Lecture Notes in Mathematics Editors: A. Dold, Heidelberg F. Takens, Groningen
1674
Springer Berlin Heidelberg New York Barcelona Budapest Hong Kong London Milan Paris Santa Clara Singapore Tokyo
G. Klaas C. R. Leedham-Green W. Plesken
Linear Pro-p-Groups
of Finite Width
Springer
Authors Gundel Klaas Wilhelm Plesken Lehrstuhl B fur Mathematik Templergraben 64 D-52062 Aachen, Germany e-mail:
[email protected] [email protected] Charles R. Leedham-Green Queen Mary and Westfield College University of London School of Mathematical Sciences Mile End Road London E 1 4NS, England e-mail:
[email protected] Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme
Leedham-Green, Charles Richard: Linear pro-p-groups of finite width / C. R. Leedham-Green; W. Plesken; G. Klaas. - Berlin; Heidelberg; New York; Barcelona; Budapest ; Hong Kong ; London ; Milan ; Paris ; Santa Qara ; r Singapore; Tokyo: Springer, 1997 (Lecture notes in mathematics; 1674) vat ISBN 3-540-63643-9
e
Mathematics Subject Classification (1991): 20G25, 20E18, llE95, 20Dl5 ISSN 0075- 8434 ISBN 3-540-63643-9 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and pennission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1997 Printed in Germany
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Preface
The present notes are the result of a research project supported by the DFG (Schwerpunkt: "Algorithmische Zahlentheorie und Algebra"), which has been going on for several years. In previous projects on classifications of p-groups according to coclass a complete structure theory has been developed. The study of pro-p-groups of finite width in these notes is a natural continuation of the coclass project. It revealed many new examples of pro-p-groups. The role of p-adic space groups has been replaced by open pro-p-subgroups of semisimple algebraic groups over the p-adic numbers. While preparing these notes we got help from many people, in particular program support from Colin Murgatroyd and Matthias Zumbroich. Lots of ideas were discussed with B. Souvignier, G. Nebe and R. Camina. The unknown referee, S. Sidki, and R. I. Grigorchuk helped us with valuable references and comments. We thank them all.
Aachen, June 1996
C. R. Leedham-Green, W. Plesken, G. Klaas
Contents
Introduction Width and just infinite pro-p-groups Ultimate periodicity and obliquity .. Four types of just infinite pro-p-groups Non-soluble p-adically simple groups . Contents and organisation of these notes
1 1
II
Elementary properties of width
9
III
p-adically simple groups (jj-groups) The Baker-Campbell-Hausdorff formula. . . . . . . . . . . . . . . . . . ii-groups and their Lie algebras . , . . . . . . . . . . . . . . . . . . . . ii-groups as open subgroups of groups of automorphisms of Lie algebras Obliquity and lattices of normal subgroups . . Uniqueness and existence of maximal ii-groups
12 12 14 16 18 19
IV
Periodicity
21
V
Chevalley groups
26
I a) b) c) d) e)
a) b) c) d) e)
VI
Some classical groups a) Basic structures: orders and Cayley maps. . . . . . b) Tables: Involutions, orders and lower central series. . c) Table of patterns for lower central series
3 4 5
6
30 30
38 53
VII
Some thin groups
55
VIII
Algorithms on fields Arithmetic in 0 . Calculating automorphisms in characteristic 0 The group of units of 0 . . . . . Fast arithmetic in characteristic 0
59 59
a) b) c) d) IX
Fields of small degree a) Extensions of «b of degree 2,3 and 4 b) Extensions of Q3 of degree 2, 3 and 4
X a) b) c) d) e)
Algorithm for finding a filtration and the obliquity The BASIS algorithm . . . . Split and Non-Split groups The groups Gli] . . . . , .. Calculating the obliquity . . Periodicity of the lower central series and obliquity
60 60 61
62
62 66 68 68 69 70
74 75
VIII
XI
a) b) c) d) e) f) g) h)
The theory behind the tables The relevant Qp-Lie algebras up to dimension 14 . Generators for the maximal p-adically simple groups. sln (K) . sU3(K Qp) . Sll (J(2( K)) . sOS(Qp)l split case. sOS(Qp)l non-split case 1
92((fJp)
XII Tables Dimension a) b) Dimension c) Dimension d) Dimension e) Dimension f) Dimension g) Dimension
.
