism
satisfies
(Pp)
we have
F pw = e ,
and
as
W
F ~ G ~ G/G~ It f o l l o w s
that
The
following
2.2
(i)
a free
that
a,b
itself
example the
subgroup
rators
f
shows
images in
where
Gab V
a2,b
in
generate
Gab
a V-free
it
(xj)
, j ~ J
. Take
that
generate subgroup
[a2,b]
that
G
to be
Gab
G
given
subgroup.
a
2
,b
generate
an a b e l i a n
in T h e o r e m
(in
~ n ~
) generate
2
subgroup.
group laws
-i)
in two g e n e [x[y,z]]
so that
However
because
[a,[a,b]][a,b]
suppose
b y the
abelian,
is c o m p l e t e .
to
the ~ - f r e e
-- e •
Hence
enough
= a(aba-lb-l)a-l(aba-lb =
the p r o o f
freely
is free
a free of
and
is not
is the v a r i e t y
[x,y] 2 . It is o b v i o u s of
of
is m o n o m o r p h i c
the
and
images
t h e y do not
70 We next
note
property
(Pp)
quently must
if
G
then
V=
Gab
satisfies
2.3.
primes.
= O
Let
V G
generate
a ~-free
of
subset
(xj)
(xj) and
such
that
£ J'
that
= O
Corollary
there
the
apply
clearly
. To
2.3
satisfying
one
Conse-
prime,
be u s e d
must
section
we
conditions
first
recall
then see
of
G
for
p
Gab
free in
Then
it
in the
abelian
G
(xj)
infinitely and
whose
imaqes
~ J
freely
every
finite
, j
.
it is e n o u g h
to p r o v e
a ~-free
exists of
(xj)
this we
p
such
, j ~ J'
from T h e o r e m recall
V
must
III.8.2.
We
that
subgroup.
a prime
follows
that be
then
But that
for a f i n i t e ~
satisfies
in
G pab
2.2
(ii) , if we can
b y the
are
remark
of e x p o n e n t
linearly
just
zero
, so
have
~ V G ® Z/pZ ~ Tor (Gab,Z/pZ)
be
zero. Thus
shall
that
the
Subgroups
apply
for a s u b g r o u p
is G r - f r e e .
with
subgroup.
the v a r i e t y
IV.3.
In this
V
(P)
be a set of e l e m e n t s
images
Corollary
vPG
group
than
will
satisfying
generates
The r e s u l t vPG
that we m a y
1.3.1
, j ~ J
, j
independent.
We
remark
V
or a p - g r o u p .
for m o r e
This
in
a free
subqroup
subset
cient
( Pp)
a variety
, j E J
By P r o p o s i t i o n
which
in a v a r i e t y
free a b e l i a n
zero.
be a q r o u p
PROOF:
above
property
be
-
(x3)
freely
qenerates
show
group
is e i t h e r
of e x p o n e n t
Let
. Let
i__n_n G a b
(Pp)
is a V - f r e e
of
COROLLARY
VG
if
be a variety
proof
many
that
of V - f r e e
the r e s u l t s of a ~ - f r e e
by Schreier's
Also,
proof
obviously
theorem the
is c o m p l e t e .
Groups
of S e c t i o n group every
varieties
2 to g i v e
suffi-
to be ~ - f r e e . subgroup
of a ~ { - f r e e
Ab
Ab
and
for a n y
7i prime
p
have
that
the o n l y v a r i e t i e s there
are m a n y
~-free cern p
group
corollaries
THEOREM
~ . We
shall
get
of T h e o r e m
3.1.
tisfyinq
(i)
(Po)
. Let
F
nerate
summand.
(ii)
Let
F
V
Mostowski
[63])
-in -
F F
be
F
whose
imaqes
Then
a variety
in
(xj)
satisfyinq
imaqes
j ~ J
freely
PROOF:
The
group
F
for
rate
proof
and
subqroup
PROOF:
of
The
PROPOSITION qroups
are
results
for at least
con-
one p r i m e
results
Let
and
V
as d i r e c t
let
be
a yariety
(xj)
, j £ J
Fab
freely
(in
, j ~ J
freely
qenerates
(P)
for
~
a certain
for
F
be
sabe a
n Ab= ) qea V-free_
prime
p
.
a set of e l e m e n t s
independent.
a ~-free
subqroup
of
Fab
from T h e o r e m
is free
in
~
F
2.2
n ~b
Then
(xj)
,
.
since and
for a ~ - f r e e
vPF
= 0
for all
.
[12])
primes.
subqroup.
, j E J
linearly
p = O
many
(xj)
are
a set of e l e m e n t s
a free
let
is i m m e d i a t e
3.2. ( B a u m s l a g
be
and
F ab p
qenerates
infinitely
j ~ J
in
the g r o u p
COROLLARY
of a
p
a V-free_ q r o u p
p
hand
.
be
whose
primes
for a s u b s e t
of t h e s e
qroup
-
Let
are
On the o t h e r
of t h e s e
(Pp)
all
be a ~-free
in
of
property
practically
[38],
set of e l e m e n t s
subqroup
Most
these
2.2.
(Hall
a direct
that
conditions
subgroup.
satisfy
[64])
varieties).
sufficient
a ~-free
which
(see
(Schreier
stating
to g e n e r a t e
p = O
It is k n o w n
of t h a t k i n d
results
varieties
or
property.
Let
Let
F
in
Then
V
F
(xj)
be a v a r i e t y be a ~ - f r e e
whose , j ~ J
imaqes
satisfyinq
qroup in
freely
and
Fab
(Pp)
let freely
qenerates
(xj)
,
qgne-
a T-free
.
proof
3.3.
is
immediate
(P.M.
residually
from C o r o l l a r y
Neumann nilpotent
[65])
Let
][-qroups
~
2.3.
be a v a r i e t y
for some
fixed
whose
non
free
empty
set
72 of p r i m e s .
Let
set of e l e m e n t s Suppose
that
in
qenerates
PROOF:
We
start an
with
latter
case
where
by
j 4 J
W
(3.1)
q
W )
(3.2)
to
x
by hypothesis
images
of
(xj)
in
W ~ z/pz
2.2
(ii) w e
(yj)
must
, j ~ J
induced
exact
obtain
of p r i m e s
in
, j ~ J
is
abelian x
a residually q
Fab
or
. In the nilpotent
. In the
sequel
we
case.
generated
where
b y the
images
of
(xj)
p
does Fab
a map = xj
.
~ ~
we o b t a i n
the e x a c t
sequence
not c o n t a i n
are
linearly
Proceeding
as
p-torsion.
Since
the
independent
their
images
in the p r o o f
f : F' ~ F~ w h e r e , with
the p r o p e r t y
F' that
of T h e o r e m
is ~ - f r e e for
p
on
£ x
the
map
fP~
is m o n o m o r p h i c .
(3.4)
:
This
F'/~'P ~ F/F~ w a y we
f*(I) : F ' / p ~ obviously
,
$ F p. /<x.> ao 3
) Fp ab
in
f(yj)
a
sequence
Z/pZ
~ J
(xj)
that free
dividing
in this
Fab
form a basis.
and
(3.3)
which
~
of
Fab/<Xj>
, j
thus
by
be
independent.
Then
is e i t h e r
is even
set of p r i m e s
linearly
It is c l e a r
Fab
F
are
, j ( J
.
a product
) Fab----9> F a b / < X j >
with
F
Thus
(xj)
~-torsion.
remark.
see that
W ® z/pz ~
since
of
being
subgroup the
(3.1)
subqroup
let
Fab
contain
~-group.
replace
the
not
and
in
following
is the
. Consider
Tensoring
the
with
~
imaqes
does
it is e a s y
therefore
Denote
whose
(abelian)
Z/qZ-module
shall
F
a V-free_ q r o u p
a y-free
a free
~-group
be
Fab/<Xj>
freely
residually
F
finally
obtain
a map
F "p -- F / p n£I[ F p(I)
is m o n o m o r p h i c ,
also.
It r e m a i n s
to p r o v e
that
for a
73
~-free
group
F
(3.5)
n p£~
Thus
let
F/N i
e ~ x
finitely
Nl ~ N2 potent
~ F
~-group.
tent
and
(p)
and
there
a
there
exists
~-free. F/N 2
Finally
F/N 3
.
. Then
generated
, x ~ N2
x ~ N3
Fp = e w
a
there
finite
exists
m
Also,
there
finitely exists
Since
i
Fp
with
afortiori,
A group
G
surjective A group
F mp + l
x ~ pQ~
~
is c a l l e d
map
f
: K ---9} G
V
is c a l l e d
in
a ~-presentation hs
= iG
LEMMA
4.1.
only
N3 4
F
finite,
F
so
that
nil-
with
p-groups
c N3
with
N2 ~ N3
are
nilpo-
certainly
'
PROOF:
If
G
it
is
G
presentation which
it
group
F
define
proof
Groups
a splittinq there
: F
be
in
V
a retract
is a
splitting
splits.
Thus
is a q u o t i e n t . . If g
the
is c o m p l e t e .
in
qroup
exists
in
~
t
: G ~ K
the
V-free
a retract
of
~ G
that
there
is
splittinq
such
if
to
every
with qroup
exists
ft
= iG
F
if
s
.
there
: G ~ F
.
Let
if
h
" Thus
Splitting
in
G
with
and
N2 ~
and
x ~ Fp m+l
IV.4.
is
and
--
x ~ NI
hence
finite
m+l
(3.6)
with
exists
£ ~
p-group.
~
F
generated, p
'
and,
Ni 4
: F ~
f
: K K
such
. Then
G
of
~-free
some
group G
is
in
that
is a the
let
, then of G
surjective square
qroup
in
V
if
qroup.
a retract
Conversely, >> G
V
a
clearly
every
be map
every
y-free
a retract in
V
~-free
group of
~ then
the we
of ~-free may
74
h
F
>$ G
g~
II
¢ - - - f> >
is
commutative.
splits It
by
gs
is c l e a r
group
If
groups
which
LEMM_A
4.2.
for
are
it
not
G
inverse
of
h
then
f
.
In o t h e r
Let
is a r i g h t
in a S c h r e i e r
is ~ - f r e e , group.
: G ~ F
: G ~ K
that
V-free
s
a
is
variety
isomorphic
varieties
V-free
be
V
V
(see
a retract
(under however
Theorem
of
every
the
retract
s ) to there
a
may
of
a V-free
subgroup exist
of
splitting
4.6).
V-free
qroup
F
. Then
Gq ab
=
is a d i r e c t
PROOF:
is
the
summand
in
(-)q ab
is
Gq ab
s,
Since
identity.
the
Hence
identity.
THEOREM p
4.3.
prime.
~-free
qroup
s,
But
Let
Let
a
>
VqG
is
Fq ab
G F
VqG
functor
it
h,
Gq ab
is a d i r e c t
VqF
h, ~ VqG
VqF
be
be
a retract
, hence
a monomorRhism
a variety
of
of
a homomorphism
f~o : F / F q >
q
= O
'
1,2
' ....
that
summand
VqG
= O
exponent
a ~-free f
isomorphisms
and
for
Gq ab
>
= 0
= 0
follows
Fq ab
V
and
and
'
a
> G/Gq
in
Also
"
'
.
q
qroup.
: F ~ G
Fq ab
= 0
o_rr
Then such
q
= p
there
that
f
is
n
a
induces
75 PROOF:
This
immediately
follows
from
Corollary
1.3. n
COROLLARY
4.4.
p
satisfyinq
prime,
qroup
such
Then
G
lity
a .
n
ni n2 = Pl P2
a variety
(Po)
a
Let
prime.
4.6.
be
Gab
4.5.
, p
THEOREM q'
that
V
contains
COROLLARY q = p
Let
Suppose
is f r e e
in
subqroup
V=
be
Then
Let
.
~
nk "'" P k
F
which
of
a nilpotent
G
r~ ~
a nilpotent
. Then
exponent
that
V
a retract
be
of
is
on is
a
V-free
a ~-free
variety G
q = p of
,
a V-free
of cardinality
on
of
or
retract
a set
variety
a retract
q = O
a
set
of
exponent
group
~
.
cardina-
q = 0
o__rr
is V-free._
of exponent
of
a V-free
group
is o f
the
form
G = FIxF2×...xF k
where
Fi
is
PROOF:
Since
free
G
in t h e
variety
is n i l p o t e n t
~i
and
= V= n B p ini
of exponent
' i = I ..... k
dividing
q'
it
.
is a
n.
direct
product
Since
G
summand be
the
of
groups
is a r e t r a c t in
free
Fab
Pi of
in
V.
exponent
a V-free =
, so t h a t
group
of
(Pi)ab such
dividing
group
F
pi 1
, (Pi)
,
must
i
= 1,...,k be
.
a direct
ab must
be
free
in
V=I n Ab==
.
Let
F.I
that
=l
(Fi)ab
We may
then
define
theses
of Corollary
~
a map 1.2
(Pi)
f for
ab
•
: Fi×F2x...xF k ~ G q = O
, thus
proving
satisfying our
the
theorem.
hypo-
76
L e t us s a y that if there
two g r o u p s
exist
hi that
:I
for e v e r y
i ~ 2
~/~
is c o m m u t a t i v e . between
In
[14]
K
G.
G
and
is in
(ii)
G
is r e s i d u a l l y
(iii)
G
has
~
V
lower
V-free
groups.
which
are
In this
same
sequence
....
square
the
existence
of a h o m o -
.
has
defined
a ~-parafree
qroup
(of rank
n )
nilpotent; lower
central
sequence
as a V - f r e e
group
n ). satisfying
central
sequence
Baumslag
has
(Po)
it is a p p a r e n t
V-parafree
given
many
groups examples
that w i t h
behave
exactly
of V - p a r a f r e e
respect like groups
not V-free. =
section
of V - p a r a f r e e prove
central
;
is a v a r i e t y
to the
lower
that
G
(of rank
, i=l.2
the
G
Baumslag
such
the
°
that w e do not r e q u i r e
(i)
If
same
hi_ll
Note
morphism
as a g r o u p
the
~ ~/~_~
hil
[13],
have
isomorphisms
I~.il such
K,G
a number
we
shall
groups
express
part
in h o m o l o g i c a l
of p r o p o s i t i o n s
about
(iii)
terms
of B a u m s l a g ' s (Corollary
parafree
groups.
5.4)
definition and
then
77
PROPOSITION ~-free
f
and
Then
g,
Given
: F ~ G
duces
Let
group,
jection.
PROOF:
5.1.
let
g
i 9 2
the
same
: G ~ G/G~
there
inducing
the
have
: VG ~ V(G/G~)
sequence
denote
zero
a V= - f r e e
F/FO+II
central
, i ~ 2
is t h e
exists
hi+i
lower
some
canonical
pro-
map.
group
-~ G/G°+II
the
as
F
. Note
and
a homomorphism
that
then
f
in-
isomorphisms
hm
The
G
: F / F m° ~ G / G °
,
I < m < i
diagram F O- o i/Fi+~
(5.2)
shows
ik i o o Gi/Gi+ ~
that
k
is an
> F/F~+ 1
~ F/F~
lhi+ i
~h i
> G/G~+ i
isomorphism.
~ G/G~
Consider
the
diagram
1
) ~ F io/ F io+ i -- F a b
O --~ V ( F / F (5.3)
l
i (h i)
is e a s i l y
seen
to be
(F/F~)ab
lh 2
) ~ O iO/ G iO+ i ~ G a b
VG g*>V(G/G
which
Ik 1
*
~
commutative.
I (h i) ~
.
(G/G~)ab
Since
h.
~ 0
~ O
is an
isomorphism,
l
(hi) * g,
also
is.
The f a c t
: V G ~ V(G/G°)I
The
following
LEMMA
5.2.
that
f
Lemma
Let
induces
g,
: VqG
an
isomorphism
is the
f
ki
zero
5.2
is an
: K ~ G
be
an
~ Vq(G/G~)
that
then
yields
that
map. extension
of our
a homomorphism
isomorphism is the
is an isomorphism,
zero
f, map
of q r o u p s
: Kq 2 Gq ab ab for
all
basic
"
Theorem
in
Suppose
i ~ 2 . Then
l.i.
~
. Suppose
also f
that induces
78
for e v e r y
i ~ I
PROOF:
The
proof
is the
difference
being
that
morphism.
and
However,
a monomorphism
same ~5
since
zero map we may
replace
group.
argument
Then
the
PROPOSITION
5.3.
VG ~ V(G/G~)
g,
abelian such
there
that
f
1
PROOF:
First
their
images
(yj)
, j ~ J
Lemma
can
5.2
: F/F
let in
be
through
group
an epi-
is the
VqG
by
the t r i v i a l
any difficulty.
of exponent
zero.
i ~ 2 . Then~
F
, the only
: VqG ~ Vq(G/G~) the group
for
i.i
is n o t n e c e s s a r i l y
without
a variety
: sle~
~ GIGq
, i
q = 0
. Choose
1
Suppose
if
Gab
and a h o m o m o r p h i s m
we conclude
f : F ~ G
that
f
, i ~ $ . The
=
1,2
Let by
induces
~ q = 0,1,2
....
elements
form a b a s i s .
and d e f i n e
~ G/G
g,
(1.6)
zero map
of Theorem
(1.6)
case
> G/G~
that is f r e e
f : F ~ G
isomorphisms
Gab
x F
~ G
3 be
setting
....
, j ~ J
such
the ~ - f r e e
group
f(yj)
= xj
that on
. By
isomorphisms
same argument
works
for
i 9 2 . But by definition
Gq
q / 0
if w e
s h o w that
g:
is the g~
zero map
factors
(5.6) It
goes
a ~-free
induces
fq
in d i a g r a m
in d i a g r a m
V
: H/H~ ~
as the p r o o f
in o u r
is t h e
exists
(5.5)
foi
Let
fq~
is thus
: V q G ~ V q ( G / G q)
for a l l
G?
I'
so t h a t
as
g:, enough
to s h o w t h a t
we have by Corollary
(5.v)
~
i--
vqs
=
g~ = O
. Since
Gab
is free
III.8.2
vG o z / q z
,
vq(s/s°/
=
v(G/s°)
® z/qz
abelian
79
so
that
Note
gq
that
is a n
= g,
the
part
arbitrary
COROLLARY
@ i = 0 of
Let
conclusion with
the
variety
5.4.
.
and
G
be
in
Gab
is
free
V
. Then
q
= O
in
V
the
remains
true
if
V
n A__b__
followinq
statements
are
equivalent,
(i)
G
(ii)
Gab
is
map
for
(iii)
has
There
5.5.
Gab
in
I__ff G (of
in
all
i i> 2
L e t'
5.6.
n)
such
(5.8)
be
fq
VG
be (of
that
From
exists
f
Since
rank
the
and
as
some
V-free
~ V ( G / G °)
is
qrou~ the
from
= 0
zero
nilpotent
. Then
of
f
G
: F ~ G
and
5.3.
qroup
in
V
zero,
it c o n t a i n s : F ~
5.1
with
i_ss V - p a r a f r e e .
exponent
, i--1,2
f
.
Propositions
n) , t h e n
:G/G q
a homomorphism
, i i> i
embeddinq
: F/F
) G
.....
satisfyinq
a V-free induces
q=0,i,2
(Po) .
qroup
F
isomorphisms
. . . . .
l
Proposition : F -- G
F° = e
F
a variety
1
PROOF:
: VG
a residually
n A_b_ a n d
is V - p a r a f r e e
rank
follows
V
g,
qroup
: F / F o ~~ G / G o
G
Let
sequence
;
a V-free foi
V
central
V_ n Ab__ a n d
immediately
COROLLARY
COROLLARY
lower
free
that
This
free
same
exists
such
PROOF:
the
5.3
with
, the
F
map
and free
fo
Lemma
5.2
in
, inducing
factors
V
it
follows
that
there
a monomorphism
as
w
(5.1o) whence the
F £ G ~ ~/G ° it
fallows
required
that
properties.
f
itself
is m o n o m o r p h i c .
Obviously
f
has
80
COROLLARY for
5.7.
infinitely
Let
V
be a v a r i e t y
many
primes
be a set of e l e m e n t s arly
independent.
o_if
G
j ~ J'
, such
(xj)
_be _~-parafree that
, j ~ J
GPab
are
then
¢ J"
1.3.1
subsets
we m a y
We m a y J'
Then
By P r o p o s i t i o n
finite
in
G
G
that
(xj)
find
linearly
that
such
, j £ J'
a prime
enlarge
(xj)
the ~ - f r e e f(yj)
Fp = e Note
p
the
set
the
But
group
on
F
= xj
Lemma
let
in
(Pp)
(xj )
Gab
a ~-free
, j ~ J
are
line-
subqroup
F
in
5.2
such
images
that
V=
Gp ab
a basis.
form
a ~-free and
yields
set
of
to a set
, j ~ J"
then
the
finite
, j ~ J'
generates
(yj)
for a n y
the a s s e r t i o n
for
(xj)
(xj)
,
, j ~ J'
satisfies
(Pp)
(xj)
~ J"
We
then
subgroup
the m a p
, j
prove
by considering
f : F ~ G
the d e s i r e d
,
result
defined
since
.
that
the c o n c l u s i o n
(It is
implicitly
state
two
PROPOSITION
G
be
part
other
i__ssy - p a r a f r e e
and
that K
that
consequences
K
remains
is d r o p p e d
of the h y p o t h e s i s
f : K ~ G
, then
5.7
nilpotent
immediate
Let
~ Gab
of C o r o l l a r y
residually
5.8.
f2° : Ka b f
imaqes
to p r o v e
that
and
(xj )
images
freely
that
G
satisfyinq
and
qenerates
such
independent
, j ~ J"
thesis
we
their
it is e n o u g h
-
by
zero,
.
PROOF: all
in
Let
.
of e x p o n e n t
G
correct from
be a homomorphism
is V-parafree_
statement.
in
same
Next
5.2.
~
nilpotent.
(of the
the
be ~ - p a r a f r e e . )
of our L e m m a
is r e s i d u a l l y
if the h y p o -
. Suppose
that
If rank
as
G ) and
is a m o n o m o r p h i s m .
COROLLARY
5.9.
o f the
same
PROOF:
We
jective,
Let
finite
only have so
is
f,
f : K ~ G rank.
Then
to p r o v e . Since
be
an e p i m o r p h i s m
f
is an
that K
and
f, G
of ~ - p a r a f r e e
qroups
isomorphism.
: Kab ~ Gab have
the
. But
same
if
finite
f
is sur-
rank,
the
8i groups
Kab
Hence
f.
and
Gab
are
free in
~ n A~
PROPOSITION
two r e s u l t s
5.10.
Let
~
of a r a t h e r
m ~ I
has a ~-~arafree
PROOF:
a £ G
such that
Choose
the n o r m a l
G
~
Q
(5.11)
VG
Since
N
Gab = m
subgroup
yields
f * > VQ
and rank
is free a b e l i a n , is e p i m o r p h i c .
Now consider
G/G~N same
= Q/Q~
has
. It t h e n
V-parafree
PROOF: C
g~
5.11.
Let
h.~
is a b a s i s by
the
f.
Ga b
N/[G,N] h.
element.
a . The e x t e n s i o n
square
> Qab ~ 0 .
is c y c l i c .
S i n c e rank
is n o n - t r i v i a l . It f o l l o w s
that
Since f.
Gab
: VG~VQ
(i ~ 2)
the d i a g o n a l
is the zero map.
to be the zero map, follows
sequence
also.
from C o r o l l a r y
as a y - f r e e g r o u p
of r a n k
m-I
Every ~-parafree
be ~ - p a r a f r e e
is the ~ - f r e e
.
VQ
g r o u p of r a n k
G
Gab
m-i
> V(G/G~)
is then ~ - p a r a f r e e
PROPOSITION
A ~-parafree
~g~
lower c e n t r a l
Q/Q~
zero.
of r a n k
generated
, the m a p
f. - - >
is the zero map,
is e p i m o r p h i c ,
nature.
sequence
g.l
g.
G
b y one e l e m e n t ,
v(G/G~) Since
of
in
m u s t be m o n o m o r p h i c .
VG (5.12)
N
6*>N/[G,N]
Qab = m-I h.
quotient
its image
the e x a c t
is g e n e r a t e d
different
be a v a r i e t y of e x p o n e n t
q r o u p of rank
N >~
f i n i t e rank.
is m o n o m o r p h i c .
We c o n c l u d e w i t h
Consider
of the same
m+i
Since
f.
But of c o u r s e
5.4
that
Q
(of r a n k m-i).
has the Clearly,
6
qroup
of rank
m
is a q u o t i e n t
of
.
of r a n k
m
g r o u p on one g e n e r a t o r .
. Consider Clearly
K = G *V C Kab
is free
where in
82 fl A=b= o f r a n k zero map
for
m+i
. Next we
i ~ 2
Since
prove
that
g~
G / G ~ _¢ K/K~l
: VK ~ V(K/K~)
we m a y c o n s i d e r
is the
the
following
= O
, the
diagram
VG
(5.13)
--
g*
where
~ ~
implies that m+l
V (G/Gi)
V G ® VC
VK
map
>
is an
g*'
isomorphism
is an
isomorphism.
that
g~
has
the
K
. Clearly
[s the same
K/K°
>
V ( K / K i)
by Theorem Thus,
the
z e r o map.
lower
III.6.1.
fact
that
It t h e n
central
g,
VC
is the z e r o map,
follows
sequence
is t h e n V - p a r a f r e e
Since
from Corollary
as a V - f r e e
group
=
of r a n k
m+l
5.4
of rank
, and has
G
as
quotient.
IV.6.
In this
section we
all
groups
Let
the g r o u p
considered G
(6.1)
i.e. the
suppose
~
be
be
V
is a v a r i e t y
finitely
of exponent
presentable
given by a finite
is the q u o t i e n t
normal
subgroup
R
x i , x 2 ..... x n
are called
relators.
number
The
We d e f i n e
maximum
defvG
deficiency
consequence
group
generated
yl,Y2,...,y r
generators,
n-r
6.1
finite that
.
F
on
xl,x2,...,x n . The
by
elements
and t h e e l e m e n t s
y i , y 2 ..... Yr
the
the p r e s e n t a -
, the ~ - d e f i p i e n g y
of the
of T h e o r e m
is c a l l e d
~
and
,
of the ~-free by
in
zero,
~-presentation
G = g p v ( X l , X 2 ..... X n l y l , y 2 ..... yr )
G
tion.
that
will
in
The Deficiency
deficiency of
the g r o u p
V-presentations the m a x i m u m
of
of
of the
G
G
to b e
. It w i l l
deficiencies
the be a of the
83
finite
~-presentations
always
exists
M
is a n y
mum
exists
a ~-presentation
whose
finitely
number
zation
indeed
generated
of g e n e r a t o r s
of a t h e o r e m
Knopfmacher
THEOREM
of
group, sM
we
. The
(see E p s t e i n
(6.3)
(ii)
defvG
~< dim
PROOF:
We o n l y p r o v e similar.
the
Let
generators
and
(6.4)
Gab-SVG
shall
defvG
denote
following [27])
. If
the m i n i -
is a g e n e r a l i -
and a t h e o r e m
of
first
R > h> F r
image
abelian.
K
of
Thus
canonical
g~ G
Consider
(6.5)
~ VG ~ K
. Since
relators
r ~ s(R/[F,R])
prime.
the p r o o f
the e x a c t
we get
group,
We c l a i m
that
(torsion abelian
statement
free) group
Tensoring
with
yields
for
and Q
let
G
with
the
Fab
is free
is g e n e r a t e d
b y the
inequality
= sVG + rank
K
= sVG + r a n k
Fab-rank Gab
Gab
,
(i).
V = Ab
rank.
of
second
sequence
since
R/iF,R]
= sVG + n - r a n k
proving
of the
h , > Fa b ~ Ga b ~ 0 .
abelian
of the
p
be a V - p r e s e n t a t i o n
is a free
R/iF,R]
images
,
inequality,
relators.
h,
,
GPab-dim v P G
O ~ V G ~ R/iF,R]
spaces
is just
there
6.1.
~< r a n k
thus
In p a r t i c u l a r
[51].
defvG
The
by
P. Hall
(i)
n
M
is finite.
deficiency
abelian
of
(6.2)
being
and
, the V - d e f i c i e n c y
To p r o v e R ~
this
let
) F ---->~G
and c o u n t i n g
G be
is n o t h i n g be
a finitely
a finite
the d i m e n s i o n s
of
else
but
generated
presentation the
the
resulting
in
~
.
vector-
84 n-r
= rank
G
.
It f o l l o w s
that
in this
that
also
is an e q u a l i t y .
(6.3)
case
(6.2)
is a l w a y s
an e q u a l i t y .
By C o r o l l a r y
III.8.2,
We c l a i m
we h a v e
for
p
a prime
dim
Gp - dim ab
vPG
=
(dim G ® Z / p Z - d i m
TorZ(G,z/pZ))-dim(VG
(9
z/pz)
(6.6) = rank
Gab
= defvG
by
DEFINITION:
A
is e q u a l i t y
is c a l l e d
sentable)
(finitely
group
G
p-efficient
in
PROPOSITION
6.2.
we have
Since
(6.8)
dim
V
G
primes
GPab = G a b
be
G
in
V
or O - e f f i c i e n t
in
V
(6.3)
for w h i c h . A
is an e q u a l i t y
(6.2)
(finitely
pre-
is c a l l e d
a finitely
presentable
qroup
in
V
. Then
p
~< d i m
GaPb-dim v P G
a prime
p
® Z/pZ
and
for w h i c h
z ~ VG (9 Z / p Z ~ T o r i ( G a b , Z / p Z )
III.8.2) , we
GPb-dim
group
for w h i c h
Gab-SVG
exists
vPG
(see C o r o l l a r y
presentable)
.
Let
always
(III.2.i4).
efficient
in
rank
there
PROOF:
V
for all
(6.7)
and
,
Z VG = TorI(Z,G)
since
- O
,
we h a y e
equality.
,
obtain
v P G = ( d i m (Gab(gZ/pZ) -dim = rank
Gab
>i rank
Gab-SVG
T o r Z ( G a b ,Z/pZ) ) -dim (VG®Z/pz)
- dim(VG~gZ/pZ) •
85 Moreover,
VG
may be written
VG
with
ni/ni+ I
k = O
or
some
We h a v e
(ii)
i ~
k ~ 0
Next we give (i)
= z~...@Z~Z/nIz~z/n2Z~...~Z/nkZ
for
if
for e v e r y
prime
p
Consider
V = Gr
.
in
PROOF:
V = Gr
Let
I ~ i < k
k
Every
The
second
II.5.13). (iii)
. We need
to m a k e
the
and
~ = A=b= e v e r y also
finitely
p-efficient
in
V
abelian
qroup
... ~ Znk
with
is e f f i -
G
summand
a generator
of f i n i t e
to m a k e order)
ni/ni+ I x~
the
,
x~
rank G+i
,
commute ~
$ ~ m
.
hand we have
=
equation
being
6.4.
easily
that we
Every
one
G
,
(2)
be a n y v a r i e t y
(6.10) is e f f i c i e n t
in
(~) r e l a t o r s
x~
sH2G
PROPOSITION
V
qenerated
direct
Gab = rank
V
in
that
... ~ Z ® Znl ~
for e v e r y
It f o l l o w s
Let
(6.6))
finitely
rank
(6.9)
if
.
. Choose
O n the o t h e r
equality
.
G ~ Z ~ Z ~
relators
(see
is e f f i c i e n t
I ~ ~ ~ r a n k G +k = m and
we have
p/n I .
seen
group
cient
. Consequently
examples.
already
6.3.
i < k
and
generated
PROPOSITION
as
deduced
indeed
qroup
G = g p v ( X l , X 2 ..... XnlY) in
V =
.
the K ~ n n e t h
get equality
of exponent
relator
from
zero.
in
(6.2).
Theorem
(see
86
PROOF:
Let
R > h)
5-term
sequence
F
g ~G
R/[F,R]
cyclic.
We
y ~ F 2o
then
be
6.
have
~
G
is
Let
~
that
any
rators (v)
It V
(vi)
is =
In
r
easy
to
[81]
see
Beyl
has
group Swan
by
has
cases:
Since
. If
associated
( F2
cases
we
.
image ~ F o2
Fab O
y
is
of
y
, then
abelian,
g.
is
y ~ F o2
and free
, it
: Fab
If h.
must
~ Gab
have
.
of
exponent
zero.
~
given
a presentation
where
that
y
0
= n-I
V=
any
by SGab
free
of
7k
and
order . Define
the
xa
G = A tends
to + ~
a cyclic
= Gr the
action
: a
~ C3
that
shown
V
k
two
in
shown
in
Let
the
- sVG
in
b ~
canonical
relators
efficient
x
The
~ n
We
is
nilpotent
will
show
in S e c t i o n
with
efficient
group
is
n+r in
~
efficient
2
,
group
that
the
cyclic of
a
that
A
be
are
the
that
efficient
on
C3 A
are
extensions
in
Gr
groups
elementary
group
C3
E
groups
following
. Consider
. It m a y , and
finite
of
G
. in
abelian
order
of
. are
not
group
3 with
A
generator
by
.
shown H2G
= 0
that .
defvG
tends
to
-ca
7
gene-
.
[17]
In
G
Fab~Ga
the
= O
variety
group
a cyclic (vii)
K-presentation.
by
In b o t h
Gab
any
and
Gr
VG
efficient
be
h~
trivial.
R/[F,R]
rank
that
(iv)
is n o n Hence
(6.11)
so
given
R/[F,R]
to c o n s i d e r
h.
: VG
6.>
is g e n e r a t e d
monomorphic.
and
the
reads
0 ~ VG
Since
be
as
87 It
seems
to be
in
Gr
. However
THEOREM tion.
6.5. Then
an
similar.
whether
nilpotent
groups
are
efficient
a qroup
in
~
, qiven
a p-efficient
: K ~ G
which
qroup
induces
an
by K
a
finite
in
V
~-presenta-
and
a surjective
isomorphism
i
give
the
Consider
corresponding
be
exists
f
i ~
We
question
have
G
there
every
PROOF:
we
Let
homomorphism
for
open
proof the
5-term
o
Denote
by
I
tation
are
xl,x
~
the
(6.13)
for
finite
= 0
, the
proof
for
p
R }h > F
~-presentation
a prime g ~ G
being
and
the
sequence
R/EF,R1
vG
image
of
2 .... ,x n
K
p
h,
. Then
= gpv(Xl,X2
k
a
~*->Fab--~*->Gab
. Suppose we
the
define
~
0
.
generators
the
group
of
K
the
presen-
by
..... X n l Y 1 ..... y j , z i ..... z k)
where
(i)
y l , y 2 ..... y j , z l , z 2 ..... z k
(ii)
the
canonical
(iii)
the
images
of
6,(VG)
Obviously that f,
f
: VK
follows . We
images
of
there
is a
induces
an
. The
from
Theorem
have
k
= sVG
in
map
isomorphism
f,
i.i.
It
in
in
R/[F,R]
R
,
I
form
a basis
form
a
of
set
of
I
,
generators
.
surjective
assertion
elements
Y l ..... Yj
z i ..... z k
with
~ VG
of
are
about remains
f
: K ~ G
: Kab the to
. Also,
J Gab
lower show
and
central that
it an
epimorphism
series K
is c l e a r
then
is e f f i c i e n t
in
88
(6.14)
and
j+k
= sVG + n - r a n k
Gab
therefore
(6.15)
rank
Gab-SVG
: n-(j+k) rank
where
the
last
epimorphic.
COROLLARY
It
We n o t e
follows
6.6.
generated
PROOF:
inequality
This
from
the
~ rank
fact
Gab-SVG
that
f,
: V K ~ VG
is
is e f f i c i e n t .
variety
of e x p o n e n t
zero
every
(finitely
is e f f i c i e n t .
is c l e a r
from
Theorem
that by Corollary
of e x p o n e n t
K
In a n i l p o t e n t
group
Kab-SVK
follows
that
~ defvK
zero,
6.6
6.5
and
we have,
a homological
(6.12).
for a n i l p o t e n t
characterization
of
variety
the ~ - d e f i c i e n c y ,
namely
(6.16)
de~G
We c o n c l u d e
COROLLARY
with
6.7.
qenerahDrs
a qroup
PROOF:
The
the
Let
and
exists
= rank
G
all
nilpotent
in
Gr
of
with
, which
K
result
be a g r o u p relators,
emphasis
presentation
following
r+k K
Gab-SVG .
n
qiven
where
fact
turns
defGrG
o f class out
and
that
. The deficiency < d-1
of
k G/G d
is at
to be
~< n + r - ( r + k )
= n-k
[18].
by a ~-presentation
n = SGab
qenerators
is on the
groups
of C h e n
.
k
with
n+r
. Given
d ~ I there
relators
with
relators
suffice
for a
in the v a r i e t y
~d+i
least
the d e f i c i e n c y
of
of G
89 Since 6.6 k
G/G d to
is g e n e r a t e d
obtain
relators.
in t h e
an
Taking
of
K
as
relators
free
group
this
and
all
THEOREM tors
section
we
primes
p
7.1.
and
elements
of
G/G d
inverse on
Given
consider . Note
Let
r
= n
the
images
same
we by
of
may n
apply
generators
these
generators
Corollary
k
we
and
relators
obtain
a G{-pre-
.
IVx~=_GrouRs
In
SGab
~d_l-presentation
absolutely
sentation
by
G
a variety
that
be
relators.
bz_Special
V
a qroup
~
satisfying
is n e c e s s a r i l y
with
Suppose
Presentations
of
(Pp) exponent
a ~-presentation
that
Gab
is
for
by
qenerated
by
p = O
zero.
n+r
qenera-
n
elements.
Then
(i)
Gab
(ii)
VG
(iii)
defvG
(iv)
Let G
is = 0
free
of
rank
n
.
. = n
p
~ in
be
, whose
V-free
abelian
particular
a prime imaqes
or in
subqroup
F
p
= O
Gp ab o_~f
is
G
in
. I__ff x I .... ,x n
form G
efficient
a basis,
whose
then
embeddinq
f
V are
elements
of
they
qenerate
a
: F >
=
an
isomorphism
: FJF = GJG PROOF:
By
Theorem
6.1
n+r-r
so
that
we
may
we
i = 1,2 .....
have
= n ~< d e f v G
conclude
,
~< r a n k
Gab-SVG
.
<~ n - s V G
> G
induces
90
defvG sVG
= 0
rank
The
remaining
for
all
tors.
p
7.2.
Suppose
We
thus
may
are
then
easily
G
r
PROOF:
The
abelian generate from
Theorem
COROLLARY abelian
PROOF:
It
prove
and
r
relators
p
that
such
since
If
vPG
7.1
Let can
in
= O
is
F
free
part
of
of
of
are
rela-
linearly
G
abelian
images
r
If
Gab
subgroup
are
and
elements.
images
the
.
of
rank
x l , .... x
a basis.
The
n in
m
result
(iv).
G
be
be
generated
V
= O
(see
generated
by
xl,x2,...,x
n
form
a basis
F = G
= Gr
, H2(G,A)
H2G
that
generators
n
Gab
hence
G
since
Let
by
whose
Theorem
is
is w e l l - k n o w n that
2.2
a group
with by
n+r
n
generators
elements.
Then
generators.
Gab
7.1
A
Theorem
n+r
a ~-free
know
[60])
, hence
group
to
we
from
with
G
7.1
n
7.4.
in
Suppose n
rank
from
qenerated
generate
(Magnus
on
Gab
is
independent
group
of
a group
a prime
relators.
is V - f r e e
;
follows
they
follows
7.3.
be
elements
then
find
= 0
.
then
Gab
linearly
COROLLARY
= n
G
Theorem
Gp ab
and
Let
are
independent,
By
Gab
VG
.
that
x l , x 2, .... x m
PROOF:
~ i.e.
assertion
primes
cOROLLARY
= n
= 0
. But
elements,
hence
generate
G
Gab
result
. The
it
~ their
is
free
images
then
follows
.
and
that
of
n
let
G
be
a knot
group.
Then
for
any
thus
enough
.
Gab
= Z
G
has
[19]) , so
that
(see
[19]).
a presentation by
Theorem
7.1
It
is
by we
l+r
generators
have
VG=H2G=O.
9±
THEOREM tots
7.5.
and
Let
r
G
Then
(i)
dim
Gp = n . ab
(ii)
vPG
= 0 .
(iii)
defvG If
a qroup
relators.
elements.
(iv)
be
= n
..,x n
are
then
for
embeddinq
every
The
f
proof
G
where
= bk'Qk(ai'a2
denotes
words
Let
in
exponent
sum
qenerate
a ~-free an
hypotheses
> G
same
as
the
G = gpv(al,a2
is q e n e r a t e d
in
whose
a ~-free
induces
an
proof
of
V
by
n
.
imaqes
in
G= ab
subqroup
F
o__ff G
,
isomorphism
p-t k
.... ' a n ' b i ' b 2 ' .... br)
i_n_n b k
(p = O
subqroup
F
Theorem
7.i.
..... a n , b i , b 2 ..... b r l Y i .... ,Yr )
a l , a 2 ..... a n , b l , b 2 , . . . , b r o__rr p
of rank
n
, k
= l,...,r
. Suppose a prime). and
the
and
that Then
Qk
Qk has
a I ..... a n
embeddinq
f:F)---9 G
isomorphism
X G/GP l
i ~ I
It is a p p a r e n t
al, .... a n
qenerate
: F )
fP : F / F ~ l every
qenera-
GJG
is t h e
7.6.
PROOF:
of
n+r
i ~ I
COROLLARY Yk
prime
is e f f i c i e n t
elements
they
fPl : FJF
for
G
' p
by
'
a basis,
whose
induces
Ga pb
'"
form
PROOF:
a ~-presentation
that
~ in p a r t i c u l a r
xl,x 2
- -
Suppose
with
. Thus
G
of Theorem
that and 7.1
Gp ab
the
is g e n e r a t e d
elements
(p = O)
by
the
al,...,a n
in
or T h e o r e m
7.5
(p
images G
of
satisfy a prime).
the
92
COROLLARY
7.7.
Let
G = gpv(al
.... ,a
,bly)
where
the
exponent
sum
s
n
o_~f
b
i__nn y
o_~f
G
. I_~f p
induces
an
is n o n - z e r o . T h e n is a p r i m e
with
every
PROOF:
p /
s
qenerate
, then
the
then
Gp ab
i 9
If
p
2
Note the
of
is a p r i m e
with
p /
for
V
so c a l l e d
this
LEMMA
= Gr
finite f,
PROOF:
s
Theorem
[G:K]
It
THEOREM
8.2.
induces
, p an
yields
of
Magnus
the
f
: F)
> G
the
q
> 0
the
by
the
result.
assertion
follows
from
V
Theorem
= Gr
embeddinq be
is g e n e r a t e d
[58].
variety
be
Let
we
< ~
, the
have
that
: HqG
multiplication
morphis.
p / n
embeddinq
F
of
a number
.
a subqroup
K ~ G
with
= i
(q,n)
o__ff
. Then
is e p i m o r p h i c .
VI.16.4
just
part
of
the
: K + G
f, o C o r
is
then
HuRpert-Thompson-Tate
, say.
+ HqG
7.5
first
consider
f n
Since
The
we
Let
index
: HqK
the
"Freiheitssatz"
section
8.1.
. Theorem
n
IV.8.
In
subqroup
1
al,...,a
that
free
G/GPl
'
images
a
isomorphism
f~[ : F / F ~
for
a i ..... a n
follows
Let
prime.
by that
K
be
If
for
isomorphism
corestriction for
~ HqK
i ~
. But
f,
must
be
a subgroup
of
= p
k
exists.
Thus
by
[43],
I
+ HqG
n
q
map
since
we
(q,n)
= I
this
is a n
iso-
epimorphic.
G have
of f,
index
n
, say.
: Kq ~ Gq ab ab
'
Suppose then
f
93
f?~ : K/K~ = G/Gq for
every
PROOF: the
We
This
map
now
i ~ i
f.
is
.
an
: H~K
easily
get
easy
consequence
~ H~G
the
is
of
Theorem
i.l
since
Theorem
on
by
Lemma
8.1
epimorphic.
Huppert-Thompson-Tate
normal
p-comple-
ments.
COROLLARY
8.3.
Suppose
G
finite.
Suppose
that
f
: Kp 2 Gp ab ab
G
that
K
n N
such
= e
"
and
Let
Then G/N
K
there ~ K
be is
(N
a
Sylow-p-subqroup
a normal is
subqroup
called
a normal
of N
G.
of --
p-comple-
ment).
PROOF:
(Stallings
a p-group
there
[74]) exists
Apply n
Theorem
with
8.2
Kp = e n
with
. Set
q N
= p
= Gp n
. Since .
K
is
CHAPTER
CENTRAL
In t h i s c h a p t e r We h a v e
tried
to
tical
interest
Ganea
term
results Let
E
we have focus
which
(Section
that
2).
structure
2 we use
this m o d u l e
an explicit tions in
integral
The
perties
of
H~[E]
coefficient
properties
results theory the
last
two
of
refrained
ring
. In S e c t i o n
the
term by means
by one
of central with
and
sections
the S c h u r
special
by homology
with
of t e r m i n a l
we
obtain
some
multiplicator
as d e f i n e d
of perfect
yield
of a finite
5 we
in S e c t i o n
in p r i n c i p l e
the
by replacing z/qz
. The
in S e c t i o n groups.
known
nilpotent
E
take
The
o f the G a n e a
of the b e t t e r
pro-
groups.
in
and unicentral
applications
using
In
the e x t e n s i o n
is g e n e r a l i z e d
then
sequence
left.
. In S e c t i o n
coefficients
Also,
the u n i v e r s a l
of
constitute
theory
its g e n e r a l i z a t i o n
H,G
case
to the
via
term.
5-term
extensions E
exhibit
free p r e s e n t a =
the
term
stem covers
two s e c t i o n s
we present
2. F i r s t
and
N
Ganea
of
us to e x t e n d E
of
so-called
in
7 that
proofs~
Pontrjagin
structure
In S e c t i o n
stating
I we
o f the m o d u l e
in t h e s e
from
In S e c t i o n
associated
6 the
theore-
or the
for t h e i r
properties
in S e c t i o n
of group
extension.
with
classes
extensions.
sequence
4 relates
characterization
in S e c t i o n order
various
5-term
techniques
to y i e l d
associated
homology
of Schur
logical
the
results
we have
term enables
: H2Q ~ N
presented
integral
the
on central
Section
of Schur.
theory
over
l o o k at s t e m e x t e n s i o n s
3, and d i s c u s s
from
of t h e G a n e a
Ganea
sequence.
on t h o s e
a central
H,G
results
sequence
structure
homology
3 we d e f i n e
a closer
be
in
Section
with
spectral
description
is g i v e n .
attention
In p a r t i c u l a r
---97Q
a module
some
are o b t a i n e d
require
: N >---> G
EXTENSIONS
assembled our
V
8 a homoIn t h e
t e r m as o b t a i n e d theorems group.
on the
Also,
an
96 estimate
on
the r a n k
of
in S e c t i o n
I0 we
that
information
yield
about The
K/[H,H]
content
people. where
many
t i o n we
introduced
of
the r e s u l t s
have
Ganea
[69].
[30]
in
ture.
(Sections
found,
avoid
the u s e
(Section
of 9)
Vermani [70],
[84].
where
for
approach the G a n e a Finally here
bach
[24]
like
[31];
[37],
[70] w i t h to h a v e
In J o h n s o n
and
other
[35]
and
obtained of
free
The m o d u l e in the
by
struc-
litera-
its a p p l i c a t i o n s applications
than
Evens'
are
since
on the o r d e r
to G r e e n
[33]
we of
and
of P r o p o s i t i o n
9.5
is to be
in R o b i n s o n
found
also.
theory,
The
is due
relation
to
of the
in p a r t i c u l a r
with
Robinson. papers
in w h i c h
and L e e d h a m - G r e e n
to v a r i e t i e s . to yield
quotient
and
The presenta-
by means
to a p p e a r
are due
given,
some
[73]
bare hands,
first
estimates
homology
[48]
is u s e d
of a c e n t r a l
are
escaped
to m e n t i o n
term
iO
and
[72],
Gruenberg
[22].
where
The
of S e c t i o n
.
of p a p e r s
his
2) w a s
of S c h u r
the p r o o f
information
of E c k m a n n - H i l t o n - S t a m m b a c h
nilpotent
are g e n e r a l i z e d
the G a n e a
multiplicator
finite
K
5, 6 and more.
is s i m p l e r
sequences.
applications in
extended.
of S c h u r
G
seems
we would
are
theory
term,
with
seem
[28],
presentation
content
presented
H2G
Finally
theorems
papers
its d e s c r i p t i o n
theory
to Evens
-Yen
further
famous
(Section
not
of
number
the
3,
i.e.
are g i v e n
subgroup
P. H a l l
d~es
of the
spectral
The
if w e
lines
methods,
I, 4)
Our
GaschOtz -Neubuser
term
type,
Eckmann-Hilton-Stammbach
are due
also.
from
is o b t a i n e d .
a large
studied
is on the
Ganea
generalization 7, 8)
from
to m e n t i o n
ideas
(Sections
to be
H2G
The
is from
H,G
The
stems
K
normal
of S e c t i o n
by topological
presentation ture
the g r o u p
and
here
also used
nilpotent of H a l l ' s
a nilpotent
chapter
adopted
G
theorems
about
has
G. R i n e h a r t
for
of all w e have
have
We
some
H
of this
Schur
[22].
for
First
proved
prove
H2G
results [53]
presented
parts
of
the
In E c k m a n n - H i l t o n - S t a m m -
information
of a d i r e c t
about
product
the
Schur
of g r o u p s .