78 78 79
80 81 85 87
89 91
92 3 . 6 8 . 9 . 10 12 14
93 94 96 97 99
99 . 105
XIII Uncountably many just infinite pro-p-groups of finite width a) The Nottingham group . . . . . . . . . . . b) Construction of uncountably many groups . . . . . . . . . . . . ..
106 . 106 . 106
XIV
108 . 108 .108
Some open problems a) Problems on general just infinite groups b) Problems on p-groups .
XV
References
109
XVI
Notation
113
XVII Index
115
I
Introduction
a)
Width and just infinite pro-p-groups
Let p be a prime. For a finite p-group P with lower central series I I (P)
:=
P
~
12(P)
:=
[P, P]
~
...
~
li(P) := [P, l i - l (P)]
~
...
define the width w (P) of P by
To study all finite p-groups whose width is bounded by some number b is a rather ambitious project. Here we attempt to approach the easier task of classifying certain infinite pro-p-groups P := lim Pi whose width w(P) := lim w(Pi ) is bounded by b +say. Though P is the inverse limit of a sequence of finite p-groups Pi, the idea of our approach is not to use this as a construction of P, but to construct and investigate P by utilising factor groups of P. As we shall soon see, classifying in this context can no longer mean enumerating the isomorphism types and developing recognition mechanisms, but rather supplying a sufficiently accurate description for one to be able to answer reasonable questions about the structure of these groups. Rather than with the width, we shall sometimes work with the average width, ultimate width or upper average width. (1.1) Definition. Let P be an infinite pro-p-group.
(i) P is said to be of finite width if the width w(P)
:=
suplOg p(l!i(P)/,i+l(P)I) i
is finite.
(ii) If P is of finite width, its average width wa(P) is defined as wa(P) := lim logp(IP/:~+I(P)I) ~-tOO
'/,
if the limit exists. If P is not of finite width or ijlogp(\P/ , i+l(P)\)/i converges to infinity, then wa(P) := 00.
(iii) The ultimate width w(P) of P is defined as
(iv) The upper average width wa(P) '/,s defined as
I Introduction
2
Note that all these variants of the width are infinite if P is not finitely generated as a pro-p-group. Whenever the average width W a (P) is defined for a pro-p-group P, it is equal to the upper average width wa(P). According to [Roz 96] the pro-2-completion P of the Grigorchuk group, cf. [Gri 80] satifies 1
log2(hn(P)/,n+l(P)I) =
3 2
{ 1
if n = 1 if 2m + 1 -:; n < 3· 2m - I + 1 if 3· 2m - I + 1 ~ n < 2m + 1 + 1
So it has width 3, upper average width 5/3, and its average width is undefined. Note that the infinite pro-p-groups P of finite coclass are by definition the groups of ultimate width w(P) equal to 1. Obviously their average width is also 1. Pro-pgroups of finite coclass are known to be soluble, cf. [Le.\T 80], [Don 87], [Sha 94]. At the present time it is not known whether the pro-p-groups of average width 1 are of finite coclass. (1.2) Lemma. If P is a pro-p-group] and N is a normal subgroup of P, then w(P) :?:: w(P/N), wa(P) :?:: wa(P/N), w(P) > w(P/N) and wa(P) :?:: wa(P/N), whenever these invariants are defined. This result is entirely trivial. Note that w(P) is always defined, and w(P) and wa(P) are defined if and only if P is infinite. As in the case of pro-p-groups of finite coclass it is natural to restrict oneself to the just infinite groups. (1.3) Definition. An infinite profinite group P is said to be just infinite, if P has no non-trivial (closed) normal subgroup of mfinite index] i.e. P has no proper infinite (topological) factor groups. It is easy to see, cf. Proposition (11.5), that any finitely generated infinite pro-pgroup has a just infinite homomorphic image.