In
97
[21],
[23],
[36],
the
left and
The
following
tions: tion
[44],
[67],
[80]
also generalized papers
are
related
[49],
Sections
2; V e r m a n i
[85],
Section
of the
we
integral
shall
5, 6;
is e x t e n d e d
to
extensions. of the
indicated
Iwahori-Matsumoto
[47],
secSec-
9.
Generalities
prove
homology
(2.7)
to the c o n t e n t
3,
V.I.
section
sequence
to n o n - c e n t r a l
Kervaire
In this
the
some
general
of a central
results
extension.
o n the
Consider
structure a central
extension
(i.1)
i.e.
E
: N > h> G
an e x t e n s i o n
(1.2)
with
m(u,x)
g~Q
N ~ ZG
= u.x
,
,
. We m a y
u
£ N
define
, x
~ G
a map
m
: N×G ~ G
by
.
q
Since the
N
is c e n t r a l ,
following
diagram
N ~ (1.3)
The map
m
(1.4)
that
NxG ~m
N >h>
G
course,
that
m
makes
>>G ~g g~>Q
m.
: H.(NxG)
~ H.G
and by
the K ~ n n e t h
Theorem
a homomorphis~,
the n o t a t i o n B
It is c l e a r
commutative
~ : H . N ® H . G >--> H.(NxG)
Here we use Note
is a h o m o m o r p h i s m .
II
induces
(see II.5.~3)
m
H.G
is a h o m o m o r p h i s m
is n a t u r a l ;
more
to d e n o t e of graded
precisely,
if
m.)
H.G
.
the g r a d e d groups
group
of degree
{HiG}
•
zero.
Of
98
N #
(±.5)
> G--~
fll
f2 ~
N'>
is a m a p
1.6)
f3 ~
) G'
of c e n t r a l
H.N
Q
~ Q'
extensions,
@ H.G
-
then
~
>
the
diagram
H.G
fl*®f3 *~
if2*
H . N t G H . Gv
~t
>
H. Gv
is c o m m u t a t i v e . In P r o p o s i t i o n terms
of the
recall
the
pp.216-217. B' (G)
is the
we
shall
inhomogeneous
definition The
give
an e x p l i c i t
standard
of that
resolution
description
resolution.
resolution;
(non-normalized)
is a Z G - f r e e
_~'(G) Bn'
1.1
for m o r e
inhomogeneous of
Let
differential
0
ZG-module n
, n ~ 1
on all
details
~
in
briefly
see
[43],
resolution
Z
[XllX21 ...IXn]
is d e f i n e d
0 n [ X l l - - . I x n]
first
standard
, 0n>Bn_ - 1 0n-i ) ... Of) ~, 0o~ Z : "'" ~ Bn o free
us
of
O
.
, x.l ( G
and
the
by
= Xl[X21-.-Ix
n]
+
n-i
+
[
(-lli[x11---Ixixi+ll---IXn
] +
i=1 +
Note tion
that
; the m a p
0
o
is t h e n
defined
to be
the
augmenta-
~ : ZG ~ Z .
A l s o we S =
B' ~ zG o
(-l)n[xll . . . IXn_l ]
shall
need
to d e f i n e
(I ..... i,i+l ..... i+j)
shuffles.
An
is a p e r m u t a t i o n
(i,j)-shuffle ~
of
S
~
with
of the
set
~(k) < ~(£)
99
whenever
k < ~ ~ i
(i,j)-shuffles. signature the map
or
The
i < k <
signature
s(~)
o f the p e r m u t a t i o n N
as
PROPOSITION
~ . By
Q
o f an
~ . With
we denote
the
set of all
(i,j)-shuffle
this
notation
~
is t h e
we may
describe
follows.
1.1.
The map
p
is i n d u c e d
b y the c h a i n
transformation
p([XllX21..Ix i] ® [Xi+llXi+21 "''Ixi+j])
=
(1.7)
[ where
x I ..... x i ~ N
PROOF:
H,G
. The map
~
ZN~ZG & CN@CG
where
CN×CG
p.288
ff.) , w h i c h
~(-)
b y the duces
denotes
shall
is t h e n
use,
I • - • I xx(i+j)
]
.
for
induced
the
embeddings the
is i s o m o r p h i c
subcomplex
Z__N c-~ C__N a n d
short,
b y the
the n o t a t i o n
following
compo-
~(NxG) in
~-~ C G
of
C__N and
. Also,
~(-)
we have
~ the m a p
Z__G c-~ C__G . We
C__G
recall
~
(see
[57],
denoted
is i n d u c e d
that
~
in-
embedding
in the K O n n e t h
sequence
y
® H.(C_G) >
(see
[43],
is the E i l e n b e r g - Z i l b e r
the A l e x a n d e r - W h i t n e y Let
product
to
of cycles
~. : H.(C_N)
Finally
% ~ C(NxG)
I CNxCG
the c a r t e s i a n
(1.9)
248).
. We
~ G
I x×(2)
of chain maps
(1.8)
by
x i + 1 ..... x i + j
= H,(Z ®G ~'(G))
C__G = Z Q G ~' (G) sition
and
(-1) s (~) [x×(1)
map~
x I .... ,x i £ N
p.172 map,
it is g i v e n and
) H.(CN
as
® C_G)
ff.). i.e.
the h o m o t o p y
follows
Xi+l,...,xi+ j ¢ G
(see . Then
inverse
[57],
p.243/
of
100
¥([xll --.Ix i] O [xi+ll ...Ixi+j])
=
(1.10) (-I) s (~)
where,
for
[XT(1) I "''IXT(i+j) ] ® [X~(1) I "''IX~(i+j)]
I ~ k ~ i+j
and
~ ~
= I X~(k) T(k)
' for
ke
, otherwise
= I X~(k) X~(k)
I ~ ~(k) < i ,
, for
[e
i+I ~ ~(k) ~ i+j
,
, otherwise
It is then clear that
m~y([xll--.ix i] ® [xi+lJ---Ixi+j]) (I.II)
= m( [ (-l) s(~)[ (XT(1) ,X (i)) I (XT(2),Xc(2)) I ...I (XT(i+j) ,Xo{i+~])
=
since,
[ (-1) s(~)
~9
for each
If we consider, (1.12) then
=
[x~(1) I ---Ix~(i+ j) ]
k , either for N ~
N
XT(k)
abelian,
or
x (k)
is
e .
the extension
) N ---9> e ,
(1.4) yields a map
(1.13)
PN
: H.N
® H.N
~ H.N
•
It follows from the properties
of the K~nneth Theorem and the map
(I.IO) of the Eilenberg-Zilber
Theorem
in (1.13) Moreover,
is associative
the diagram
(see [57], p.242)
and commutative
¥
that the map
(in the graded sense).
101 e>----> N
(i .14)
I
~J
N>
yields,
>}N
I
) N ---9> e
by naturality
(see
(1.6)),
H. (e) ® H . N
~=i
H.(e)
PN
is c o n c e n t r a t e d
follows
t h a t the e m b e d d i n g
We h a v e
thus p r o v e d
PROPOSITION
1.2.
We r e m a r k that, Pontrjagin
and c o m m u t a t i v e
the ring
1.3. The m a p
l
and
is a u n i t
N
qraded
H.N
Ho(e)
= Z
, it
for the m a p
, the m a p
N
makes
N .
H.N
r i n q w i t h unit.
c a n be
identified
space
K(N,1)
N : H . N ® H.G ~ H . G
over the rinq
Naturality
zero,
with
. For the
(I.I) we have
applied
e >--9 G
iF
N ~---> G
yields
~ H.N
ring of the . E i l e n b e r g - M a c L a n e
qraded module
(1.16)
H.N
in d i m e n s i o n
H.(e)
obviously,
extension
PROPOSITION
>
For a n y a b e l i a n q r o u p
into an a s s o c i a t i v e
PROOF:
H.N II
H.N ~ H.N
central
~
I
(1.15)
Since
the d i a g r a m
H.N
.
to the d i a g r a m
---9> G
Ig g~>Q
makes
H.G
into a
the
i02
H.(e)
(9 H.G
B=I
) H.G
(I.17) H.N ® H.G
so that
B
is unitary.
N
)
Associativity,
H.N (9 H.N (9 H.G
~N ~
H.N (9 H.G
~
H.G
i.e. the fact that the diagram
> H.N (9 H.G
1.i8)
is commutative,
again
and of the map
y
follows
(1.10)
>
H.G
from a s s o c i a t i v i t y
of the KOnneth Theorem
of the Eilenberg-Zilber
Theorem
(see [57],
p.242) .
PROPOSITION H.(BxC)
PROOF:
1.4. Let
~ H.B (9 H.C
B , C
be torsion-free
abelian groups.
Then
as rings.
We first remark that the integral h o m o l o g y of a torsion-free
abelian group is clearly torsion-free.
The K~nneth Theorem then yields
an isomorphism
(1.19)
H.B (9 H.C 2 H. (BxC)
and it is then obvious
that the diagram
(H.B(~H.C) (9(H.B~{.C)
i®To~.
) (H.B~H.B) (9(H.C~H.C) NB®~C
(i . 20)
~J
H.B (9 H.C
~5 H. (BxC)
is commutative.
Here
of graded groups.
NBxC
T denotes
H. (BxC)
the switching map for the tensor product
i03 COROLLARY
1.5.
Let
A
the e x t e r i o r
a!qebra
PROOF:
assume
First
the n u m b e r vial. B
For
and
q
EzA
A
over
both
A
finitely
of generators.
q i> 2
C
be a t o r s i o n - f r e e
we have smaller
generated. q = 1
A = B×C q
Then
H.A
is
.
Let
than
a b e l ian q r o u p .
with
We a r g u e
then the
induction
the c o n c l u s i o n
numbers
, so that b y
by
on
is t r i -
of generators
induction
and
of
Proposition
1.4
(1.21) For
H.A = H.B ® H.C
A
non-finitely
= E z B ® E z C = Ez(B×C)
generated
a direct
limit
= EzA
argument
.
easily
yields
the r e s u l t .
PROPOSITION : H.C
PROOF:
1.6.
Let
C
® H.C ~ H.C
Since
is
H C ~ 0 n
be a finit e cyclic trivial
, for
in p o s i t i v e
n = 0
and
n
q r o u p ,. T h e n
the map
dimensions.
odd,
only,
it s u f f i c e s
to
consider
: HnC
for
n,m
9 1
, odd.
® HmC ~ Hn+mC
But
then,
the proposition
is p r o v e d .
We
t h a t w e may,
finally
homology
note
with
coefficients
the t e n s o r
products
associated
with
PROPOSITION qroups.
(1.22)
be
1.7.
Let
K
is e v e n
instead
in a f i e l d
taken
an a b e l i a n
n + m
over
group
K N
b e a field,
so t h a t
of taking K
integral
becomes
let
NI,N 2
Then
H . ( N I X N 2,K)
~ H. (N I,K)
the ring
a graded
~ K H. (N 2,K)
being
two
take that
H.(N,K)
algebra
be
, and
homology,
, the o n l y c h a n g e
. Of course,
and
Hn+m C = O
over
abelian
K
.
i04
a_~s K - a l g e b r a s .
PROOF:
Let
The
proof
is
N>---> G ---~Q
(2.1)
as
first
V.2.
The
be
restriction show
how
R~--9 F---~> G
¥
a central
to
HIN
may
be
. Recall
(2.2)
H2G
Hopf's
2.1.
is a r e p r e s e n t a t i v e
(2.3)
y(u
PROOF: duced
By
we
x
We
define
the
Ganea
term
of
~ in
: H,N
Q H,G
terms
of
a
~ H,G
free
. We
shall
presentation
p.204) .
is ~ G
1.1
a representative
of
u
~ N
and
z
£ F
, then
=
the
[v,z][F,R]
map
¥ =
p
: HIN
® HIG
~ H2G
is
in-
by
(2.4)
~([u]®[x])
consider
the
short
Ra b>
associated
with
R )
a
with
fs
function
=
[ulx]-[xlu]
exact
K ) ZG
> F = I
~
, u
( N
, x
(
proof
of
Proposition
G.
sequence
IF
f~ G and
~)
IG
(see s(e)
[43],
= e
p . 1 9 8 ) . If
, then
we
may
G the
1.4.
have
® x[G,O])
Proposition
Proposition
n R/[F,R]
~ F
of
of
Term
extension.
described
[43],
I__ff v
proof
Ganea
® HIG
[F,F]
(see
the
~ H2G
that
~
formula
PROPOSITION
Now
to
y : N ~ Gab
the
by
analogous
II.4.4,
a commutative
diagram
s
: G ~ F
construct,
is as
in
i05
~2 (2.5)
q~i ,[,
qo21 Rab
by
~I
)
II
>
ZGQFIF
%01Ix ] = I @
(SX-l)
>> IG
setting
e2[xly]
Applying
the
H2G = [F,F]
= sxsy(s(xy))-I[R,R]
functor
Z ®G -
to d i a g r a m
n R/[F,R]
= ker(l
® ~)
¥(u @ x[G,G]) (2.6)
=
since
u x = xu that
G
of
2.2.
(2.7)
X,y
E G
.
and u s i n g
the
fact
that
obtain
(~2([ulx]-[xlu])
[F,R] )-I)-I[F,R]
susx (su) -i (sx) -i IF ,R]
su = v
proved
(Ganea
[30])
Then
, sx = z
that
the c o n c l u s i o n
the m a p
representatives
y
as d e f i n e d
and of the
free
follows. by
(2.3)
is
presentation
PROOF:
and
Only
R >
)F
Let
E
: N >h_h> G g-~> Q
b e a central
exten-
th e s e q u e n c e
N~Gab ~H2G
is e x a c t
~
6E H2Q ~ N
g*> ~
Gab
-~ 0 Qab
natural.
the e x a c t n e s s )>G
duced
presentation
of
. By H o p f ' s
N
also
the c h o s e n
of qroups.
let
,
(2.5)
( G
.
THEOREM sion
. Setting
we h a v e
independent of
X
= susx(s(ux))-l(sxsu(s(xu) =
Note
we
,
be of
at
H2G
remains
a presentation Q
formula
. Of c o u r s e for
H2G
of
G
to be proved. and
R~--9 S , H2Q
To
this
end
S ~-~ F ---~>Q
the
in-
>)N
we have
is a p r e s e n t a t i o n
i06 ker(g,:H2G~H2Q)
= ker([F,F]nR/[F,R]~[F,F]nS/[~S])
(2.8)
=
R
n
[F,S]/[F,R]
= [F,S]/[F,R]
since
[F,S]
the o t h e r
~ R
hand
(2.9)
Thus We
it f o l l o w s
of the
Theorem
--
show that the map G
the
N = S/R 2.1
quences
On
that
second
¥
is c l o s e l y
center
be of
related
NI>---+ G --e>Q a second G
to t h e
central
extension.
also
N.
. Note
ordinary
be a c e n t r a l
that
'
are p o s s i b l e
is c e n t r a l .
= [F,S]/[F,R]
. Let
N2/NI>--9 G --9)Q/N 2
N 2 c Z2G
that
is p r o v e d .
in the g r o u p
commutator
let
2.2
fact
from Proposition
im(¥:N(DGab~H2 G)
finally
and
, in v i e w
,
Note
= Z.G
1
choices
for
NI,N 2 • Consider
then
the
extension, that i=I,2
1
associated
se-
in h o m o l o g y .
H2G
(N2) ab (9 G a b
(2.10)
N2/N I ®
(G/Nl)ab
¥ )
H2 (~/~i) ~ ~2 (~/N2) 6,1 Ni
i
Here
~
denotes
PROPOSITION
2.3.
the obvious
projection.
6,¥~(u[N2,N2]
PROOF:
Consider
ciated
presentations
® x[G,G])
a free p r e s e n t a t i o n of
= [u,x]
R~--+ F ---+}G
G / N i , i = 1,2
, u
of
£ N2
G
, x
and
~ G
.
the a s s o -
I07
(2.&l)
Note and We
S >
that let
v
then
)F ---->)G/NI ; T~--) F - - - ~ G / N 2 .
S/R
~ NI
( T
, z
, T/R 6 F
~ N2
be
, T/S
~ N2/N I
representatives
. Let
of
u
u
, x
, x
~ G
respectively.
have
6,¥~(u[N2,N2]~[G,G])
= 6, y ( v S ~ z [ F , F ] S )
= 6.([v,z][F,S])
(2.i2)
-- [ v , z ] R
But
~ N2
in
F/R
J G
we
[v,z]R
have
=
[u,x]
~ S/R
, so t h a t
our
~ ~i
proof
is
complete. Note
that
it
follows
(2.13)
from
Proposition
[ , ] : Z2GxG
is a b i l i n e a r
map.
Of
2.3
that
~ ZiG
course,
this
is w e l l - k n o w n
and
is e a s y
to p r o v e
directly.
If w e
REMARK:
consider
an a r b i t r a r y
variety
V
instead
of
Gr
we
extension
i9
obtain
PROPOSITION Then
the
2.4.
yt N@Gab~>
is e x a c t
the
: N ~-h> G
g-~> Q
be
a central
Z
"
sequence
(2. iO)
PROOF:
E
Let
and
Let
h.
g*) )Gab
~ 0 Qab
natural.
f : F
associated
commutative
E ~-~N
VG~VQ
~G
V-free
diagram
be
a V-free
presentation
with
exact
rows
presentation of and
Q
. Then columns
of
G
and
we obtain
the
gf
: F-9> Q
following
i08
H2F
= H2F
L Y > H2G ~ H2Q ~ N ~ Gab ~ Qab ~ 0
N~Gab
\
It is t h e n o b v i o u s exact.
L
L
O
0
V.3.
Various
Classes
3, 5, 6 we
for n o t a t i o n a l
any variety H2(-, -)
[72],
~
is r e p l a c e d
be
extensions
of
Q
We
group.
found
is an
such that
E (3 .i)
:
N) f2 ~
E'
H 2-
at the e n d
sequence
[35]).
We
isomorphism
(2.10)
of sequence
is c o m m o n l y
is a l s o
(2.7).
known
as
the
in t h e v a r i e t y
V = Gr
w e do, h o w e v e r ,
is t r u e
is r e p l a c e d indicating
the n o t i o n
say t h a t and fl
E'
by
how
of S e c t i o n s
introduce
: N~--) G - - ~ Q
two
V-
, in
and
to e f f e c t
5, 6. of
isomorphic
(not n e c e s s a -
: N'>---) G' - - ~ Q
: G ~ G'
inducing
the d i a g r a m
> G
---@>Q
fl~
il
.. N' >---->G' --+>Q
is c o m m u t a t i v e .
Accordingly~if
morphism
[[Ell
class
i,
Extensions
this
. Remarks
(see G r u e n b e r g
if t h e r e
do
if
first
E
what
of what
Much
zero,
We
the
of C e n t r a l
shall
7(-,-)
to be
that
the n a t u r a l i t y
reasons.
by
11
~ N ~ Gab ~ Qab ~ O
develop
extensions
isomorphic : N ~ N'
are
a fixed
rily central)
[73].
of e x p o n e n t
translation Q
shall
,i
2.2
from
mainly
f2
~ VQ
follows
of S c h u r
are
VG
from Theorem
theory
Let
L
Naturality
In S e c t i o n s
the
1
Q
is g i v e n ,
of an e x t e n s i o n
E
we may
speak
. It is c l e a r
o f the
that
iso-
equivalent
±09 extensions
belong
to the
same
isomorphism
class,
thus
explaining
our
notation. In the ciate
sequel with
we shall
E
: N >h>
the h o m o m o r p h i s m of
HA[El
H2(Q,N)
(3.3)
where
associated
is the with
(3.4)
G
extensions.
We m a y
then
asso-
g~Q
: H2Q ~ N
(see E = 6,
hA[El
6E ,
central
the e x t e n s i o n
(3.2)
sequence
consider
given
II.5.1).
b y the u n i v e r s a l
By P r o p o s i t i o n
coefficient
II.5.4
we h a v e
,
"connectlng
map"
in the
5-term
homology
sequence,
(3.2) E H2Q ~ N
H2G ~
We use p r o p e r t i e s
of
h~
HA[E]
Gab ~
to d e f i n e
Qab ~ O
various
o
classes
of c e n t r a l
extensions.
DEFINITION:
The
central
extension
(i)
a commutator
(ii)
a stem
extension,
if
(iii)
a stem
cover,
hA[E]
It is c l e a r of
Q
class.
from
of b e i n g
e x t e n s i o n , if
in one
tator
extensions.
if
naturality in one
N e x t we g i v e
to be
(3.2)
of the
~A[E]
HAlE] is
that
of these
is c a l l e d = O
is e p i m o r p h i c ;
isomorphic. the p r o p e r t y
classes
of a c e n t r a l
depends
various
characterizations
classes
(i) , (ii) , (iii) . We
only
on
its
for a c e n t r a l first
extension isomorphism extension
consider
commu-
110 PROPOSITION lowinq
3.1.
For
statements
a central
are
(i)
E
(ii)
N>--> Gab--%> Q a b
(iii)
g,
(iv)
N
PROOF:
~
n [G,G]
The
H&[E]
g~ Q
the
fol-
is e x a c t ; ,
.
equivalence
(3.4)~ since
: N~-~h> G
extension;
[Q,Q]
= e
E
equivalent.
is a c o m m u t a t o r
: [G,G]
extension
of
E = 8,
(i) and
. Given
(ii)
(ii)
follows
from
it f o l l o w s
the
from
5-term
sequence
the d i a g r a m
[G,G] ~ [ Q , Q ]
N>-->
N ) that
g,
course
: [G,G] N
g >>Q
> Gab - j % ) Q a b
2
n [G,G]
G
[Q,Q] = e
. Thus
(iii)
. Finally
if
holds.
(iv)
If
holds
(iii)
then
holds
then
of
N~--9 Gab----~> Qab
is exact.
PROPOSITION Q
3.2.
The
equivalence
by
Ext(Qab,N)
are c l a s s i f i e d
PROOF:
The
universal
coefficient
Ext(Qab,N)
shows if Note
that
~[E]
lies
that
tension
E
if
[E']
Z~xt(Qab,N) 1 denotes
N ~--> G a b - - ~ > Q a b
extensions.
the
then
of c o m m u t a t o r
extensions
of
sequence
> ~> H2(Q,N)
is a c o m m u t a t o r in
classes
H~Hom(H2Q,N)
extension,
i.e.
HAlE]
= O
if and o n l y
o
equivalence Z[E']
= AlE]
class
of the
. We n o w
abelian
turn
ex-
to stem
ill PROPOSITION lowinq
3.3.
For
statements
a central
are
extension
E
is a stem
(ii)
E 6.
: H2Q ~ N
is e p i m o r p h i c ;
(iii)
h.
: N ~ Gab
is the
(iv)
g*
: Gab ~ Qab
(v)
N ~
lence The
fol-
z e r o map;
;
of
(i) and
(ii) , (iii) , (iv)
PROPOSITION
of
3.4.
(iv)
For
statements E
is a stem
(ii)
E ~ 6. : H2Q ~ N
(iii)
g.
continue
in the
subsequent
(v)
follows
from
the
from
5-term
(3.3).
The
sequence
equiva-
(3.4).
is trivial.
extension
E
: N >--~ h G
9~Q
the
fol-
,
cover;
easily
We w i l l
and
equivalent
;
: Gab 2 Q a b '
This
(ii)
is c l e a r
a central
are
(i)
PROOF:
the
extension;
equivalence
equivalence
lowinq
g~Q
[G,G]
The of
: N >--~h> G
equivalent.
(i)
PROOF:
E
and
follows
the
g.
: H2G ~ H2Q
from
discussion
the of
5-term stem
is the
zero map.
sequence
extensions
(3.4). and
of stem
covers
sections.
V.4.
Indecomposables
Let
(4.1)
E
be a c e n t r a l graded
module
: N> h) G
extension. over
the
g-~> Q
By P r o p o s i t i o n graded
ring
H.N
1.3
we k n o w
. We m a y
that
thus
H.G
ask w h a t
is a the
ii2 relations H.G We
are b e t w e e n
. In this associate
the
section with
(4.2)
we
the
extension shall
An e l e m e n t
of
H G n
posable,
and
We
discuss
shall
give
H.N-module
In = c o k e r ( ~
whose
an e l e m e n t I.
(4.1)
:
some
H.G
whose
results
structure
in this
of
direction.
group
I
defined
by
® H,G ~ HnG) 3
is n o n - z e r o
image
the m o d u l e
the g r a d e d
@9 HiN i+j =n i>71
image
and
is zero
in low d i m e n s i o n s .
in
I
is c a l l e d
n
is c a l l e d
indecom-
decomposable.
In d i m e n s i o n
n = I
we h a v e
to c o n s i d e r
(4.3)
By the h.
: HIN @ H o G ~ HIG ,
explicit
: N ~ Gab
PROPOSITION In d i m e n s i o n
description
. We m a y
4.1.
thus
I l ~ Qab
n = 2
of
is c l e a r
~IH2N
® HoG
is the G a n e a
from
we h a v e
map
N~
with
N ~h'., G
we
obtain
this
agrees
with
to c o n s i d e r
h.
¥ . Applying
>N
(1.7)
"
the e x p l i c i t
agrees
in
state
: (H2N ® HoG)
It
N
)>e
g)>Q
@9 (HIN ® HIG)
description : H2N ~ H2G
of
-- H 2 G
~
•
in
(1.7)
. By d e f i n i t i o n
the n a t u r a l i t y
of
¥
that ~IHIN
@ HIG
to the d i a g r a m
li3 Y
N ®N
Y
N ® Gab
From
this
we m a y
(4.5)
infer
P(H2N
Thus
in o r d e r
PROPOSITION
pROOF: ately
The
> H2G
Q HoG)
12
COROLLARY E
4.3.
(2.7)
E
Let
E
is e n o u g h
to c o n s i d e r
7 : N(DGab~H2 G-
: H2Q ~ N)
with
the e x t e n s i o n
(4.1)
immedi-
~s a stem
decomposable~
in S e c t i o n
immediately 3 and
We do not k n o w ever we m a y
prove
. Note
that
the
only
follows
extension.
if all e l e m e n t s
in d i m e n s i o n
if all
elements
in d i m e n s i o n
elements
I ar____ee
in d i m e n s i o n
2
. if and o n l y
from
if
12 ~ H2Q
the d e f i n i t i o n
.
of these
extensions
4.2.
far-reaching following
Proposition
I
.
and all
12 = O
extension
Proposition
of a n y
I i ~ Gab
I I ~ Gab
i.e.
is a c o m m u t a t o r
This
if and
i.e.
be a c e n t r a l
if and o n l y
i.e.
= ker(6,:H2Q~N)
HA[E]
6,E =
: N>--9 G --9> Q
cover
E
im(g,:H2G~H2Q)
we h a v e
indecomposable,
are E
¥
extension
indecomposable,
I,
it
® HIG)
associated
II.5.4
is a stem
are
PROOF:
...
yields
But b y P r o p o s i t i o n
iii)
~ H2Q ~
p(HiN
12
12 = c o k e r
ii)
~
ker(HA[E]
sequence
(4.6)
i)
...
O
that
to c o m p u t e
4.2.
~
~
lh.
l(~h. ~
(4.4)
of
> H2N
results
results 4.4(i)
in h i g h e r
which
dimensions;
simplify
generalizes
the
(4.5).
how-
calculation
ii4 PROPOSITION
4.4.
im(B (ii)
If
N
n ~ 2
: HIN ® Hn_IG ~ HnG)
is t o r s i o n - f r e e ,
im(B PROOF:
(i) For e v e r y
then,
the
for
~ im(~:HiN
_
square
¥®1
~
N ® Hn_IG Since
y : N~
(ii) We show
~ H2N that
im(N We have
2 ~ i < n
: Hi_iN
@ Hn_iG
is t o r s i o n - f r e e , ® N ~ HiN
is c o m m u t a t i v e
also
is, w h e n c e
the result.
,
-l) i m ( ~ : H i N
@ Hn-i G
~®i> Hi N @ H n-1.G
Hi_i N ® H n _ i + i G N
@ Hn_ I.G ~ HnG)
square
Hi_IN @ N
Since
,
HnG
¥ ® i
: H i - I N @ H n - i + I G ~ HnG)
a commutative
~ HnG)
H2 N @ Hn_2 G
}
is surjective,
for
2 6 i ~ n
following
N @ N @ Hn_2 G
@ Hn_2G
! im(~:H2N
: HIN @ Hn_IG ~ HnG)
(i) By n a t u r a l i t y
we have
B
>
H,N = EzN
is s u r j e c t i v e
HnG by C o r o l l a r y
and the r e s u l t
1.5.
Thus
follows.
~
H
n G)
115
V.5.
The
results
is k n o w n
in t h i s
a s the
role which
H2Q
(or a g r o u p of
Q
paper
[37],
E
terms
5.1.
U
~A[E]
: H2Q - N E
constitute
[73].
group
extension
in p r i n c i p l e
It is b e c a u s e
plays
is o f t e n
be a subqroup
with
N = H2Q/U
with
stem
it)
Stem Covers
in t h i s
called
the
what
of the cru-
theory
that
Schur multipli-
and
stem cover
come
from Hall's
of
H2Q
there
[35].
Let E
[72],
homology to
and
section
of S c h u r
isomorphic
Let
associated
Extensions
the n e x t
second
see a l s o
stem extension
with
the
. The
PROPOSITION
PROOF:
and
theory
cial
cator
Stem
U = ker
. Choose the
then
exists
a
HA[E]
any central
canonical
yields
. Then
extension
projection.
that
U = ker
E
The
: N~-9
5-term
6E , = ker
HA[E]
G
~Q
sequence and
that
is a s t e m e x t e n s i o n .
By definition only
if
of a stem
U = 0
cover,
° We r e m a r k
that
is i s o m o r p h i c
to a s t e m c o v e r
To prove
let
and
this
consider
E'
E
inverse
of
cover
: N~--9 G --4>Q
: H2Q~---) G' --9) Q
b e the
~
~
~.l H 2 ( Q , H 2 Q)
obviously that
is a s t e m
with HA[E]
if and of
Q
H A [ E ' ] = I H 2 Q. : H2Q ~ N
~he d i a g r a m
(5.1)
a way
any stem cover
¢ : N ~ H2Q
H2(Q,N)
Then
a stem extension
n(~,a[E])
ALE']
= ~,A[E]
Hom(H2Q,N)
IV. n
~
= IH2 Q
Hom(H2Q,H2Q)
, so t h a t w e m a y c h o o s e
. Proposition
II.4.1
then
E'
yields
in s u c h the d i a g r a m
I16
E
:
N>--9
~L
(5.2)
showing The
that
name
stem
PROPOSITION some
stem
PROOF:
H2Q
E t
:
E
and
cover
5.2.
G
~ Q
L
,
---~G' ---9> Q
E'
belong
is m o t i v a t e d
Every
stem
(G.
Rinehart by
epimorphism.
[69]) AlE]
Now
=
Let
following
the
E
of
isomorphism
Q
class.
result.
is e p i m o r p h i c
: N>---> G ---9-Q
~ { H2(Q,N)
consider
imaqe
of
the
. Then
be
a stem
~ = E({)
extension,
: H2Q
~ N
is an
diagram
>-
E
>
H2
rI
(Q, H2Q)
>> H o m (H2Q, H2Q)
L~.
(5.3)
E x t (Qab ,N)
To prove
our
Z(~)
Then
proposition
: IH2 Q
Let
n(~-~.(9'))
=
}~ ~ E x t ( Q a b , N )
with
is e p i m o r p h i c ,
since
exists viously
9
has
the
f : G'
we
> H 2(Q,N)
first
find
( H 2 (Q,H2Q] : ~-~
n(~)-~.n(n')
X~ = ~ - ~ . 9 ' 9
: H2Q
~ N
with
required
with
~ G
Z
>
~'
u ~ Ext(Qab,H2Q)
an extension
find
by
extension
E x t (Qab ,H2Q)
be
same
cover.
characterized
and
to t h e
A[E']
such
: 0
that
. Thus
9.
Now
: H 2 Q ~---> G' --9) Q
E
:
(5.4) --->> Q
with ff(~')
there
let
It
~.(~) = IH2 Q
~ Ext(Qab,N)
follows
that
D = Z(u)+~' E'
: H2Q ~-eG'
II.4.l
=
exists
: Ext(Qab,H2Q)
= ~ . By Proposition
E'
> G
be
is e p i m o r p h i c .
properties.
Hom(H2Q,N)
~ ~ H 2 ( Q , H 2 Q)
~.(u) = ~ . D e f i n e
with
N>
. Now
ff >>
we may
there Ob--e> Q
then
ii7
commutative.
Since
~
is s u r j e c t i v e ,
f
also
is.
Thus
the p r o o f
is
complete. From
the d e f i n i t i o n
PROPOSITION
5.3.
stem
of
covers
PROPOSITION of
Q
In this
Q
Let
PROOF:
as m a n y
speak
can b e
[69])
Consider
obtain
= ~ ~ H 2 (Q,N)
the
be
of
if
>
covers
.
cover
and
f'
let
= 0
: G -- G
, every
.
the d i a g r a m
H ~ Hom(H2Q,N) f*l
Z > H2(~,N)
. Then
of stem
Ext(Qab,N)
to a m a p
first
Q
a stem
f*l
_ H ~ Hom(H2~,N)
we m a y u s e
= f*(K(~))
: H2Q ~ N
to
diagram
Ext(~ab,H2~)
)
E > H 2 ( ~ , H 2 ~)
~.i
(5.6)
q : A[E]
~(~)
= IH2 Q
this
property.)
clear
that
tion
II.4.3.
)
~ H 2 ( Q , H 2 ~)
. (Otherwise But t h e n
e,(q)
= f*(~)
~ Hom(H2~,H2~)
~.~
Ext(~ab,N)
Since
class
cover
~ Z ) H2(Q,N)
Ext(~ab,N)
of
.
Then,
lifted
f*~
A[E]
stem
extension.
Ext(Qab,N)
Let
o f the
classes
Ext(Qab,H2Q)
isomorphism = 0
infer
isomorphism
in
: H2Q>--9 G --~J Q
f : Q ~ Q
(5.5)
different
Ext(Qab,H2Q)
E
easily
are e l e m e n t s
be a stem
(G. R i n e h a r t
we m a y
is o n l V one
if
shall
: N>--> G - - 9 > Q
homomorphism
are
There
5.5.
covers
as t h e r e
only
c a s e we
PROPOSITION E
There
5.4.
if and
of stem
E
) H2(~,N)
is a stem
replace e,H(~) . The
9.1
E
~
cover,
b y an
we may
isomorphic
= e . Since conclusion
Hom(H2~,N)
suppose extension
Ext(Qab,N)
then
follows
= O from
with
, it is Proposi-
I18 COROLLARY jective
5.6.
f'
PROOF:
[72])
representation
there exists that
(Schur
of
a complex
induces
Consider
Let Q
Then,
Since
C*
G
be a c o m p l e x
is a stem cover f'
pro-
of
: G ~ GL(n,C)
such
f .
the e x t e n s i o n
is i n j e c t i v e
H2Q >
)> PGL(n,C)
(it c o n t a i n s
we m a y a p p l y our P r o p o s i t i o n
(5.8)
if
linear representation
C* ~---9 GL(n,C)
(5.7)
f : Q ~ PGL(n,C)
)
1
5.5
G
to y i e l d
f'
= O
: G ~ GL(n,C)
and
with
>>
If'
C* >
all roots) , E x t ( Q a b , C * )
If
> GL(n,C)-~ PGL(n,C)
commutative.
REMARK: purely for
We n o t e formal;
H2(Q,N)
an a n a l o g o u s
t h e y o n l y use , see
(II.5.1).
universal
we h a v e r e s u l t s We r e f r a i n
that the p r o o f s
for
positions
5.1
The s i t u a t i o n
through
H 2-
for v a r i e t i e s
H2(-, -)
7(-,-)
sequence
not be e p i m o r p h i c , that
Qab = Qq ab
V
coefficient
V-
sequence
since
and
exact
~
5.5
sequence
(see
5.1
by
zero
(III.8.8)) ,
through
t h e y are o b t a i n e d
H2(-, -)
are
of e x p o n e n t
holds
to P r o p o s i t i o n s
of e x p o n e n t
that one h o p e s
However, for
exact
through
7(-,-)
5.5.
auto-
in P r o -
5.5.
It is c l e a r
efficient
by
5.1
for any v a r i e t y
them explicitly
complicated. by
Since
corresponding
from s t a t i n g by replacing
the u n i v e r s a l
coefficient
~
matically
of P r o p o s i t i o n s
V(Q,N)
in g e n e r a l .
be p r o j e c t i v e
there
q > O
to r e p l a c e
seems H 2-
to be m o r e by
V q-
is no short e x a c t u n i v e r s a l
~ in p a r t i c u l a r ,
H
Under
the a d d i t i o n a l
over
z/qz
we h a v e
in
(III.8.1)
hypothesis
and cowill
on
(see (III.8.11))
Q
ii9
E : V(Q,N)
so that
in this
positions
5.1
Ext(Qab,N)
case we through
= 0
projective.
will
Note
~ Hom(vqQ,N)
immediately 5.5.
Note
that
automatically
finally
that
obtain
if
be q
results
analogous
the h y p o t h e s i s satisfied, is s q u a r e
to P r o -
corresponding
when free,
Qab Qab
to
is z / q z will
always
z/qz-projective.
be
V.6.
The
theory
quotient in m i n d the
of c e n t r a l
group we
Q
shall
particular
sarily
Central
strong
extensions
is p e r f e c t , prove
i.e. w h e n of r e s u l t s
the=hypothesis
shall
of P e r f e c t
usually
Grou~s
is p a r t i c u l a r i l y
a series
results we
Extensions
when
Q = [Q,Q]
. With
in this
section.
that
use w e a k e r
nice,
Q
be p e r f e c t
hypotheses
the
this
case
Since
for
is u n n e c e s -
in the
statements
of our p r o p o s i t i o n s .
~ROPOSITION E'
r
6.1.
h !
: N t >
>
G'
: N ~ N'
and
Ext(Qab,N')
= 0
(i)
There
Suppose g
s
~
Q'
: Q ~ Q'
E are
: N ~-h~ G two c e n t r a l
be g r o u p
g~ Q
and
extensions.
homomorphisms,
and
Let suppose
that
.
exists
inducinq
t : G ~ G'
H2Q
(6.1)
!
that
6E ~
r,s
if and
only
if
N
~r
s.l 6E' H2Q '
*
N t
is c o m m u t a t i v e . (ii)
If
t
exists~
it is u n i q u e
if and
only
if
Hom(Qab,N')
= O
.
120 PROOF:
If
converse
t
exists,
consider
then
clearly
(6.1)
)
E )
O
H2(Q,N)
>
and of
H"
(6.1)
is an
we
II.4.3
yields
(ii)
exists,
then
t
N
(6.2)
> h >
r$ N'
is c o m m u t a t i v e . tative.
Then
versely
if
diagram
(6.2)
Hom(Qab,N')
COROLLARY
t' =
t'
Proposition
implies
g ~> Q
t~
s~
G T
gt>
and
the
for
of
be
another
t
= s*(~')
j so that
.
map making
some h o m o m o r p h i s m
is a h o m o m o r p h i s m Thus
r,(~)
Q'
: G ~ G'
commutative.
II.5.4
E' = s*(6, ) = s * ( H ' ( ~ ' ) )
the e x i s t e n c e
t'h'fg
f : Q ~ N'
t
then
is u n i q u e
(6.2)
commu-
f : Q ~ N'
. Con-
t' = t - h ' f g if and o n l y
makes
if
= O .
6.2.
of P r o p o s i t i o n is u n i q u e
Let
this
G
) h t)
n' "\ // H o m ( H 2 Q ' ,N')
obtain
E' = rS,E = 6, s,
isomorphism
Ts*
H 2 ( Q , , N ,)
. Using
then
Proposition
If
7'>
~' = ALE']
r,(K(~))
Since
~ > Hom(H2Q,N')
Ts*
Ext(Qab,N, ) >
commutativity
the
E >> H o m ( H 2 Q , N )
H 2 ( Q , N ')
T
~ = A[E]
To p r o v e
the d i a g r a m
Ext(Qab,N)
Let
is c o m m u t a t i v e .
(Eckmann-Hilton-Stammbach 6.Z
with
if and o n l y
Q
perfect
if d i a q r a m
[22]).
the map
(6.1)
t
Under
the h y p o t h e s e s
: G ~ G'
is c o m m u t a t i v e .
exists
and
i2i We
recall
that
Ext(Qab,H2Q) give
= O
a more
or
PROPOSITION of
stem
tive)
Let
to
subgroup
extensions
Conversely
let
U ~ H2Q
H2(Q,N)
= 0
an
map
sions
E'
we may
and
T : H2Q
the
. Then
stem
case
we
stem
the
covers are
isomorphism
V
are
is
a map to
two
to
of
Q
to
with
the
subqroups
the
of
surjec-
stem
exten-
extension. of
Set
N
H2Q
Associate . It
with
is c l e a r
that
subgroup
of
H2Q
= H2Q/U
and
consider
. Since
Ext(Qab,H2Q)
coefficient
exact
E
.
= O
the
we
have
sequence
for
~ Hom(H2Q,N)
a unique
r
~[E]
of
H2Q
is a n o t h e r : N ~ N'
) H2Q
equivalence E = 6,
making
with
extension the
[E]
is s u r j e c t i v e ,
associated stem
class
E
of
E is
associated
exten-
is a s t e m U
. Finally with
diagram
6E >I N p
(6.3)
commutative.
II
I Ir
) H2 Q
E' ~ 6, ))N'
II U )
It
.
classes
(necessarily U
if
able
extensions
correspondence
and
same
the universal
subgroup
find
U >
the
~ N
~ . Since
: N'~--9 G' ----~ Q clearly
of
of
isomorphism
=
the
~ N)
given.
determines
~[E]
with
extension if
thus
T
a stem
: H2Q
be
: H 2(Q,N)
The
U
if there
yfeld
, so t h a t
yields
if
be
U = ker(HA[E] stem
Ext(Qab,N)
In this
correspondinq
G ---))Q
projection
= O
class
.
isomorphic
canonical
only
V
).
in o n e - t o - o n e
. Moreover
extension
: N~-9
5.3
description
are
if a n d
stem
E
Q
H2Q
U ~ V
Let
isomorphism
Ext(Qab,H2Q)
of
of
the
one
complete
correspondinq
PROOF: the
U
from
sion
less
extensions
, then
is o n l y
(see P r o p o s i t i o n
6.3.
subqroups H2Q
there
follows
that
r
is
isomorphic,
and
Proposition
6.1
U
122 asserts
the
lie
in
Now
let
existence
the
same
the
t
commutative
:
given.
the
N
Ir bE t *-))
so
to
It
enough
N
is
thus
= H2Q/U
that
U
converse,
isomorphic
an
, N'
E'
Then
E 6. --9>
H2Q
prove
and
II
II
To
E
--->> Q
t~
be
H2Q
commutative,
. Hence
N V ~ - - > G v --9 Q
extensions
is
r
diagram
r~ E v :
i n d u c ing
class.
N >----> G
(6.4)
stem
: G ~ G'
isomorphism
E
of
of
we
N'
= ker
first
extension to
recall
E
with
consider
= H2Q/V
E 6. _c k e r
we
that
Thus
have
= V
.
every
HAlE]
those.
. Then
E' 6.
the let
stem
extension
canonical U ~ V ~
a commutative
is
projection.
H2Q
. Set
square
of
canonical
projections T H2Q
(6.5)
>>
tl
~r
a --~>
H2Q
Now E'
if
E
: N~--> G --9> Q
: N' >---> G' --9> Q
Proposition inducing
6.1 r
N
an
diagram
. Since
r
Nv
is a n
extension
extension (6.5) is
with
implies
with ZA[E']
the
surjective,
HAlE] =
E t 6. = a
existence
t
is
E = 6. =
also:
of and
T
and
, then t
by
: G ~
the
proof
G' is
complete. We the
remark map
that t
in
if
Q
(6.4)
is p e r f e c t is u n i q u e l y
it
follows
determined
from by
Proposition r
6.1
that
123 PROPOSITION
6.4.
stem extension.
Let
Q
be perfect
and
let
E : N~--> G --9>Q
be a
Then
(6.6)
E 6, > N ~ 0
0 ~ H2G ~ H2Q
is e x a c t .
PROOF:
Note
This
that
perfect,
immediately
it f o l l o w s the
second
precisely
the
COROLLARY
6.5.
central HIG
REMARK:
and
homology
Let
Q
= O
It e a s i l y
shows
that
In g e n e r a l , example, is e a s i l y
seen
is a s t e m c o v e r
The
following
the
stem cover
o f the
F/F ° n
proposition Q
if
with
Q
of
and
let
E
: N~--@ G --97 Q
E.
is Q
are
of
Q
be a
if and o n l y
from P r o p o s i t i o n
3.4
H2G = O
is a s t e m c o v e r .
then
is t r u e
we may have
H2(F/F~+I)
of
that
stem extensions
is the s t e m c o v e r
the c e n t r a l
of
associated
and 6.4
it
if,
that
if
in a d d i t i o n ,
stem covers
with
Q
E
if
: N~-~G-9, Q Proposition
is p e r f e c t .
H2G ~ O
(absolutely)
. For
free g r o u p .
Then
extension
: F °n /" F °n+± >--9 F / F ~ +i --)> F / F
E
(2.7)
.
perfect E
6.3
a non-commutative
that
(6.7)
(6.8)
be
groups H2Q
the c o n v e r s e
F
sequence
.
with
however,
let
be
follows
is a stem e x t e n s i o n 6.5
of
Then
H2G
from
from Propositions
subgroups
extension.
= 0
follows
,
n ~ 2
But w e h a v e
= Fn o + I /-F no+ 2
~ 0
gives,
Q
in t e r m s
for
of a free
.
perfect,
a description
presentation
of
Q
.
of
it
i24 PROPOSITION
6.6.
Let
Q
S >--) F
H2Q
is the
stem cover
PROOF:
Since
Q
)
of
> [F,F]/[F,S]
Q
being
g~ Q
, the m a p
is p e r f e c t
isomorphism
let
Then
g
we h a v e
[F,F]/[F,F]nS
the
and
~>Q
be a free p r e s e n t a t i o n .
(6.9)
be p e r f e c t ,
induced
beinq
[F,F]S
~ [F,F]S/S
by
induced
= F
~ Q
by
and
f .
therefore
,
f . We m a y
thus
consider
the c e n t r a l
extension
E :
[F,F]
nS/[F,S].~
It
is e a s y
:
to see
H2Q
that
nS
I;
FI
E
)> [ F , F ] / [ F , F ]
> [F,F]/[F,S]
>
>
G
it is c h a r a c t e r i z e d
by
HA[El
= 6E .
=
i
H2 Q Thus The
it is the following
"
stem cover. proposition
is a r e s u l t
of Eckmann-Hilton-Stammbach
[22].
PROPOSITION let k
6.7.
f : X ~ Q
: X ~ G
with
(6.10)
If
k
Let
: N ~ h> G
be a homomorphism f = gk
f,(H2X)
exists,
E
if a n d
g~ Q with
only
be X
if
c g,(H2G)
it is u n i q u e l y
determined.
a central perfect.
extension
Then
there
and exists
125 PROOF: set
If
k
=
im
Q'
exists, f ~
Q
then
clearly
and
S
)
> X
S
= ker
(6.10) f
. We
holds. may
To
then
prove
consider
the
converse
the
diagram
>) Q'
(6.11) S/IX,S]
Set, f'
for
short,
: X'
~ Q
to
the
rise
S'
the
~
X/IX,S]
= S/[X,S]
map
-9>Q'
, X'
induced
by
diagram
of
5-term
H2X
-~
H2Q '
= X/IX,S] f
whence It
immediately
remains
that
the
to
-~
f~H2X'
= k'
diagram
(6.11)
gives
S/[ X, S ]
-~
...
~
•••
~
S t
f,H2X
c
g,H2G
~ G
such
: X'
_q H 2 Q
.
that
f'
= gk'
~
Qab
, i.e.
such
diagram
S t >___3, X t I
(6.12)
___)} Q t
I
s t I
kt
j
I
l
v
N )
is
. The
by
I[
H2 Q v
construct
: X ~ Q
denote
sequences
II H2 X v ~
. Also,
commutative.
To
g
> G
>>Q
this
end
consider
H2XV
~
H2Qt
~
S f
~
O
H2G
g*>
H2Q
~
N
~
Gab
obtain
a unique
(6.13)
Since
f~H2X'
(6.13)
commutative.
fect. the
Hence
existence
~
g,H2G
we Since
Ext(Qab' ,N) of
X = 0
a uniquely
is p e r f e c t , = Hom(Q~b,N) determined
map
s'
it
follows
. Thus k'
: X'
: S'
~ N
that
=
0
making Q'
is
Proposition
6.1
~
(6.12)
G
making
peryields
126 commutative, The reader space
completing
m a y compare
theory
that the
REMARK: being
sequence those
plays here
The proofs
placed
by
vious
hold
V-
and
V.7.
In the next
tuting z/qz
, where
q
b y the
ZqG =
the terms
(7.2)
We call
Zq n
homology
lower
an e x t e n s i o n
exact
analogous
if
H 2-
to
is re-
. We leave
the ob-
of the Theor Z of Schur
present
ideas due
to Evens and
[28].
in Section
integer.
(I.i.4)
{Z q}
The
basically
lower
series
. It is defined
.
by
, ZqG/Z q n n-l-G = Zq(G/Z q_IG)
central
{G q}
will have
recursively
In 8
problems. in substi-
the h o m o l o g y w i t h c o e f f i c i e n t s
series
are defined
tools
to the reader.
central(q)
{x~ZGlxq=e}
theory.
coefficient
zero
V(-,-)
apparent
the main
that results
theory of Schur consists
series
ZqG = e
formal,
to g r o u p - t h e o r e t i c a l
central
central(q)
in that
the theory of Schur,
is any p o s i t i v e
The upper
(7.1)
Then
of the
integral
then be replaced
the upper
generalize
~I
by
in c o v e r i n g
It then becomes
of e x p o n e n t
is replaced
we shall
the theory
for the
(I.I.2).
H2(-, -)
theorems
and the u n i v e r s a l
~
A Generalization
The g e n e r a l i z a t i o n
to
. It is thus obvious
two sections
apply
spaces.
are p u r e l y
sequence
of the statements
7 we shall
we shall
sections
in any v a r i e t y
translation
Section
of this
H2(-, -)
stated
topological
a role a n a l o g o u s
5-term h o m o l o g y for
6.7.
this result with w e l l - k n o w n
for c o n n e c t e d
H2
the
the proof of P r o p o s i t i o n
in
series will
as defined
to be r e p l a c e d
as follows.
Let
in by
127
(7.3)
E
central(q) follows
, if
from
: N
N < ZqG
(II.3.13)
(7.4)
HqG
From
~ h > G
(II.5.3)
. Note that
g,)
that
gives
6E _L,_~N
O ~ Ext'../qz(Qqb'N)a
as
in S e c t i o n
(7.3)
a
Z / q Z - h o m o m o r p h ism
3, we m a y
IIA[E]
: Hq2Q -- N
By P r o p o s i t i o n
II.5.4
we m a y
(7.4).
of c e n t r a l ( q )
DEFINITION:
Proceeding
as
extensions
using
The
central(q)
a stem(q)
extension,
if
(iii)
a stem(q)
cover,
~A[E]
ing p r o p o s i t i o n s
that
PROPOSITION
For
statements
7.1. are
E
(ii)
N ~-e G q q ab --~) Q a b
(iii)
g,
(iv)
N
q
q
G ~ Q #
(7.4)
G = e
Q
;
sequence
Qq ~ 0 ab exact
sequence
~ Homz/qz(Hq2Q,N ) ~
with
the c e n t r a l ( q )
...
extension
IRA[El
with
3 we m a y of
the h o m o m o r p h i s m
define
various
classes
HAlE]
(7.3)
is c a l l e d
~A[E]
= 0
;
is e p i m o r p h i c ;
then
immediately
to P r o p o s i t i o n s
extension
extension;
is exact: q
g,>
It
is i s o m o r p h i c .
central(q)
is a c o m m u t a t o r ( q )
n G ~
KALE]
equivalent.
(i)
: G •
if
correspond
the
Gq b
properties
extension,
, sequence
to a 5 - t e r m
E H2(Q,N)
in S e c t i o n
(ii)
E = 6,
rise
identify
a commutator(q)
KALE]
is a Z / q Z - m o d u l e .
coefficient
extension
if
N
m
(i)
Since
h,~
associate
(7.6)
in
then
the u n i v e r s a l
Thus,
6. E
>> Q
(7.3)
HqQ
we r e c a l l
(7.5)
g
3.1
(7.3)
yields , 3.3
the
the
follow-
, 3.4.
followinq
i28 PROPOSITION statements
7.2. are
For
the c e n t r a l ( q )
E
is a stem(q)
(ii)
6,
: H Q ~ N
is e p i m o r p h i c ;
(iii)
h,
: N ~ Gq ab
is the
(iv)
g*
: Gqab ~ Qqab ;
(v)
N ~ G ~
PROPOSITION statements
G
7.3. are
(ii)
6,
: H Q ~ N
(iii)
g,
: G q ~ Qq ab ab
do not
example For
q
the
and
'
to be true.
an
there
exists
An a r g u m e n t
the
following
: H q G ~ H~Q 2
z e r o map.
to P r o p o s i t i o n s
However,
free,
is the
if
Qq ab
then w e a g a i n
5.1
, 5.2
, 5.3
is z / q z - p r o j e c t i v e obtain
meaningful
, for
results.
isomorphism
analogously
7.4.
g,
analogous
n : H2(Q,N)
PROPOSITION
(7.3)
cover;
is s q u a r e
then we h a v e
Proceeding
extension
map:
central(q)
statements
seem
if
zero
equivalent,
is a stem(q)
5.4
followinq
extension~
For
E
the
the
.
(i)
In g e n e r a l
(7.3)
equivalent,
(i)
q
extension
HOmz/qz(H~Q,N)
to the p r o o f
Suppose
a stem(q)
analogous
~
Qq ab
of P r o p o s i t i o n
5.1
we o b t a i n
is z / q z - p r o j e c t i v e . --
Let
U
extension
E
with
to the one
used
ker
HA[E]
in the p r o o f
: U
( H~Q
. Then
.
of P r o p o s i t i o n
5.2
establishes
PROPOSITION
7.5.
isomorphism
class
of
Q
I__ff Q a b q __is Z / q Z - p r o j e c t i v e , of
stem(q)
is an e p i m o r p h i c
image
covers
of
of that
Q
there Every
stem(q)
is p r e c i s e l y
stem(q)
cover.
one
extension
129 PROPOSITION E
: N~>
7.6.
Let
G --)>Q
morphism
class
spondence are
two
7.7.
The
the
subqroups
(necessarily U
Let
H~Q
V.8.
We r e c a l l
from
from Section
of
f'
Then
the
in o n e - t o - o n e
U ~ V
if and o n l y
if t h e r e
correspondinq
to t h e p r o o f
V
7 the d e f i n i t i o n
corre-
and
V
is a m a p to
.
of P r o p o s i t i o n
and U n i c e n t r a l
U
iso-
corresponding
extension to
.
5.5.
if
stem(q)
let
= O
: G' ~ G
. Moreover,
the
Terminal
to a m a p
are
and
Extlz/qz(Qaq,N)
Z/qZ-module.
Q
cover
o__ff H ~ Q
, then
extension
is a n a l o g o u s
lifted
if
of P r o p o s i t i o n
extensions U
b e a stem(q)
Then~
be a projective
subqroups
surjective)
to the stem(q)
proof
Qq ab
of
c a n be
to the p r o o f
o f stem(q)
with
extension.
f : Q' ~ Q
is a n a l o g o u s
PROPOSITION
: H q Q ' ~--~ G' --~> Q'
be a stem(q)
every homomorphism
The proof
E'
6.3.
Grou~s
o f a stem(q)
extension.
The
extension
is c a l l e d
a stem(q)
(8.2) where
N
q
to d e n o t e q = 0 as
ZqG
o G~
if
,
is a n y p o s i t i v e
integer.
stem
then we may
, also.
extensions We
shall
generalize
If w e u s e include
the n o t i o n
the t e r m this
stem(O)
extension
case by allowing
o f a stem(q)
extension
follows
DEFINITION: if
extension
The
extension
(8.±)
is c a l l e d
an m-stem(q)
extension,
m ~ 1
i30
(8.3)
N
( ZqG
n Gq m+l
--
Note
that
that
if
a i-stem(q)
(8.1)
is a n m - s t e m ( q )
(8.4)
The
gk
following
m-stem(q)
: G/G~
8.1.
,
yields
A stem(q)
if a n d o n l y
k
then
= I .....
extension.
g
induces
m+l
a homological
where
6.E =
PROOF:
Consider
HA[E]
extension
Note
also
isomorphisms
i
c h a r a c t e r i z a t i o n of
6, + k e r
: H2Q
the
: N~--> G -~>Q
is a n m - s t e m ( q )
T
~ N
= H2Q
m
and
Tm
: H2Q
~ H2(Q/Q
)~
diagram
N ~
G
N G q 7--9 G m associated
E
if
E
ker
the
a stem(q)
extension,
= Q/Q~
proposition
(8.5)
and
is j u s t
extensions.
PROPOSITION extension
extension
-->> Q
-->>Q / Q q m
5-term
sequences
ker
Tm ....> N n G
+I
E
H2qG
(8.6)
-~
H2qQ
II
6.
>
N
-~
I 19
T i m m
A simple m-stem(q)
diagram
chase
extension,
shows
i.e.
if
that N
~
map,
follows. so
that
Conversely, N ~ Gq m+i
"
if Thus
the
II
II
ab
ab
then
m+l
(8.5)
-~ Q qab ~ O
is e p i m o r p h i c .
c Gq --
(8.5)
m~±
G aqb
'
holds,
proof
then
N
Now
n Gq m+l u
is c o m p l e t e .
must
if = N be
E
is a n
and the
zero
i3i PROPOSITION U ~ H~ L
and
extension the
Q
, then
U
+ ker
U
of
Of
T
let
Q is
also
U
6,
E
or
are
is
Q
ker
If
of
= 0
is
or
no
if
H2Q
with
non-trivial
Qq ab
a non-trivial of
H Q
is --
m-stem(q)
z/qz-projective
m-stem(q)
with
extension
U fi H Q
of
and
7.4
It
T
enables
HA[El
= U
, then
us
¢ H ?~ Q 2
find
a
U ~
are
H2Q
no
with
stem(q)
Proposition
if
there
U
to
. By
is n o n - t r i v i a l
= 0
Given
8.1
exit
is
.
non-trivial
m-stem(q)
m
.
Let
if
Z/qZ-projective.
E 6, = k e r
ker
Tm ~ p H ~ Q
Since
m-stem(q) + ker
H~Q
there
of
q
Q
be
a
finite
, there
are
no
= O
if
Qq ab
or
p-qroup.
Suppose
z/qz-projective
o__rr q = p
extensions
m-stem(q)
non-trivial is --
q = O
the
Tm = H ~ Q pH~Q
Conversely,
jective m-stem(q)
is a
extension
rood
U ~ H~Q
Q
if
and
finite
then
U
. If w e , hence ker
Proposition extension
ker
so
HA[El
x
m
Q
pH~Q
we
= H~Q
. If
yields
the
.
is
H~Q
is a
Tm --¢ p H ~ Q
it n e c e s s a r i l y
8.2 of
= ker
had
Tm~
U + ker
p-group,
may
would find q = 0
existence
E
subgroup , then be
a
. If
converse
of U
the
is
if of
Qq ab
an
H~Q
would
whole
subgroup or
is
of
U ~ is
H~Q
with generate H~Q
.
with
z/qz-pro-
a non-trivial
k
of
true.
PROOF:
U
Q
8.4.
. Moreover
also
given
be
extension.
8.3.
extensions
ker
q
a subgroup
Qq ab
- -
If
if
, Proposition
with
an m-stem(q)
COROLLARY
. Then
U
= HqQ z
m
COROLLARY
subqroup
true.
if w e
= ker
is no
= H~Q L
m
course,
Tm = H ~ Q
tension
T
there
. Moreover,
q = 0
+ ker
Suppose
U + ker
converse
PROOF:
Now
8.2.
.
i32 REMARK:
Note
weakened. H~Q
is
that
It w o u l d finitely
s e n t a b l e . For
DEFINITION: if t h e r e
Note
q = 0
i Q
Q
8.5.
ker
their
Let
that
is a l w a y s
group
Q
so
m-stem(q)
Q
. Note
direct
G
there
not
m
also
product
that
seem
QI×Q2
of
Q
QI
then
Q
pre-
terminal(q)
.
then
and
Q2
is t e r m i n a l ( q )
of c l a s s
for w h i c h
to be p o s s i b l e .
is c a l l e d
if
may be
is f i n i t e l y
is n i l p o t e n t ( a )
be nilpotent(q)
8.3
does
Q
extensions
an~
8.4
is a n y g r o u p
if
of class
is m o n o m o r p h i c ~
By Corollary
m
are , too.
I_[f
is t e r m i n a l ( q )
are no n o n - t r i v i a l
m-stem(q)
extensions
.
PROPOSITION q = p
Q
Gqm+i = e
H F Q ~ H ~ ( Q / Q q) z m
Q
to s u p p o s e
such a generalization
implies
PROPOSITION
PROOF:
of Corollary
is t e r m i n a l (q)
, then
T m
, the h y p o t h e s i s
This
A nilpotent(q)
if
k
be e n o u g h
generated.
terminal(q)
of
q = p
a r e no n o n - t r i v i a l
that
G/G~+I
if
k
Tm
8.6.
Then
Let
Q
C pH~Q
be
a finite
the c o n v e r s e
I.l.l
Q
consequence
of C o r o l l a r y
DEFINITION:
A group with
8.7)
m
q = O
, say. q = 0
Q
or
if
Qq
is
a b
is a l s o
true.
is n i l p o t e n t ( q ) . T h e
rest
is a n
immediate
8.4.
is c a l l e d
N ~ ZqG
N ~--9 Z q G
o__rr
If
"
Z/qZ-projective,
N~--> G - - ~ Q
Suppose
Moreover,if
'
By Lemma
p-qroup.
is n i l p o t e n t ( q ) , o f c l a s s it is t e r m i n a l ( q )
--
PROOF:
Q
the
unicentral(q)
if for e v e r y
extension
sequence
~} ZqQ
is e x a c t .
Our next
proposition
of Evens
[28].
is an
improvement
due
to M e i e r
[62]
of a r e s u l t
133 PROPOSITION
8.7.
Let
Q
(8.8)
~ : ~Q
is m o n o m o r p h i c
then
Qq ab
is --
PROOF:
be
E
I__ff
~ H~(0/zqQ) Q
is u n i c e n t r a l ( q ) .
z/qz-projective,
Let
nilpotent(q).
the
: N >h_h> G
converse
g~> Q
be
Moreover~
is
also
if
q = 0
or
if
true.
a central(q)
extension
and
let
-± M
= g
(ZqQ)
. Since
M ~ N
HqG
--
HqG It
is e a s y
follows
to
that
have
G
Since
trivially
see
that
Nn(G
#
@ q M -¢ N
q
L
T,
6
, whence ZqG
~ M
N >--> Z q G
is e x a c t , To
prove
or
if
so the
Qq ab
that
Q
second is
G
) Nn(G
6E -- * )
H2qQ
M)
the
0
H q (Q/ZqQ)
~
obtain
0
ker
(8.9)
, we
#
q
N
~ M/G
M)
--
. But,
we
by
conclude
, the
Gqab
~
# q M ~ Gqab ~
is e p i m o r p h i c . = e
diagram
Thus
the
if
(~abq
0
( Q / Z q Q ) aqb
ker
construction
that
~
G •q M = e
~ O
a = O of
it M
, i.e.
, we M ~
ZqG
sequence
-@ZqQ
is u n i c e n t r a l ( q ) . part
of
z/qZ-projective
the
proposition then
there
we exist
recall stem(q)
that
if
covers
q = O of
Q
Let
(8.10)
be
such
diagram
E
a
stem(q)
with
exact
.
: H ~ Q >-e G
cover, rows
so and
--~} O
that columns
6E
is
an
isomorphism
Consider
the
.
i34 0
0
k er
J'
o
ken
1
1
Hq G
(8.11)
0 -~
Hq(G/ZqG)
ZqGnG q
Q
phism.
Hq(Q/ZqQ)
-~
ZqQnQ q
The
sequence
- coker
ker o E
is
zero
the
is
COROLLARY
8.8.
and
Let
z/qz-projective.
PROOF:
Let
Q
2 Q/ZqQ
applied
extension, hence
Q
be
Then,
if
be
G/ZqG
to
so
that
(8.11)
Q
then
is an
isomor-
yields
o ker ~ ~ HI0 ~ ZqGG] ~ zq0~0~ - O
a stem(q) map,
0
0
, then
Since
6E , > H2Qq
~
is u n i c e n t r a l ( q ) ker
,
Hq Q
0
If
~ H2qQ
a
ker
the
map
~ = O
nilpotent(q)
is m o n o m o r p h i c .
Suppose
is u n i c e n t r a l ( q )
of
Thus
.
nilpotent(q). Q
w
class
m
q = O , it
is
. Since
-o-r
Q qa b
-is -
terminal(q).
Qq ¢ ZqQ m --
we
have
T
4°
m ~
H~i0/0m~)
(8.12) H q (Q/ZqQ)
If
Q
is u n i c e n t r a l ( q )
ker
T m
= 0
Note
that
need
not
sum
, SO if
be.
QI To
decomposition
that
Q
' Q2 see
, then
ker
o = O
is t e r m i n a l ( q ) are
this,
unicentral(q) let
q = 0
by by
Proposition Proposition
, their
. Then
direct
there
is
8.7.
Hence
8.5.
product a natural
Q direct
135
H2Q = H2QI
H 2(Q/ZQ)
Thus,
in o r d e r
clear
that
to
~
we may
Q Q2ab
~
need
not be m o n o m o r p h i c .
So,
need
not be u n i c e n t r a l ( q )
.
V.9.
In this
we
although
shall
of the
give
multiplicator
of a finite
rank
for a n i l p o t e n t
of
H2G
PROPOSITION
9.1.
(9.m)
PROOF: H2G find
(Green
= 0
We
argue
~ thus
a central
corresponding (2.10)
applied
by the
yields
[33])
Q = QI×Q2
G
Let
is t e r m i n a l ( q ) ,
it
Multiplicator
on the G
and
order
of
the
an e s t i m a t e
Schur
on the
.
IG 1 = pm
. Then
m (m-±)
assertion N
group
to
> N
--9>e
N>~-h> G
g>>Q
the d i a g r a m
(Q2/ZQ2)ab
group
group
It is then
P
subgroup
N>
®
two e s t i m a t e s
induction on
factor
'
its c o m p o n e n t s .
Schur
nilpotent
2
IH2G I
® Q2ab
study
(Ql/ZQl)ab
On the O r d e r
section
~ Qlab
= H 2(QI/ZQ1)~H 2(Q2/ZQ2)®(QI/ZQI)ab@(Q2/ZQ2)ab.
study
Qlab
~ H2Q2
m
. If
is true
m : 0
in this
of
G
of order
G/N
. Then
, then
case. p
G = e
For
. Denote
naturality
of the
and
m ~ I
we m a y
by
the
Q
sequence
i36
N (9 N
I" H2 N
h,~
1 ®
(~.2)
~h,
N (9 Gab I H2 G ~ H2 Q ~ N ~ G a b ~ Q a b ~ 0 i ®g,l
. ~,,
N ® Qab
I O
Since
N
y"(l (9 g,)
is c y c l i c , = ¥
H2N : 0 . H e n c e
Using
duction hypothesis
the
there
fact that
IQI
exists = p
m-I
the p r o o f
COROLLARY
with
and a p p l y i n g
the
in-
we o b t a i n
IH2G[ and
m-I
¥"
<
IQabI'[H2Q
j < p
(m-i) (m-2)
-p
~ m(m-i)
= p
is c o m p l e t e .
9.2.
Let
mI m2 m~ n = Pl "P2 ...p£
G
be a n i l p o t e n t
q r o u p of o r d e r
. Then
I ~ m i (mi-i) IH2G j < i:i Pi e
(9.3)
PROOF:
Since
subgroups
Pi
G
is n i l p o t e n t
it is the d i r e c t
product
' i = i,..., ~ . The K ~ n n e t h - s e q u e n c e
of its P i - S y l o w
(II.5.13)
then
yields
H2G ~ whence
the a s s e r t i o n
We o b s e r v e
H2P i
by Proposition
that the e s t i m a t e
G = C x...×C P P (II.5.13)
~
i=i
, m-times,
(9.1)
then
easily yields ± m(m-l)
IH2GI
= p
9.1. is b e s t p o s s i b l e .
IG I = p
m
For
if
and the K ~ n n e t h - s e q u e n c e
137 by
induction
enter
into
We will
on
m
. However,
consideration,
state
one
if w e
then
of these
allow
better
the
structure
estimates
in P r o p o s i t i o n
9.5
are , but
of
G
clearly first
to
possible.
need
some
preliminaries. Let
G
be
Z G = G n
nilpotent
. Consider
(9.4)
E
Since
G/Zn_zG
in
induces
G
then
then
9.3.
, Zn_IG
J G
~ Gn
and
the
embedding
of
Zn_IG
® Gab
~ H2G
be
the
Ganea
term
(2.1)
of
o : G n Q Zn_zG/G 2 ~ H2G
zero
PROOF:
Consider
a stem
H2G
we
map.
= N
= e
cover
g(N)
(ii)
g-i(G°) -- K ° N
_~ z2~
(iii)
g(ZnK)
-
(i) , (ii)
are
) Zk_IG
we may
M C Zk_IK
: H 2 G >h_h> K
g-~> G
with
HALE']
argue
by
= i
,
! Zn_iG
g(ZkK)
E'
have
(i)
that
= e
Then
is t h e
Inductively
' G n° + i
extension
o G 2 ~ Zn_IG
¥ : G n° Q G a b
Let
yot,
to prove
Gn ° ~ e
--9> G / G ° n
: G on ® Z n - I G / G 2
(9.4).
(9.5)
Statements
~ then
central
is a b e l i a n ,
extension
Setting
n
prove
PROPOSITION the
the
: G ° ~---> G n
L,
We may
of class
assume
. Thus
we
trivial. for
k = i , . . . ,n
g(Zk_~) have
To prove
)_ Z k _ 2 G
(iii)
we
. Clearly . Set
induction
g(ZIK) ~ ZoG=e . -i M = g (Zk_2G) , so
i38
ZkK/M ± Z(K/m ~ Z(G/Zk 2~) = Zk_IG/Zk_2G whence
g(ZkK)
!
Zk_IG
(K~N)ab
- Next
® Kab
H2K
~I (9.6)
consider
~g.=O
G n° ® G a b
]
H2G
"'"
~6.:i H2G
Recall
from
since
(see
Proposition [15],
that
zero
Note
that
yot$
= 0
we
have
in
= e
, and
using
(iii)
we may
the
¢ G
that
statement, namely
that
: G °n ® Z n _ I G / G ~ ~ H 2 G
proved
a slightly
stronger
: G n° Q L / G ~ ~ G °n ® G a b
is the
image
under
g
of
the
centralizer
of
is e x a c t .
in
K
K
.
n
9.4.
(Vermani
[84])
Let
G
be
nilpotent
of class
sequence
(9 • 8)
. But
infer
--
COROLLARY
K
, where
t~
L
commutator
map.
(9.7)
and
is the
= e
[K~N,ZnK]
yot.
is the
6,¥~
p.3)
[K~,ZnK]
it f o l l o w s
that
2.3
G °n
®
G/Zn-IG
I'
H2 G ~ H 2 ( G / G ~ ) ~ G n° ~ 0
n
. Then
139 PROO_____FF:T h i s
immediately
PROPOSITION
9.5.
be n i l p o t e n t
follows
(Gasch~tz- Neub~ser-
of c l a s s
where
s(Q)
denotes
PROOF:
From
sequence
Yen
9.3.
[3111Vermani
[84])
Let
G
n . Then
IH2G I ~ IG~
(9.9)
from P r o p o s i t i o n
s(G/ZG)-IIH2Gab
the m i n i m a l
number
(9.8) we m a y
,
of g e n e r a t o r s
infer
for
of
Q
.
n ~ 2
IH2GI ~ IH2(G/G~)I'IG ~ ® (a/Zn_IO)I/IG~l. NOW trivially
IG°~en (G/Zn-I G) I " <
IG~ Is(G/Zn-aG)
, so that
1%GL ~ 1% (Glc~) I l~k s (GIZG)-I since
s(G/ZiG)
G / G ~ _ I , etc.
~ s(G/Zn_IG)
. Repeating
the a r g u m e n t
for
G/G~
,
and u s i n g
s(~/G~/Z(~/G~)) ~ s(G/ZG) we obtain
=
We c o n c l u d e G
this
nilpotent with
iH2Gab I . i G2]S (GIZG)-i
section with finite
of a n i l p o t e n t
group
(9.10)
hG =
G
a proposition
Hirsch number.
• on the r a n k of
Recall
H2G
for
t h a t the H i r s e h n u m b e r
is d e f i n e d b y o
o
rank Gi/Gi+ i i=i
if all of the wise.
( f i n i t e l y many)
summands
are
finite,
and
hG = ~
other-
140
PROPOSITION
9.6. L # t
(9.11)
be n i l p o t e n t w i t h
r a n k H2G ~ [rank G a b - l ] - h G
PROOF: sion
G
Let
G
be n i l p o t e n t
of c l a s s
E : G ° ~--> G ---~>G / G ° n n
(9.12)
. Then
- rank H 2 G a b
n . Consider
and the s e q u e n c e
the c e n t r a l
(2. 7 ) a s s o c i a t e d
extenwith
E
6E f~ G ° n ~ O ,
GOn ® Gab I H2 G ~ H2(G/G~)
Counting
hG < ~
r a n k s we o b t a i n
Repeating
r a n k H 2 G ~< r a n k
H 2 ( G / G ~)
+ rank(G~
rank
H 2 ( G / G ~)
+
this a r g u m e n t
G/G~
for
@ Gab)
- r a n k G On
(rank G a b - i ) ' r a n k
, GIG~_ 1
, etc.
G°n
we o b t a i n n
rank H2G ~ rank H2Gab = rank
Finally,
since
rank H2Gab
rank
H2G ~
proving
our p r o p o s i t i o n .
We note
that the e s t i m a t e
G
abelian,
hG = rank G
= ~(rank
nilpotent, the same [70] .
then so is
type.
Gab-l) r a n k G a b
(rank G a b - l ) h G
(9.11)
K
- rank H 2 G a b
for
of H a l l ' s ~3{R~
Let
of this c h a p t e r
H 4 K . If
H
and
. A l s o we s h a l l o b t a i n a n u m b e r
The c o n t e n t
since
equality.
the m e t h o d s
theorem:
, we h a v e
is c l e a r l y b e s t p o s s i b l e
Theorems
In this s e c t i o n we shall u s e ce~ebrated
o o (rank Gab-l) "i~ 2= r a n k G i / G i + I
H2Gab+(rank Gab-1) hG-(rank Gab-1) rank Gab.
and w e o b t a i n
V.IO.
prove Hall's
+
of this
section
is to be found
in o r d e r
to
K/[H,H] of r e s u l t s
are of
in R o b i n s o n
141 We c o n s i d e r acts G
a Z-group
as a g r o u p
, i.e.
G
, i.e.
a group
of a u t o m o r p h i s m s .
Let
a normal
subgroup
of
£ Z . We d e f i n e
a series
of n o r m a l
(10.1)
The
NI = N
quotient
denoted
by
of two F
i = 2,3,...
l
successive
. Clearly
F. l
the c e n t r a l
(IO. 2)
We m a y
, Ni+ I =
draw
N i c G o2 , h e n c e
the
N
which
on w h i c h
the g r o u p
be a n o r m a l is m a p p e d
E-subgroups
[G,Ni]
, i = 1,2 ....
terms
Ni/Ni+ I
is a Z-module.
Z-subgroup
into of
I
G
itself by
by every
setting
, i = 1,2,... Now consider
of
will
be
for
extension
E i : Fi_l >
thus
G
G
> G / N i ---9) G / N i _ 1
following
diagram
(G/Ni) ab = Gab)
(note
that
for
i >I 2
,
:
H 2 ( G / N i + 1) = H 2 ( G / N i + 1)
(Ni-I/Ni+l) a b @ ( G / N i + l ) ab
¥
(10.3)
>
Fi_ I ® Gab
H 2 ( G / N i)
-~ H 2 ( G / N i _ I) ~ F i _ l ~ . . .
~6,
.1.6, 0
Fi
~
Fi_ I
0
Note
that
we may map
in
apply
lies
Proposition
~ = 8,¥
by definition
surjective.
in the
2.3
G / N i + I . In this
(10.4)
Since
Ni_I/Ni+ 1
second
to s h o w
section
0
center
that
we w i l l
of
8. y~
consider
G/Ni+ I
so that
is the c o m m u t a t o r the m a p
: Fi_ I ® Gab ~ F i ,
Ni =
[G,Ni_I]
=
[Ni_I,G]
it f o l l o w s
that
~
is
142 Every
term
in t h e
tensor
product
obvious
that
Hence
~
, as
P-qroup
PROOF: is
being every
PROPOSITION a
diagram
defined
for
The
a
map
~ hence
assertion
since
: N/[G,N]
Q Gab
Suppose
is t o r s i o n
The
map
is p e r i o d i c ,
its
all
Fi
factors
is p e r i o d i c ~
that
acts
the
free
range
N
every tensor
with
Gab
~ •
= G
; then
obvious
induction
be
on
10.4.
such
ever V extension
Suppose
that
the
G
, N ~
is
By
a
G
abelian
i
be
Z£G
image the
for
of
i__ss
domain
establishes
Hence
that
B
it
= e
in
ZG
its
domain
[G,N]
•
that
~ N
. Thus
if
i .
follows
tensorial is
~ ~
N ~
N2 =
[G,N]
A ®
some
Since
class ~ •
of modules. for
N ~
A,B
~
, where
Z
o o ~ Gi/Gi+ I
establishes
result.
the
Let
~-qroup form
by
If
G
is a
i = 2,3, ....
o o : Gi_I/G i Q Gab
~-qroups
Gab
diagonal.
o o Gi/Gi+ I
an
is
homomorphism.
Its
Then
induction
periodic.
nilpotent. of
It
. If
induction
periodic.
finite
a tensorial
%
[ i> I
is e p i m o r p h i c .
Z-modules
, then
COROLLARY that
Let
finite
is c a l l e d
~-qroup
An
A
of
¢
Z-module
is e p i m o r p h i c .
•
via
p.212).
the
[G,N]
so
Let
N
is
torsion-free
product
a
some
is
I0.3.
Set
N 4
are
epimorphic
[43],
on
.
Gab
is.
so
~ N2/N 3
~ N2/N 3
also
THEOREM
PROOF:
and
for
then
Let
, i ~ 2,3 ....
A class
implies
[6])
is
operation
Z-homomorphic.
range.
Z~_i+ I
: N/[G~N]
but
DEFINITION:
on
(Baer
~
its
Ni ~
10.2.
PROOF:
is
the
(see
(10.3)
, N _c Z £ G
primes,
so
diagonal
is
of
PROPOSITION N
4 G
Z-module,
diagram
(I0.4)
N P
is a
the
the
in
set
via
in
Let
a P-group,
the
given
map
I0.i.
(IO.3)
~
be
an
~-qroup
a tensorial
is
a class is
class.
epimorphic.
of
qroups
an
~-qroup.
Then,
if
ZG
.
±43
Gab
is
PROOF:
in
This
central
As
~
, so
is
G
immediately
series
specific
of
G
.
follows is
examples
of
from
finite
10.3
terminates
with
~
of groups,
that
perty
required
(i)
finite
(ii)
periodic
groups;
(iii)
finitely
generated
(iv)
groups
satisfying
the
maximum
condition;
(v)
groups
satisfying
the
minimum
condition.
Let PC
C : The
be
P-groups
a class
class
length P'~:
The
class
GpC:
The
class
class
ascending
ordered
by
G
all
, and
of
of
are
having
A
, A
Z-groups
normal
unions
of
satisfy
following:
the
following
the
pro-
each
series C
of
submodules
notations.
of
finite
; series
member
its m e m b e r s .
submodules unions
a normal
an
of
to
C
ascending to
C
of
submodules
A
well
ordered
by
its m e m b e r s . of
finite
length,
. normal
series
whose
.
is a s e t normal
of
series
belong
belong G
of all
and
having
of
use
.
having
and
series
inclusion,
.
the
an a s c e n d i n g
, and
abelian
abelian
to
is a s e t
O
are
C
shall a
having to
Z-groups
factors
factors An
of
We
belong
belong
containing
lower
of p r i m e s ~
Z-modules.
Z-modules
series
whose Gp'C:The
of
mention
e
the
groups:
factors
factors
ascending
inclusion,
of
, we
a set
Z-modules
whose
whose An
of
10.4
, P
, since
and
classes
in C o r o l l a r y
Theorem
of
subgroups
in t h e
next,
of
G
well
containing
e
,
144
LEMMA P~
10.5.
and
PROOF:
P'~
Let
modules
are
A
. The
by
it
, j ~ J with
set
i < i'
series
the
and
if
= ira(
first
Then
the class
(ascending) and
series
Bj+I/B j
is w e l l - o r d e r e d i = i'
, j < j'
~ IxJ
of
of sub-
in
under . We d e f i n e
A ® B
an
by
~ A k ® B~ ~ A ® B) (k,~)~(i,j)
element
o f the
set
{(re,n) I (i,j)<(m,n) }.
is e a s y to see t h a t
Hence
the
(i,j)
~ I×J are
THEOREM
factors of
of the A ® B
® B)(i,j)=Ai+I/Ai
(ascending) belong
to
series
® Bj+I/B j
((A ® B) (i,j))
C . Thus we have
shown
'
that
P~
,
tensorial.
Let
•
under
submodules.
If
that
K/[H,H]
10.6.
is in
Suppose
modules,
Hab
factors
of the
belongs
to
now apply
(i) L e t
be
' (i,j)
(A ® B ) ( i , j ) + I / ( A
PROOF:
of Z - m o d u l e s .
Ai+I/A i
IxJ
(A ® B) (i,j)
(i,j)+i
(10.6)
We
, (Bj)
(A ® B) (i,j)
Denote
P'C
if
class
also.
respectively
, j ~ J
(iO.5)
Then
tensorial,
, i ~ I
(i',j')
(ascending)
be a t e n s o r i a l
, B
i £ I
<
•
(A i)
of
for all (i,j)
Let
C
be
a tensorial
H
is a n i l p o t e n t
GpC
K/[H,H] is in
Gp~
. For
Theorem be the
is the c l a s s
is
PC
lower
(Gp'C)
in
GpC
Gp'C
10.6
class
series
to s p e c i f i c
of nilpotent
GpC
is in
•
H
belong
to
is c l o s e d K
such
(Gp'C)
is c l o s e d
and Theorem
, also.
under
sub-
IO.3
all
the
PC
. Hence
K
is a n a l o g o u s .
examples
modules
groups.
which
K
I0.5 of
Z-modules
Z-subqrou p of
. Since
the p r o o f
of trivial
of
normal
, then
. By Lemma
central
class
Thus we
for the
(Z = K) obtain
class
. Then
~ . GpC
i45 COROLLARY nilpotent.
(ii)
Let
10.7.
(P. H a l l
Then
K
•
the class
be
is t h e c l a s s
COROLLARY K
[40])
10.8.
Let
is n i l p o t e n t ,
H ~ K
nilpotent,
K/[H,H]
also.
of trivial
of hypercentral
Let
, H
modules
groups.
H ~ K
, H
class
of m o d u l e s
(Z = K)
. Then
Gp'C
We obtain
nilpotent,
K/[H,H]
hypercentral.
Then
is h y p e r c e n t r a l .
(iii)
Let
C
abelian
be
the
group
is c y c l i c .
Then
(Z = K) GpC
whose
underlying
is the c l a s s
of s u p e r s o l u b l e
groups.
COROLLARY K
10.9.
Let
H ~ K
, H
nilpotent,
K/[H,H]
supersoluble.
Then
is s u p e r s o l u b l e .
(iv)
Let
C
abelian
be t h e c l a s s group
o f modulkes
is c y c l i c .
Then
(Z = K) Gp'~
whose
is t h e
underlying
class
of h y p e r c y c l i c
groups.
COROLLARY K
lO.IO.
Let
is h y p e r c y c l i c .
H ~ K
, H
nilpotent,
K/[H,H]
hypercyclic.
Then
CHAPTER
LOCALIZATION
In t h i s
chapter
we present
rationalization P. H a l l
[39],
Baumslag
is e s s e n t i a l l y
[16],
homologicall
Section
i is p r e p a r a t o r y l
P-local
abelian
contains groups
show that
series We
successive
if
G
if
G
HPL-group
~
we
some o f
group zation
groups
can be
which
show that
the
the
integral
quotients
of
then
G
of a
Section
the
2 i.e.
with unique
then
the
are P-local.
has unique
We
lower
groups
P'-roots,
series
adopted
dimension.
3 we consider
central
have
of HPL-groups,
successive
if a g r o u p h a s u n i q u e
[52],
homology
dimension.
properties
we
and
[41].
in p o s i t i v e
is n i l p o t e n t ,
Lazard
approach
is P - l o c a l
In S e c t i o n
localization
[6±],
to H i l t o n
in p o s i t i v e
o f the u p p e r
associate
with Gp
the
Malcev's
famous
in its
subgroups, the
that are
direct
limits
asserts
theorem
Also,
P'-roots
which
that
G/ZkG
that
if and
Also,
we
. In S e c t i o n functor.
5
Among
nilpotent
show that
locali-
to the c a t e g o r y
groups.
then
an
and quotients.
as a corollary
is f i n i t e
G
G
a torsion-free
functor
of nilpotent has
of
group
P-localization
subgroups,
localization
a result if
of the
normal
extend
nilpotent
P-localization
localization.
6 we
7
an arbitrary
the properties
embedded
preserves
in S e c t i o n
[41]). T h e
homology
. We call
we prove
In S e c t i o n
Hilton
of
is a n H P L - g r o u p . 4 we
derive
theory
(see M a l c e v
some basic
P-local.
quotients
In S e c t i o n
others,
we
and
show that
show that
only
are
the
it o w e s m u c h
for a n H P L - g r o u p
P'-roots.
we
integral
from
GROUPS
groups
is P - l o c a l
the d e f i n i t i o n
whose
central
group
OF N I L P O T E N T
results
of nilpotent
VI
the
o+ i Gk
Finally
of
we prove
theorem
o f Baer,
is a l s o
finite.
148 Much
of the m a t e r i a l
in H i l t o n
[41].
the p r o o f s
contained
Although
have
been
influenced
contained
other
papers
on the t h e o r y
[10],
[11],
In this
chapter
particular for
familiar
in S e c t i o n
[41],
7 owes
with
5 is to be
from
even been
much
of l o c a l i z a t i o n
[41],
taken
to B a u m s l a g
we m e n t i o n
found
many
of
from
[41].
[15].
Among
[61],
[52],
[39],
[86].
we w i l l
[43],
differs
b y or h a v e
be u s i n g
the
theory
the L y n d o n - H o c h s c h i l d - S e r r e
example
i through
our p r e s e n t a t i o n
The m a t e r i a l
[42],
in S e c t i o n s
Chapter
some b a s i c
VIII). facts
of s p e c t r a l
(L-HS)
Also,
about
we
spectral
shall
sequence
presume
localization
sequence,
(see
the
in (see
reader
for e x a m p l e
[2]) .
We
first
recall
at a f a m i l y
groups.
Let
denote
the
family
if
m
a
of p r i m e s
localized
of the e l e m e n t s in case ring
P
Zp
An a b e l i a n if A
A
(I.I)
basic
facts
about
and
not
then
in
of p r i m e s at
P
is empty, is flat group
abelian
P
, i.e.
A
integral
m
with
By
Zp
of the j
. Recall
homology
P'
is c a l l e d
ring
subring
k/j
of a b e l i a n
of primes.
. The
Zp = Q
of
we
local
denote
a P'-number
is the r i n g rationals
a P'-number. that
groups
consisting
Note
for a n y
of
P
that the
group.
is c a l l e d
P-local
the
Zp-module
group
we m a y
Ap = Zp Q A
the
integer
P' the
as
as a b e l i a n
localization
family
. The in
we h a v e
GrouRs
discuss
empty)
expressible
is P - l o c a l ,
is a n y
Abelian
(possibly
is a p r o d u c t
integers
Local
of primes,
abelian P
some
VI.l.
.
if it is a Z p - m o d u l e .
structure
associate
is u n i q u e l y
with
A
the
Note
that
determined. P-local
group
If
i49 Clearly Z
A I
> Zp
)
induces
(1.2)
is a functor.
The
obvious
ring
homomorphism
a map
e : A ~ Ap = Zp ® A
called
the
property. there A
Ap
(P-) To
any P - l o c a l
exists
a unique
is P - l o c a l
LEMMA
l.i.
localization
then
map.
group
f'
It s a t i s f i e s
B
and
: Ap ~ B
~ • A ~ Ap
the
following
to a n y h o m o m o r p h i s m
with
f'£
= f . It
is an
isomorphism.
Then
B'
universal f : A ~ B
follows
that
if
L e t the d i a q r a m
A)
> B
--9) C
Ap~--) B' --)> Cp
be c o m m u t a t i v e zation
map.
PROOF:
This
the
fact
LEMMA
exact
immediately
that
1.2.
with
Zp
Let
rows.
follows
by tensoring
A,B
be a b e l i a n
qroups.
~,
: A ® B ~ Ap @ B
(1.5)
~,
: Tor(A,B)
localization
maps.
is the
locali-
with
Zp
and u s i n g
If
Then
~ Tor(Ap,B)
C
is a P - l o c a l
~*
: Hom(Ap,C)
~ Hom(A,C)
(i.7)
~*
: Ext(Ap,C)
~ Ext(A,C)
isomorphisms.
(1.3)
h
,
(1.6)
are
and
is flat.
(1.4)
are
J Bp
abelian
,
qroup,
then
150 PROOF:
The
first
R~---> Q - - ~ A that
Zp
assertion
be
and
a free
hence
0 ~ Tor(A,B)
presentation
Qp
~
~l
~l
~ Rp~B
rows
and
that
the
upper
sequence
with
(Tor(A,B))p
The
assertion
Zp
In o r d e r
~
let
(1.4) that
and
the
the
fact
diagram
~ 0
are
is a n
localization yields
the
maps.
fact
Tensoring
that
isomorphism.
directly
to p r o v e
. Using
conclude
immediately
follows
(1.5)
~ ApC~B ~ 0
the maps
~ Tor(Ap,B)
(1.6)
: A ~ Ap
~ Qp@B
prove
A
flat we may
~ A@B
exact
:
of
~ Q@B
has
~,
is
To
~ R@B
~.~ 0 ~ Tor(Ap,B)
is o b v i o u s .
from
(1.7)
we
the
universal
construct
an
property inverse
of
of
~*
Let
(1.7)
C ~-9 D
represent
an
element
(1.8)
Ext(A,C)
It r e p r e s e n t s
It
is c l e a r
of
(1.8)
1.2
PROPOSITION £,
PROOF:
We
--@>Zp ® A
an element
the map
inverse
1.3.
that
of
Let
A
~ H n ( A P)
in
Ext(Ap,C)
associates
6*
with
: Ext(Ap,C)
consider
. If
p ~ P ~,
, then
an a b e l i a n
the
since
(1.7)
~ Ext(A,C)
Ap
where = 0
~ H n ( A P)
= H n ( A P)
qroup.
localization
case
, then
: HnA HnA
be
is t h e
first
cases
A = Z
. Then
the
Zp @ C = C equivalence
. Thus
the
class
proof
is c o m p l e t e .
: HnA
A = z/pkz in b o t h
that
is a n
of Lemma
If
of
Zp ® C ~-~ Zp ® D
is e x a c t .
map
--)>A
= 0
. If
is t h e for
Then t for
n ~ I
, the
map.
A
is f i n i t e l y p
~ P
, then
localization n ~ 2
generated.
, and
map the
Ap
= A
for
Let
. Thus n ~ I
assertion
is
i51 true
in t h i s
then
it
exact
case,
is t r u e
for
sequence
complete
for
A
direct
functors
Zp~-
in t h a t
COROLLARY n ~ I
If
the
assertion
and
product
Lemmas
generated.
limit
its
case,
An
of
Hn-
A
A
for
and
, 1.2
If
finitely
commute
is t r u e
of
l.i
finitely
and
1.4.
the
direct
(II.5.13)
it is t h e
true
also.
B
Thus
and
by
the
direct
B
subgroups.
limits
the
,
K~nneth
the p r o o f
is n o n - f i n i t e l y
generated
with
A
is
generated, Since
both
assertion
is
also.
abelian
group
A
is P - l o c a l
if a n d
only
HnA
if
,
i__ssP - l o c a l .
VI.2.
We h a v e if a n d
seen only
n ~ I
in C o r o l l a r y if
interesting
GrouRs
H A n
PROPOSITION
a group
2.1.
Local
that
an
1.4
is P - l o c a l
to c o n s i d e r
. Such
with
groups
will
The
for
be
class
G
abelian
n ~ i
group
. It
, for w h i c h
called
of
Homolo_qz
is
A
is P - l o c a l
thus
H G n
certainly
is P - l o c a l
for
an H P L - q r o u p .
HPL-qroups
is c l o s e d
under
extensions
by
HPL-~roups.
PROOF: the
Let
E
: N>--) G --9) Q
HPL-group
of groups
N
to t h e
is P - l o c a l are
except
P-local,
apply
extension
E r2 , s
(2 .i)
H N
. We
be the E
an
extension
L-HS
, and
of
spectral claim
the
HPL-group
sequence
that
its
this
we
for
Q
the
starting
by
homology
term
= Hr(Q,HsN )
for
r = s = 0
except
for
. To
s = O
see since
N
1
since
note
that
the
is an H P L - g r o u p .
groups Thus
S
the
groups
homology s = 0
r,s E2
groups
, then
are
P-local
in a c o m p l e x
~2-r's
for of
is P - l o c a l ,
s
P-local except
abelian for
they
are
groups.
r = 0
computed Finally,
, since
Q
is
as if
152 P-local.
It
P-local,
except
n
= 0
follows
2.2.
(2.2)
PROOF:
This
the
free
Since
second
and
K
such
Then
2.3.
under
The
the
. Hence
, m
~
H G n
2
and
thus
is P - l o c a l ,
E r's
except
are for
PROPOSITION
2.4.
o o Gi/Gi+ I
P-local
We
, we
...
the
an
of
with
Let
U
by
2.1
is a n
an
G/G~
induction is
by
an
, i = 1,2,...,k
sequence
* U K)
K)
~
HPL-qroup.
on
if
G
an
the
extension
under
it
free
is e n o u g h
of
the
(Proposition
Hn_iU
is P - l o c a l
is a n
induction.
products
HPL-subqroup.
HPL-group
*U
obvious
is c l o s e d
a HPL-subgroup
~ Hn(G
be
and
. Thus, consider
G
Ni/Ni+ I
HPL-qroup.
be
Hn(G
NI = G
quotients
amalqamated
HnG~HnK
Let
~ N2 ~
HPL-qroups
group
that
proceed
o o = GI/G 2
series
Mayer-Vietoris
yields
i ~ 2
E r's m
Proposition
class
assertion.
is
is
trivial
immediately
HIG
G
products
the
. Then
a
that
from
• "" ~
PROOF:
have
= Nk+ I ~ N k ~
follows
PROPOSITION
PROOF:
G
subqroups
HPL-qroups.
and
terms
r = s = 0
Let
e
normal
are
for
the
.
COROLLARY
of
that
Then,
to
prove
HPL-groups II.6.±)
~ Hn_IG~Hn_IK
for
for
G
n i> i
~
...
.
i ~> i
every
,
HPL-qroup.
i . For
i = i
we
have
HPL-group,
o o GI/G 2
is
P-local.
For
,%joo By
Proposition
then
the
extension
2.1
we
conclude
G °. 1
>
)
G
that
G/G ° 1
is
~ G/G ° 1
and
the
an
HPL-group.
associated
Consider
5-term
i53 s equence sequence
(2.4)
As
H2G
a cokernel
of
o o Gi/Gi+ I
group thus
~
H2(G/G
o
O O ) ~ Gi/Gi+ I
a homomorphism is P - l o c a l ,
between
also.
Let
P
be
a
unique
there has
exists
3.1.
central
unique
P'-roots m
(3.1)
remains
to
(3.2)
we
(3.3)
> G
proof
of
g
abelian
groups,
Proposition
the
2.4
is
and
---9> Q
I
be
(x-z I prove
xlmN = x 2 N
N
of
x
only
of
if
it
m
We
say
every
that
P'-
. Note
that
a group
number an
m
~
abelian
G I
group
is P - l o c a l .
with
P-local
being
and
x = y
qroups
be
primes.
~ G
with
unique
abelian
extension
a P-local
abelian
Then
P'-roots
is
closed
qroups.
a central
a P'-number.
=
there
with group.
exists
xi
Q Let ~ G
having x
~ G
with
(xlN) m = x ~ N
for )m
some
z
, since
N
that
they
= x2
, so
xi = x2 y
every
by
Roots
family
¢ G
class
and
x~
have
if
m x = xl-z
m m x = xl-z i =
Then
N >
~
y
The
xN
that
for
extensions
Let
let
if
P'-roots
PROOF:
so
P-local
Unique
empty)
a unique
PROPOSITION under
(possibly
P'-roots
unique
and
The
~ 0 Gab
complete.
VI.3.
has
-~ Gab
,
that
for
~ N
unique.
xi,x 2
some
zI
is c e n t r a l .
are
by
. Let
~ G
Thus
~ N Thus
with P'-roots
It
.
of m-th
roots
y
follows
that
x~ -~ (x 2 y)m : x~ ym
exist.
, then
suppose
uniqueness ~ N
z = zT
. It
in
G/N
= Q
154 whence
ym
: e
xi = x2
, so
PROPOSITION
Let
shall
An
obvious
yxy
x -1
Let
G
that
= x
and
in
be
G
it
are
a qroup
. Consider ZG
induction
~ ZG
is P - l o c a l ,
follows
that
y = e
. Thus
unique.
with
unique
P'-roots.
Then
is P-local.
i = I
prove
N
P'-roots
, i ~ I
We
Let
that
3.2.
ZiG/Zi_iG
PROOF:
Since
; taking
central
: ZIG/ZoG
on
let
the
i
m
~
and
then i
be
(unique)
extension G/ZG
completes
have
the
a P'-number.
m-th
roots
we
Z G ~-~ G --9> G / Z G unique
.
P'-roots.
proof.
For
all
y
~ G
we
have
obtain
-i YXlY
where
x~
m-th
= x
roots
roots.
, so
and
To
prove
=
The
z
ments
for
It
follows
that
is c l e a r
that
m-th
unique
roots
,
Xl,X 2
~
ZG
in
G
are
G
indeed
(i)
G
an
(ii)
o o Gi/Gi+ I
establishes and
G/ZG
has
have
x I
unique m-th
£ G
.
If
z
=
. Since
z I
zi
,
we
6 ZG
we
=
x 2
z I
have
be
I
in
G/ZG
a close
.
relationship
between
groups
HPL-groups.
a nilpotent
HPL-qroup; , i ~
.
unique
equivalentL
is
has
= x2ZG
P'-roots
Let
z
roots
theorem
3.3. are
ZG
let
some
xlZG
THEOREM
P-local.
= x2ZG
of m-th
following
with
. It
m
x 2
uniqueness
that
is
~ ZG
m
x I
(3.5)
so
xi
uniqueness
xlZG
m
by
that
hence
(3.4)
Then
: xI
is P-local;
qroup.
Then
the
followinq
state-
155 (iii)
G
has unique
(iv)
o o ZiG/Zi_IG
P'-roots;
, i ~ I
is P - l o c a l .
PROOF: (i) ~
(ii)
(ii)
~
follows
(iii).
If
(ii) h o l d s ,
central
extensions
abelian
group.
(iii)
(iv)
(iv)
~ ~
(i).
(central)
follows If
(iv)
We
remark
with
Let
of
Zp
unique
by Proposition
Cp
a
. Let , Cp
. Then
G
is k n o w n
(see
[I0])
is o b t a i n e d
generator
exist
3.5
kind
property
free p r o d u c t s
nilpotent
HPL-groups.
LEMMA
3.4.
there
is a u n i q u e
We
PRQOF:
in
Cp
G
first
G
existence
. If
abelian
c = I
group,
and
first
prove
be any nilpotent map
note
is an m - t h
To prove of
Let
We
will
f : Cp ~ G
with
root
of
some p o w e r
we proceed , then the
G
by
is
additive
as Z p - m o d u l e . 2.3.
However
to w h i c h
establish with
there
some
respect
to
following
f(t)
is c l e a r , of
t
induction
is an a b e l i a n
assertion
G
Cp'S
HPL-qroup
that uniqueness
in
nevertheless
the
Cp
by Proposition
of
and
copy of the of
elements
Proposition of
Thus
A counterexample
written)
is a H P L - g r o u p there
groups.
of HPL-groups
in g e n e r a l .
be a f i x e d
that
3.1.
by successive
abelian
the c o n c e p t s
are n o P ' - r o o t s . of freeness
by Proposition
by P-local
coincide
G
by a P-local
3.2.
(multiplicatively
G = Cp
P'-roots
by successive
2.1.
true that
t ~ Cp
is o b t a i n e d
P'-roots
the g r o u p
P'-roots
G
unique
of an HPL-group
it is n o t
Let
with
Proposition
holds
unique
follows.
group
it
that
has
from
extensions
is a n H P L - g r o u p
as
G
2.4.
the g r o u p
of a group
Thus
G
groups
from Proposition
is o b v i o u s
and
let
= x
.
since
where
x ~ G
every m
. Then
element
is a P ' - n u m b e r .
on the n i l p o t e n c y HPL-group, in this
hence
case.
class
c
a P-local
Let
c ~ 2 .
156 Consider
the
central
a unique
have
a central
map
extension f'
:
extension
Cp
E'
~
E
: G°'--gc G
G/G °
with
C
with
~[E']
--gG/G2
f'
(t)
. By
= xG °
.
C
= ft*A[E]
induction
Consider
H2 ( C p , G )cO
~
we
then
and
the
diagram
O
E'
: Gc>-----> K
JJ
(3.6) E
---97 C p
g~
f'~
: G ° ~---> G ---~ G / G ° C
Next
we
compute
H 2 ( C p 'G°) that
H 2 ( C p 'G°)c
H2C P = O thus
It
follows
. By
apply that
have
a map
Let
f"
: Cp
~ G°c
phism Let L
be
= L(S)
1.2
gs be
defined
and has S
first
recall
a
the set
(Zp-)
defined
by
: Cp the
by
map
' where
= t
, s
E'
by
(Cp) s . There
~ S
the
free
is a c o p y is a n
, y
s
~ Cp
We
= O
: Cp
~ K
u
£ G Oc
some
. Then
group
obvious
have
P-local.
by for
= u
of
is
we
~ Ext(Z,Gc°)
= x-u
-i
1.3
GOc
splits,
f"(t)
= g s ( y ) - (f"(y))
F = F(S)
that
gs(t)
defined
and
ts
Proposition
E x t ( Z p , G °)
that
with
that
know
that
property.
let
By we
~ sO
~ G
f(y)
generator
k(s)
prove
required
-- s ~ s ( C p ) s
guished
to
2.4
H 2 ( C p ,G c °) = O
We
~ G
Proposition
Lemma
say.
: Cp
. We
~ E x t ( Z P 'G°)c • H o m ( H 2 C P , G ° )
may
f
C
the
map
is a h o m o m o r -
on
S
Cp
with
map
k
. Let distin: F ~ L
,
.
S
PROPOSITION ~< c
. Let
phisms
f'
3.5. f
LeE
: S ~ G
: F ~ G
G
be
be
a
, f"
an
HPL-qroup
function.
: L ~ G
such
which
Then that
there the
is n i l p o t e n t are
unique
diagram
.
of
class
homomor-
,
157
S
3.7)
f, ....
F
9
.-7
k~ L
is
/
commutative.
Moreover
the
diaqram
F
kl
(3.8)
5/
G
T,
is c o m m u t a t i v e ,
PROOF:
The
of
free
fs
:
For
(Cp) s ~ G
about
with
then
c+l G°
= e
hypothesis,
: H F ~ n
PROOF: 1.3.
H L n
For For
II.6.3.
S S
s
yields
The , n ~
f" we
map I
consisting arbitrary
it
Lemma
f(s)
the
of
3.4
yields
is
element an
(3.7)
, 5"
has
easy
universal
property
of
commutative.
making
the
property
a unique
universal
localization
one
then
the
making f'
: F ~ L
from
. The
: L ~ G obtain
k is
follows
~ S
fs(ts ) =
product
3.6.
f'
every
free
k,
L/L~+ I
also.
groups.
PROPOSITION
lkc+ I
>
statement
by
o F/Fc+ I
>
property
(3.8)
the Since
commutative.
that
map.
this
follows
consequence
from of
Proposition
Corollary
±58
VI.4.
In t h i s
section
we
group
G
nilpotent formation we
the
4.1) . We
then
induced
THEOREM
4.1.
Let
into
such
(41).
~,
with
the
is t h e
For
we
short
Theorem We
4.1
shall
map
see
map
zation property groups.
map
not
~ HnK
with
of
,
n ~
a natural
several
make
of
class
class ~< c
into
a
4.6).
c
and
first
(Theorem
-p
(Theorem
a
trans-
steps;
~ : G ~ Gp
which
qroup
Gp
,
map
call (P-)
in t h i s
for
in
transformation
. Then
there
a homomorphism
i
necessarily
nilpotent
groups
k
: G ~ K
the
respect
a
(P-)
in T h e o r e m
4.1
specific
localization and
enjoys abelian
n ~ i
is c a l l e d
~ : G ~ Gp
shall
Z : G ~ Gp
and
with
that
: HnG
the
is
homomorphisms
H G n P
localization
the
Gp
associates
map.
between
nilpotent
that
that
property
(P-)
functor
Grou_Rs
homomorphism
a nilpotent
: H G ~ n
k,
a
HPL-group
the
HPL-qroup
localization
A homomorphism
and
be
Nilpotent
construction
a natural
G
a nilpotent
: G ~ Gp
. The
define ~
the
a nilpotent
Gp
and
of
construct
group
functor
is
shall
[ : G ~ Gp
define
exists
Localization
the
to
map
subsequent
properties groups:
map
(of
G
of
in t h e
to
that those
it e n j o y s G
proof
G map.
of
).
sections
in p a r t i c u l a r
ma D . For
is a P - l o c a l i z a t i o n
constructed
analogous
homomorphisms
localization
into
the of
localization the
locali-
a universal
(nilpotent)
HPL-
i59 PROOF: of
(Hilton
G
. If
lization Let E
[4±])
c = I
map
: N~--9 G --9>Q
Let
•
~'
: N ~ Np
tion.
We
conclude
A[Ep]
~"*(~) map
=
=
: Q ~ Qp
We may
our
~ : G ~ Gp
: N~
thus
the
loca-
the
central
. Consider are
the
given
coefficient
extension
localization
to use
theorem
by
induc-
(II.5.1)
and
define
) G
a central
extension
---9) Q p
the
5'~
(4.4)
1.3
c
~ H 2(Q,NP)
~ H 2 (Qp,Np)
Proposition
making
E
consider
which
the universal
(~"*)-Io~.I(~)
~(~)
and
class
properties•
~ { H 2 (Q,N)
=
on hhe n i l p o t e n c y
By Proposition
required
, Q = G/G ° c
: N p )----> G p
~ =
the
: H 2(QP,NP)
isomorphism.
Ep
induction
that
~"*
(4.3)
has
~"
from
by
is a b e l i a n •
AlE]
and
, 1.2
(4.2)
by
G
N = G° c
maps
is a n
, then
Set
i.I
proceed
6 : G ~ Gp
c ~ 2
Lemmas
We
II.4.3
. Since
by
establishes
construction the
existence
of
a
diagram
---9 Q
~
~"J,
Ep : Np~-----)Gp---~ Qp commutative. order we
to prove
consider
follows i.i
By
from
and
1.2
(4.5)
is t h e E2-term
Proposition that
the
~,
: HnG
L-HS-spectral
the
universal
the g r o u p
~ HnG P
, n ~> i
sequences
coefficient
Gp
of
is a n is t h e
E
theorem
and
HPL-group.
In
localization Ep
map,
. It e a s i l y
(II.5.2)
and
Lemmas
that
Hr(Q,HsN )
localization and
2.1
hence
e~)
map, the
Hr(Q,HsNp)
except
E -term
for is
~,, .~. > H r ( Q p , H s N p )
r : s = O
localized
by
. It
follows
the m a p
that
induced
by
the 11
!
~.$.,
160 except map,
It
for
except
is
clear
class
c
Theorem
We
r = s = 0 n
from
the
then 4.1
remark
potent
is
it
Thus
~ : G ~ Gp the
Note
map
that
6
no
G
a localization
such any
proof
that
5.3)
of
k,
finite that
the
to
use
PROPOSITION
localization
with
PROOF: = I
case.
xm
We
~ ~G
Let
is
the
localization
of
class
if
G
~ c
is
nilpotent
. The
proof
of
of
Proposition
that
isomorphism. the
4.1.
map,
i.e. H K n
In
certain
Let
G
map.
If
x
G
Gp
and
results
K
We
and
shall
for
see
the
group
~ Gp
exists
there
is
k
we
will
are
used group
: G ~ K
map
the
an
HPL-group
a nilpotent
later
of
a nil-
G/G ° c
case
a map
proved
a nilpotent
is
series
for
independent
series
this
localization
to be
G
and
central
obtain,
the
if
a nilpotent in
HPL-group
indeed
is
course,
to
is
that
P-local
lower
order
central
be
G
Of
whatsoever. is
is
if
the
, n ~ i
series
lower
of
an
G° c
groups
properties
2.4
we
may
use
(Corollary chosen
later,
it
central
is c o n -
construction.
and
let
% : G ~ Gp
a P'-number
m
~
be
I
.
proceed
, then
nilpotent
conclude
an
for
4.2.
that
then
Theorem
the
Gp
~ c
construction
venient
HnG P
of
from
identify
central
However,
c
is
: H G ~ n
series.
the
may
special
the
~
complete.
class
to
in
also
follows
we
: HnG
.
is
of
6,
construction
thus
HPL-group
then
= O
Gp
that
HPL-group.
use
for
. Hence
G
c ~ 2
by is
induction abelian,
. Consider
on and
the
the our
nilpotency assertion
diagram
class
c
is w e l l - k n o w n
of
G in
. If that
161
E
:
(4.7) Ep
where Let
we x
have
E Gp
N
>---9 G
---9> Q
~'~
~I
6"~
:
used
Np~-~
the
, then
Gp
notation
there
Qp
introduced
exists
mi x
~
in
a P'-number
the
proof
mi ~
I
of
and
Theorem y
~ G
4.1.
with
mI Np
=
(XNp)
= ~"(yN)
=
(£y)Np
mI Thus
there
exists
P'-number
m2
z
and
~ Np
with
x
=
(%y) z
. But
£ N
with
zm2
=
~'(u)
. Thus
u
ml°m 2 (4.s)
so
that
m
4.3.
6 : G ~ Gp
be
Let
P'-torsion m
= e
x
the
the
c
of
G
. If
in
that
known tion
diagram
(4.7)
P'-torsion with
be
xm
c
also.
Hence torsion
xm
of
case.
m-th
element.
be
=
~(y
a nilpotent
Then
c ~ proof
have
xN
in
Q x
there
proceed
Let
. Since is
find
a
m2 )-e'(u)
x
element.
roots
, then
in t h e
~ N
G
map.
Thus
we
= I
element ~ N
~(y
may
-u)
,
properties.
Let
a P'-torsion
converse
we
m2 =
required
[4i])
we
~ ker
qroup
e
if
and
and
let
only
if
element.
~ G
introduced
m2 -z
localization
. Uniqueness
prove
the
(Hilton
element,
To
(%y)
has
is a P ' - t o r s i o n
PROOF:
y
=
= ml-m 2
PROPOSITION
x
m2
x
then
G
~ ker
a P'-torsion
by is
2 of
Gp
~"
x
, it
by
the
~
i
= e
.
with
xm
to xN
£ ker
itself
class
the
nota-
the is a
a P'-number
x
a
is w e l l -
. Using
refering
that
that
m
result 6
is
nilpotency
induction
exists
so
[(x)
y
the
and
follows
element,
on
~ ker
4.1
. Thus there
yields
and
let
y =
a P'-number
then
abelian
Theorem
that ~
exists
induction
, and
~ ker , so
in
Then
m
~
I
6' is
a P'-
i62
COROLLARY
4.4.
: G ~ Gp
PROOF: We
is
This
note
Let
that
G
be
injective
for any
immediately for
a torsion-free
follows
P = ~
P
nilpotent
qroup.
Then
.
from
Proposition
, Corollary
4.4
: G ~ K
a localization
4.3.
is a f a m o u s
result
of Malcev
[61].
PROPOSITION
4.5.
Let
G° O i/Gi+l
O
,
(4.9)
ki :
(4.10)
~i
induced Note
by
that
PROOF:
We
follows
£ we
do
not
proceed
from
the
localization
the maps
,
maps
(or K)
on
for all
i .
nilpotent.
i . For
i : i
the
assertion
that
: HIG
localization
G
induction
fact
Then
~ K/K? 1
suppose
by
~,
is t h e
the
map.
O
~ Ki/Ki+ i
: G/G? 1
are
be
map
~ HIK
by
hypothesis.
Let
i ~ 2
. Consider
the
diagram
o (4.11)
o
ki_l~ o
~i ~
Ki_I/K i )
where
ki_ I
struction
of
localization
and the map.
~i_l ~
o
~i-i
>
-->>
are
localization
localization Consider
of
then
G/G: the
maps
by
it f o l l o w s
diagram
induction. that
~i
By conis t h e
i63
G?
)
1
(4.12)
G
; ~
l
the
~ G/G?
1
~.l
k;
K?
and
>
associated
l
) K
~
5-term
K/K?
1
sequences o o Gi/Gi+ I
o
H2G (4.13)
[.
and
T}{EOREM 4.6.
are
map,
thus
g
: G ~ K
Let
is p r e c i s e l y
followinq
~
(6i) .l
([i).
localization
There
}{2 ( G / G )
~.l
Since the
~
square
one
~
0
k i~
localization completing
be
maps, the
it
gp
that
ki
is
proof.
a homomorphism
homomorphism
follows
of
niipotent
: Gp ~ Kp
such
qroups.
that
the
is c o m m u t a t i v e g --->
G
K
(4.14)
gp Gp
PROOF: follows m
~ i
gp(X)
We
from y
must
be
: F --~> G
note
that
Proposition
and
It r e m a i n s f
first
>
£ G the
to p r o v e of
G
with unique that . Then
Kp
if
it e x i s t s
4.2
that
xm = ~y m-th gp
if
x
. Thus
root
of
exists.
Proposition
must
gp
~ Gp
Choose 3.5
there
g p ( X m) ~g(y)
be u n i q u e .
=
~g(y)
in
it
a P'-number
, so t h a t
Kp
a free
yields
exists
For
presentation
a commutative
diagram
i64 f!
F
~
F/Fc+ I s
(4.15)
k~
K
kc+ll
~,, L
It f o l l o w s
from
Also,
it
order
to define
f[(x)
= e
find
~
Proposition
follows
from gp
if
x
a P'-number
: Gp ~ Kp
m
= f" (x m)
f'(y)
is a P ' - t o r s i o n
so
that
complete.
f~
It
VI.5.
We
shall
THEOREM
4,
5.1.
: G ~ Gp K
unique
f'
PROOF:
Since
COROLLARY
Let has
HPL-qroup
clear
and
map.
Then
k'
with
k'~
K
there = k
.
that 4.2
to
e F/Fc+ I . Then
from
Proposition
4.3
is a P ' - t o r s i o n
that
ker
f" ~ k e r
the map
that
element, f[
induced
is by
of NilLx~_tent_G[ouRs
property
of
the map
6
as
defined
of consequences.
qroup.
homomorphism
and
y
to b e
universal
we may,
Proposition
use
In
is c o m m u t a t i v e .
a number
f'~
we
gf' (y)
map.
is s u r j e c t i v e . to p r o v e
conclude
proof
localization
enough
with
Localization
draw
with
G
this
Thus
(4.14)
of
to a n y
localization : Gp ~ K
is t h u s
: G p -- K p
followin~
K = Kp
Let
f"
b e a nilp0tent
: Gp ~ K
5.2.
that
. The
fp
that
then
the
and
4.2
. We
a universal
G
is t h e
= kc+l(y)
= e
= f[(x)
state
kc+ I
prove
element.
Properties
first
in S e c t i o n
xm
We define
is t h e n
that
it
. To
(f" ( x ) ) m
e = ~gf'(y)
thus
f"
with
~, f' (y)
=
4.5
Gp
Proposition
6 ker
~l
-->
L/Lc+ i
The
localizati0n
property. f
: G ~ K
To
any
there
map
nilpotent exists
a
= f .
and
be
indeed
must
nilpotent
exists
define
and
a uniquely
let
f'
k
= fp:Gp~Kp
: G ~ K
determined
be
.
a
isomorphism
165 PROOF: k'
Since
K
~ K
with
: Gp
are
the
is
localization
then
With
follows
this
proof
of
given
a
we
Theorem
of
Theorem
4.1
we
: G ~ K
. It
then
thus
5.3.
to
~
k~
HIK
a map and
: H2G P 2 H2K
that
substantiate
be
a nilpotent
of
an
: HIG
IV.I.2
in
follows
G
k,:H2G~H2K
from
is
K
the
remark
this
isomorphism. after
Suppose
central
and
5.2
: Gp
~ K
with
nilpotent.
Let
k
: G ~ K
we
are
in
instead
the
proof
a localization
that
k'
the
series
described
, say
Corollary
an
group.
. Using
HPL-group
.
k'
construction
the
isomorphism
5.4.
Suppose
Thus
let I
K
be
that
K
has
unique
of
G
, and
that
~ hG
: Gp
~ K
with
h'
~ = h
K
has
Theorem
5.1
£ = h
It
y
with
and
subqroup
~ ker ym
h'
~ ~G
unique there remains . By . Let
series
isomorphism
xm
Since
h'
G
central
with
PROOF:
with
finite
~ I
h'
By
any
m
phism
3.3.
be
determined
Let
P'-torsion
P'-number
G
usinq
a uniquely
homomorphism.
~
,
series
obtain
yields
have
G
series
Let
constructed
COROLLARY
m
able
k,
5.1
there k'6
map
is a
= k
. We
T
may i
state
exists
the
we
Let
central
determined
COROLLARY map
are
central
the
uniquely
. Since
Corollary
4.1.
of
k
Theorem
: HIG P 2 HIK
result
lower
= k
maps,
from
finite
HPL-group,
k'~
k~
It
an
to
that
Proposition
4.2 , z
~ G
h
there
be
by
.
a
i__ss
there
h'
exists isomor-
Theorem : Gp
~ K
isomorphism.
exists
~ then
= k
h
a unique
HPL-group
an
k'~
ker
c K
exists
there
: G ~ K
that x
is
localization
with
homomorphism h'
a
. Then
let
every
is a n
a unique
= ~z
and
there
to p r o v e
ym
G
P'-roots,
. Then
it
of
: Gp ~ K
nilpotent,
P'-roots exists
k'
be
a P'-number
±66 hz
so
that
z
ez
= e
But
Thus
h'
exists
remark
that
approach
(h'y) m
To
of
h'w
for
some
= u
by
4.3
P'-roots
surjectivity
= hv
,
Proposition
uniqueness
prove
um
By
= e
let v
have
yields
u
~ G
we
~ K
y
. Then
. Consider
uniqueness
of
= e
.
there
w
E Gp
P'-roots,
proof.
Corollary
5.4
constitutes
the
basis
of
Hilton's
[41].
PROPOSITION
5.5.
morphism.
Let
Then
PROOF:
Let
m
with
I
and
cleariy
the
=
element.
= e
with
. Then
completing
~
ym
injective.
= ~v
= h' (ym)
P'-torsion
P'-number
wm
thus
a
then
is
a
with
We
is
: h'~z
define
u
x
G
be
nilpotent~
gp
: Gp
~
~ Qp
. By
Proposition
xm
= ~y
£ Gp
by
Qp
for um
is
some
= ~v
and
ie~
g
: G ~
Q
be
an
epi-
surjective.
4.2
y
~ Q
. Then
there
. Let
clearly
exists
gv
= y
gp(U)
a
P'-number
, v
= x
~ G
, and
. Thus
gp
is
surjective.
PROPOSITION morphism.
PROOF:
Let
Then
hp
Suppose
with
so
5.6.
xm
that
x
= ~y
hy
x
= e
. By
COROLLARY Then
the
: Np
~
for
some
y
[hy
= hp~y
be y
is
uniqueness
Let
sequence
a
nilpotent
Gp
hp
must
5.7.
be
~ ker
is m o n o m o r p h i c , m
G
is
there
= h p ( x m)
:
N ~ h
G
exists
by
element.
roots
g-~> Q
(hpx)m
element
a P'-torsion m-th
h
: N
~
G
be
a mono-
a P'-number
m
~
i
. Thus
P'-torsion
of
let
injective.
. Then ~ N
and
be
in
an
Np
: e
Proposition Hence we
y
have
extension
4.3.
~ ker x
with
=
~
Since , so
h that
e
G
niipotent.
167 hp Np
is
an
is m
By
Propositions
that the
~
gp
is
trivial
i
with
whence
map. xm
it
u
some
=
to
~ ker v
~y
~ G
Qp
5.6
Next x
some
we
y
that
the
. Then
. In of
exists
that
hp
the
there
is
injective
composition exists
gphp
a P'-number
. Thus
: gphpX m
kernel
that
. Then
~ N
= e
there
know
show
E Np
: gphp~y
gphpx
we
(gphpx)m
=
order gp
to is
complete
the
contained
a P'-number
m
in
with
proof Np
um
. Thus
= ~v
for
. Since
element
P'-number
for
that
show gp
and
let
= 6ghy
~gv
the
D
5.5
Thus
follows
remains
let
Gp
surjective.
e
it
>
extension.
PROOF:
and
gp
)
gv m'
= gp~V
= g p ( U m)
is
P'-torsion
£ Q with
a
(gv) m '
= g(v
=
(gpU) m
= e
element.
m'
) = e
,
Thus
. It
there
follows
exists
that
a
there
m.m t exists
y
( N
with
hy
= v m'
(hpx)m-m'
so
that
hpX
PROPOSITION let is
U
to
5.8. a
= hp(xm-m
, as
required.
Let
G
subqroup
of
be G
a
~ Np
' ) = hp~y
finitely
. Then
x
Up
= [hy
generated 2
Gp
by
x
=
= ~ y
= u m-~'
[(v m ' )
nilpotent
if
and
Since
G
only
. Then
qroup if
and
[G:U]
a P'-number.
PROOF:
the
be
= u
. Define
Suppose
normalizer consider
[G:U] of
the
U
case
= m in where
is G
a
P'-number.
is b i g g e r U
is
normal
than in
is
U
, so
G
. Thus
nilpotent,
that let
it
suffices
= m
168
U >
be an e x t e n s i o n By Corollary that
for
with
5.7
this
--9> Q
) G
Q
localizing
part
of
generated.)
Conversely,
let
in
G
Up ~ Gp
. Consider
U~
By Corollary
5.7
of P'-torsion Q
is
finite
PROPOSITION nilpotent
PROOF: N
H2(Q,N)
Qp
= e
whence
o f an o r d e r
m
where
Let
E
need
Again
the
(Note
fact
we may
Q
it f o l l o w s
m
that
G
suppose
U
a finite
that
generated,
Q Q
consists is.
Hence
is a P ' - n u m b e r .
---~)Q
by
induction
b e an e x t e n s i o n
qroup
of o r d e r
H2(Q,N)
on the n i l p o t e n c y
= O
. For
it is of e x p o n e n t
N
U p ~ Gp
m
with
N
where
m
c
N
is a
splits.
then
= O
not
is f i n i t e l y
: N >---~ G
and
E
Bnd
we did
is a P ' - n u m b e r .
the d i a g r a m
G
We proceed
m
isomorphism
isomorphism.
Since
is a b e l i a n ,
Thus
an
elements.
Then
is P - l o c a l ,
be
we have
5.9.
an
where
Op--)> Q p
HPL-group
P'-number.
yields
m
--~> Q
> G
UpS>
of order
the a s s e r t i o n
is f i n i t e l y
normal
finite
. Let
)
>
c ~ 2
G
m
H2(Q,N)
, since
. Consider
~
g~
Q
class
of
is P - l o c a l ,
. If
since
is of o r d e r
m
.
the d i a g r a m
Q
II
N/Nc>---> G / N c --->> Q
By i n d u c t i o n sider
slQ
the and
lower
sequence
g -i (slQ)
. This
splits, yields
by
s I : Q ~ G/N c
the e x t e n s i o n
say.
Con-
N
169 -i E'
Since
N
(slQ)
: Nc>--- > g
is P - l o c a l
and
~ slQ
is o f
siQ
order
m
, the
extension
E'
C
splits
by
s
E
of
s2
: slQ
s
thus
the
COROLLARY
5.10.
nilpotent
HPL-qroup
PROOF: be
a
on
By
Since such
E
Suppose
Proposition
. It Q
is to
that
Q
: N >---) G
g
Composition
-i(s i
Q
---)> Q
a
finite
N ~
ZkG
. Then
5.9
we
know
conjugation
prove
operates
s2 -->
siQ
and
The
, say.
Q)
yields
! G
a splitting
,
proof.
Let
splitting. N
sI -->
: Q
completing
P'-number.
~ g-l(sIQ)
that
trivially
operates
an
extension
with
qroup
of
order
where
G = NxQ
that
in
this
be
G
E
induces
trivially
ZiG
n N
on
Z.G
is
m
i__ss
.
splits.
operation
on
m
N
an
Let
: Q ~ G
operation
trivial.
there
s
Suppose
exists
n N
but
x
~ Zi+±G
of
i <
Q not.
i < k
non-trivially
on
l
Zi+IG
n N
[x,sQ] tion
. It
~ e q
follows
Since
: Q ~ N
that
there
[Zi+IG,G ] ~
defined
by
exists
ZiG
q(y)
=
it
is e a s y
[x,sy]
, y
of
is
n N
to
see
~ Q
with that
the
func-
is a h o m o m o r p h i s m . r
Since
q
subgroup N
is n o n - t r i v i a l , of
N
is a n i l p o t e n t
PROPOSITION Then
the
P'-number
PROOF: by
of
5.11.
set m
Using
J ~
order
the
m'
HPL-group
Let of
I
the
all
with
U
image , say,
this
be
x
a subqroup
£ U
P-localization
where
a non-trivial
m'
finite
is a P ' - n u m b e r .
Since
is a c o n t r a d i c t i o n .
elements m
Q
x is
map
~ G
of
the
such
a subgroup
~
the
nilpotent that of
set
there G
j
qroup
G
.
is a
.
may
be
described
170
j = ~-i(Up
It
is t h e n
The
group
clear J
that
~ ~G)
J
is a s u b g r o u p .
is u s u a l l y
called
the
P'-isolator
of
U
in
G
(see
[i5]).
In t h i s
section
necessarily
start
for
an
zation map.
the
i.e.
by
terms
first
for
all
admit
Let
be
lower
we
of
(not
localization
maps
all
G i
are
which
, n ~ I
the
uniquely
that
We
the
are
looking
is a
locali-
localization
determined.
homomorphisms
study
at are
in t h i s
in a n a t u r a l of
groups
G
least
the
quotients
determined. section
it
For
of
K
the
is p o s s i b l e
way. which
are
direct
limits
of
Gi
instead in
to
map
, i.e.
image
4.5
series
Gi
that
is n o t
central
groups
is o b v i o u s
: H G ~ H K n n K
a group.
~ : G ~ K
. Hence
intend
= lim
G
i
class
G
that
classes
~ K / K ?i
the
(nilpotent)
two
members
Proposition
consider
(6.2)
lity
from
the
~,
in g e n e r a l
maps
of
remark.
with
a localization
nilpotent
whose
on
a homomorphism
: G/G~
of groups
to d e f i n e
It
that
localization
classes
We
a map
~i
the
and
it f o l l o w s
(6.1)
are
K
is c l e a r
However,
a few words
groups
following
HPL-group map,
say
way.
with
It
shall
nilpotent)
in a n a t u r a l We
we
G
of
Gi
. Hence subgroups
we
may
we m a y of
G
consider
suppose , and
its
without
that
G
canonical loss is the
of
genera-
direct
171 limit Gp
of
of
(all of) G
its n i l p o t e n t
subgroups.
We d e f i n e
the
localization
by
(6.3)
Gp
= lim
(G i) P
~i : G i
The m a p s both we
functors
H
immediately
PROPOSITION so that
Let
G
and
~ (Ki)p}
- as w e l l
as
The m a p
~,
~ : G ~ Gp
K
homomorphism. and
yield
be
6 : G ~ Gp
Zp ® - c o m m u t e
: HnG ~ H n G P
with
f
induces
a well-defined
. Hence,
we
, n ~ I
is a l o c a l i z a t i o n
two g r o u p s
Then
thus
a homomorphism
direct
Since limits,
have
6.1.
map,
{K i}
n
(G i)
obtain
in our a map map
and
of the
let
f : G ~ K
directed
directed
by universality
localization
map.
class
of the
is the
systems
systems
a map
fp
be a {Gi}
,
{(Gi)p}
,
: Gp ~ Kp
. We
have
PROPOSITION
6.2.
: G ~ Gp
PROOF:
It is o b v i o u s
PROPOSITION unique
6.3.
5.6
(6.4)
the
fp
as d e f i n e d
with
and
above
satisfies
the r e q u i r e d
the
we h a v e
xm
~ ~G
localization.
that
to
. Finally,
x y
~ Gp ~ ker
Then
Gp
there
exists
~
has a
if and o n l y
if
element.
Gi ~ Gj
induced
% : G ~ Gp
Moreover,
m ~ I
Let
is a f u n c t o r
transformation.
that
Let
is a P ' - t o r s i o n
PROOF:
G ~--> Gp
relations.
Pt-roots.
P'-number
localization
is a n a t u r a l
commutativity
y
The
map
be two n i l p o t e n t (Gi) p ~
(G i ) p ~ G p
.
(GJ)p
subgroups
of
is i n j e c t i v e .
G
. By P r o p o s i t i o n
It f o l l o w s
that
172 Now x
let
x
~ (Gi)p
P'-number be
~ Gp
= lim
(Gi) p
and
(Gi) p
has
m >~ i
a P'-torsion
Thus
y
if
element
y
£ ker
is a P ' - t o r s i o n now
turn
nilpotent there
free
6.4.
groups. just ker(f,
of
the
with
Also
there
exists
~ 6G
. Finally,
i
follows
that
~i
c I
for
Up
two qroups~
U
be
a subqroup
U
*Up
with y
let
y
£ G
Gi
y
E ker
some
a
~ . Con-
i ~ I . Hence
y
~
of
obvious
with
exact
~
rows
their
K
. Then
sub-
6 : K ~ Kp
have
that
Kp
~ U
. The
. It existence
universality. by Proposition
~
map.
nilpotent
: K ~ Kp)
from
of
localization
and
and
Hn(G.uK)
and
its
it f o l l o w s
Gp
unions G
is a
~ : G ~ Gp
of both
HnG@HnK
are
of both
n U = ker(~
is t h e n
diagram
that
the union
as kernels
is a s u b g r o u p
subgroups.
and
maps
: G ~ Gp)
*Up Kp
Kp
is a l s o
elements
is a c o m m u t a t i v e
amalgamated
be
that
= ker(6
~ : G, U K ~ Gp
there
i ~ I
exists
~ ker
with
localization
P'-torsion
that
xm
, K
Let
note
: U ~ Up)
follows
y
6 : G, D K ~ Gp
we
Since
the
G
subqroups.
First
. It
, then
products
Let
is a m a p
PROOF:
P'-roots.
there
(Gi)p)
~
exists
~G l . Thus
Then
•
there
element.
to
PROPOSITION
unique
xm
e ker(5 i : G i ~
versely,
We
with
~ then
Hn_IU
~
...
~ Hn_!U P ~
...
Finally, II.6.1
(6.5) ... ~ H n G P ® H n K P ~ H n ( G p * u p K p )
Since For
~
free
, ¥
products
PROPOSITION ~aps. also.
are
Then
6.5. the
localization Proposition
Let
Obvious
maps, 6.5
~ : G ~ Gp map
~
may
and
6 : G,K
is. be
sharpened.
5 : K ~ Kp
~ Gp*Kp
be
localization
is a l o c a l i z a t i o n
map,
i73 PROOF:
This
is o b v i o u s
(6.6)
is
~,
the
: Hn(G*K)
localization
COROLLARY
6.6.
since
map
Let
G
= HnG@~HnK
for
, K
n > I
be
in
~
PROOF: 4.5
By
V
= N
the
assertion
In
this
that We
section
has
start
Suppose
as
with N ~
A
we
an
G is
VI.7.
*V K
then
shall
a number
For
x
~ G
let
x
set
P
of
primes
the
Extensions
prove
. By
map
Propositions
k
, so
denote
maps
Xp
on
Homomorph~sm
extensions
a well-known
remarks.
~ N
of
a result
corol~ry
some
: N
obvious
map.
= G*K/(G*K)°+I
on
of
for
the
and
6.5
obvious.
Result
immediate
ZkG
. Then
=c
i s the l o c a l i z a t i o n
definition
= Hn(Gp,Kp)
.
-
~ : G *V K ---, Gp *V Kp
HnGp~HnK P
N
be
that
in
particular
: Np
induced
~ Np
of
theorem
Let
the
~
homomorphisms of
a normal
Baer
[7].
subgroup N
of
G
is n i l p o t e n t .
automorphism.
For
any
with
x N
-
>
N
>
Np
(7.1) Xp Np
commutative is
then
LEMMA of the
define
an
compatible
7.1.
Let
with
M
be
automorphisms. G-action
on
action this
and
f
G
on
Np
. The
localization
map
G-action.
a nit.potent
Let N
of
HPL-qroup
: N ~ M
be
M
the
. Then
on which
a homomorphism map
f'
: Np
G
acts
compatible ~ M
with
as
qroup
with
.
i74
N
>
Np
/ f$
/
M
~//
f'
commutative
is a l s o
compatible
with
PROOF:
u
then
there
exists
~ N
. For
u
m
=
Let 6v
for
£ Np some
v
(Xof'U) m
Taking
(unique)
m-th
THEOREM
7.2.
some
. Suppose
k
Let
P'-number.
Let
f
: N ~ M
be
M
. Then
be
that
M
be
exists
N
we
=
f'~ (XoV)
the
a normal
subqroup
Q
is
= G/N
h
>
/
I
with
=
f(xoV)
= f' (XoU m)
=
=
(f' ( X o u ) ) m
result.
of
finite
G
of
with
order
N ~ m
HPL-qrou p on which
compatible
a unique
= XofV
f' (Xo£V)
establishes
m ~
have
=
f~
(7.2)
~ G
a nilpotent
)
a P'-number
= Xof'6v
a homomorphism
there
G-action.
= X o f ' (u m)
roots
N
x
the
with
the
homomorphism
ZkG
for
where G
acts
G-action
f'
: G ~ M
is
enough
m
is
and
on
N
let and
with
G
11 kJ
f'
M
commutative.
PROOF:
First
result
for
note that
To
that
(7.3)
M
remark
= Np
the
is
the
that
and
uniqueness
f(x)
prove
we
f
by : N
is c l e a r , unique
existence
NI = N
we
m-th
Lemma ~ M
the
since root
consider
, Ni+ I =
7.1
it
localization
for
x
£ G
of
f ( x m)
the
series
[G,Ni]
map.
we in
to
have M
, i = 1,2 .....
.
prove Next x
m
the we
E N
, so
a
175 Since
Ng ~ Z k _ ~ + I G
series
{N i}
it m a y
be
Note and of
that
to construct
5.6
on
it
we may
Mi/Mi+ i
series
{Ni~
H2G
the
define
. We
H2Q
that
series
of
subgroups
now
Nk+ i = e N
localization
every
. If
~
follows
central
by construction
hence the
is a f i n i t e
used
Proposition
, ~ ~ i
x
~ G
M
by
c = i
, then
N
6 --+ N
h,> Gab /
= Np
by
M
on
(N i)
on
the
is c e n t r a l .
• Using =
i
trivially
induction
~ Qab
5.3
6 : N ~ Np
operates
proceed
the
. By Corollary
map
of
. Thus
P
Ni/Ni+ i
length
We m a y
c
consider
O
(7.4) / f,, Np
Since
Q
66
. Thus
= 0
coker
is o f o r d e r
f"
: Gab
commutative. let
(7.5)
and
[ , {'
the
F
:
Nc >
>
E
:
~o%-->
given i.e.
by
induction
that
f"h"
commutative, in t w o the
steps~
property
Qab
such
= ~"
~ \\
shall
that
the
the to
that
bottom
dividing
such
that
in c a s e
m
there
(7.2)
is
c = i
%--~> G/Nc\
M
various
first
G ~ Np
through
>> N / N c
>
. We have
we
order
assertion
N
that
in p a r t i c u l a r
factors
so t h a t
diagram
: Mc5
denote
f':
is P ' - t o r s i o n ,
N ~ Np
is o f
the
<\
, ~"
~:
hence
. Consider
H2Q
map
establishes
Fp
where
group
. Since
~ Np
This
c ~ 2
, the
localization
6 = im h, ~ G a b
exists
Now
the
m
localization right
most
find
f : G ~ M
the
(7.5)
maps.
triangle
fh = 6 - T h e
assert of
)> M / M e
Suppose
such
commutes.
of
is
is c o m m u t a t i v e ~ that
construction
existence
f"
(7.5)
of
f
f : G ~ M
Secondly
we
shall
is is
with use
±76 to c o n s t r u c t The m a p s
of
f : G ~ M
(7.5) w h i c h
such that
are a l r e a d y k n o w n
H 2 ( G / N c , N c) (7.6 )
~>
bottom
to a s s e r t
of
(7.5)
~
(G/N c , M c) ~
commutes
(7.6) we k n o w
f"*
(7.5)
we h a v e
commutes.
the c o m m u t a t i v e
diagram
H 2 ( M / M e ,M c) II
H 2(N/N c,M e ) <
the e x i s t e n c e
~(A[E])
(7.7)
yield
of
~h" *
H 2 ( N / N c , N c)
From
H2
h" * [
In o r d e r
the w h o l e
H 2 ( M / M e ,Me )
of a m a p
f : G ~ M
to v e r i f y
that
s u c h t h a t the
= f"*(A[Fp])
that
h"*~(~[E])
= ~h"*(~[E])
= ~"*(A[Fp]) = h"*f"*(~[Fp])
To p r o v e
(7.7)
it is thus c e r t a i n l y
morphic.
But since
N/N c
CoroRes
is m u l t i p l i c a t i o n m
is i n v e r t i b l e
follows (7.7)
that
Res
by in
m
= h"*
Consider
. Since
H2(G/Nc,Mc )
to s h o w that
commutative.
is of index
m
to s h o w that in
G/N c
h"*
is m o n o -
, we h a v e t h a t
: H2(G/Nc,Mc ) ~ H2(G/Nc,Mc )
and e s t a b l i s h i n g
It r e m a i n s
enough
M
c
and h e n c e
and
CoroRes
is m o n o m o r p h i c , the e x i s t e n c e there exists
of
H2(G/Nc,M
is P - l o c a l , It
is an i s o m o r p h i s m .
thus c o m p l e t i n g f : G ~ M
f : G ~ M
) c
the p r o o f of
.
such that
(7.5)
is
i77
Nc )
>
N
II
hl
Nc ~
>
M
~
M
~) M / M
)
map.
=
Thus
,q -iM
that ~ " ( x N c)
there
C
i i
Mc >
f h ( x ) -M c
G/N c
f"l
II
first
~
51 C
Recall
h" 1
G
~'I
(7.8)
D N/N c
)
M
f"h"
=
=
II Z __M/Mc
~"
: N/N c ~ M/Mc
g(x)-Mc
exists
y
where
~ M
. Let
g : N
~
x
M
is
y
~ Mc
e N the
. Then localization
with C
(7.9)
(fh(x))'y
Define
q
central, = qo~
: N q
~
is
M
6hat
~(x)
q(x)
=
homomorphic.
where
follows
by
=
q q
: M ~
M
vanishes
y
e M
Since
. Since M
is
. Clearly on
M
q
P-local, vanishes
. Define
f
and
q
factors
on
: G ~
Mc
Nc M
by
remains
to
is as
, whence
it
C
f(z)
It
is
(7.5)
easy is
to
see
=
f(z)-q(f(z))
that
f i.e.
commutative,
, z
e G
is h o m o m o r p h i c . that
fh
=
g
It . Thus
let
by
(7.9) ,
x
show
that
E N
, then
of
q
have
fh(x)
and
the
proof
is
=
fh(x) "qfh(x)
=
fh(x)
=
fh(x)"q£(x)
=
fh(x)-q(x)
=
~(x)
complete.
" q ( ~ (x) . y - i )
, by
, since , by (7.9)
the
y
£ Mc
definition
, ,
we
178 COROLLARY
7.3.
(Baer
is a P ' - n u m b e r .
PROOF:
Let
G ko / G k o n ZG follows sider Set
that
N : Z~G
(7.1o)
are
G/Z~G N
. We may
N
~
= Q
~
G
thus h
>
£I
just
f
the m a p
index
Hence
ker
at m o s t Q ko + l
k
f
of order
m
where
m
of P'-order.
. For
k = O
we know
is f i n i t e
generated.
, the by
and
assertion
induction [G~,Z
G]
It is t h u s e n o u g h
is
that = e
it
to c o n -
generated.
. Denote ---~ Q
construct
the o r d e r
. Theorem
of
Q
7.2 y i e l d s
by
m
. Con-
a homomorphism
the d i a g r a m
g ~> Q
G
given
by
>> Q
is f i n i t e we have
f
subgroup
in a f i n i t e l y
by a subgroup
P'-order.
finite
~[
the P ' - t o r s i o n
finite
be
G/ZG
finitely
N p ~--> N p ×Q
with
k
G/Z ~ G
is f i n i t e l y
the e x t e n s i o n
f : G ~ Np
on
Since
that
and
is f i n i t e
. Considering
o Gk+ I G
G/ZkG
induction
is f i n i t e .
groups
sider
by
k ~ I
Let
o Gk+ I
Then
We proceed
obvious.
[7])
if
g N
generated
of P'-order.
(Np) k° + l of
and
ker
= e f'
. Note
. But
N
that
kerf
, being
= ker
~ is
a subgroup
group,
is f i n i t e l y
Since
Np
of
generated.
is n i l p o t e n t
of class
, so t h a t
G k° + l
is an e x t e n s i o n
of
It f o l l o w s
that
G ko + I
of
is f i n i t e
179
BIBLIOGRAPHY
[i]
Andr6,
[2]
Atiyah,
[3]
B a b a k h a n i a n , A.: C o h o m o l o g i c a l D e k k e r Inc. 1972.
[4]
Bachmann,
[5]
Baer,
R. : E r w e i t e r u n g y o n G r u p p e n Z. 38 (1934) , 3 7 5 - 4 1 6 .
[6]
Baer,
R. : R e p r e s e n t a t i o n s o f g r o u p s as q u o t i e n t g r o u p s . I , I I , I I I , T r a n s . A m e r . Math. Soc. 52 (1945) . 2 9 5 - 3 ~ 7 , 3 4 8 - 3 8 9 , 3q0-419.
[7]
Baer,
R. : E n d l i c h k e i t s k r i t e r i e n 124 (1952), 1 6 1 - 1 7 7 .
[8]
Barr,
M.:
[9]
Barr,
M. , Beck, J. : S e m i n a r o n t r i p l e s a n d c a t e g o r i c a l h o m o l o g y t h e o r y . L e c t u r e N o t e s in M a t h e m a t i c s , V o l . 8 0 . S p r i n g e r 1969.
[10]
Baumslag,
G. : S o m e a s p e c t s o f g r o u p s 104 (1960), 2 1 7 - 3 0 3 .
[ii]
Baumslag,
G. : S o m e r e m a r k s o n n i l p o t e n t g r o u p s w i t h Amer. Math. Soc. 12 (1961) , 2 6 2 - 2 6 7 .
[12]
Baumslag,
G.: S o m e (1963),
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