The just infinite pro-p-groups playa similar role in the theory of pro-p-groups that is played by simple groups in the theory of finite groups. The similarity is quite marked. In both cases, most examples are of Lie type, with interesting exceptions. The soluble just infinite pro-p-groups are easily seen to be irreducible p-adic space groups. That is to say, groups P with a normal subgroup T, where T is a finitely generated free Zp-module, and PIT is a finite p-group that acts faithfully and irreducibly on T. To investigate them some of the machinery developed to study pro-p groups of the finite coclass can be used. Here we concentrate on insoluble just infinite pro-p-groups of finite width. Their average width can come arbitrarily close to 1, which might be unexpected because pro-p-groups of finite coclass are soluble.
(1.4) Theorem. Let b > 1 be a real number. Then there e.rists an infinite insoluble pro-p-group P of finite average width with 1 < wa(P) < b.
b) Ultimate periodicity and obliquity
3
This result is proved in Chapter VII. To prove the theorem with 'insoluble' replaced by 'soluble' is left as an exercise to the reader.
b)
Ultimate periodicity and obliquity
The second issue of these notes is the ultimate periodicity of the sequence (lri(P) : I'HI (P) l)iEN for insoluble just infinite pro-p-groups P of finite width. The only example known up to now not to have this property is the pro-2-completion of the Grigorchuk group, cr. [Roz 96], [Gri 80]. If the property holds with
for i 2: i o and some tEN, the average width is defined and equal to the arithmetic mean of logp(ll'i(P) : I't+I(P)I) for i = i o, ... , i o + t - 1. Again proving ultimate periodicity in the case of soluble pro-p-groups of bounded width is an easy exercise. In all cases investigated up to now, ultimate periodicity of the sequence (Irt(P) : I't+I(P)I)iEN is only part of the periodic pattern. One usually has for some t > 0 an isomorphism of the lattice of open normal subgroups of P contained in l'i(P) onto the lattice of open normal subgroups of P contained in I'Ht(P) mapping I'j(P) onto I'j+t(P) for j 2: i, cf. Chapter III. Closely related is the question of how restricted the lattice of normal subgroups is. More precisely, we want to measure how far one is removed from the most restrictive situation, where each normal subgroup N of P satisfies {HI (P) :::; N :::; 1'1 (P) for some 1: = i( N). The concept measuring this is called 'obliquity'. (1.5) Definition. Let P be a pro-p-group of finite width, and let i > O. Define f.li(P) to be the intersection of l'i+I(P) with the intersection of all normal subgroups N of P with N i l'i+l(P), The i th obliquity 0i is defined as Oi(P) := logp(II'i+I(P) : f.li(P)I). Define the obliquity o(P) of P to be max{ Oi(P) I ~ E N} if the max~mum exists; otherwise set o(P) = 00. It is easy to see that if P is a pro-p-group of finite width with a normal subgroup N #- (1) of infinite index in P then P is not of finite obliquity. Thus the obliquity of P can be seen as a numerical invariant that gives more precise information than the simple question of whether or not P is just infinite. However, we do not know whether or not every just infinite pro-p-group of finite width has finite obliquity. It may occur to the reader that it would be more natural to define the width of a pro-p-group in terms of arbitrary central sections, rather than lower central sections. This approach has theoretical advantages, but suffers from the difficulty that the width would then be harder to calculate. We shall see that if P has finite width (in our sense) and finite obliquity, then there is a uniform bound to the orders of the central sections of P, cr. Lemma (11.3). However, if P is a just infinite pro-p-group of finite width, and P is not finitely presented as a pro-p-group, then the p-covering group P is of finite width, but has an infinite centre.
I Introduction
4
(1.6) Lemma. Let P be a pro-p-group of finite width, and let N be a closed subgroup of P. Then o(P) ~ o(P/N). Proof. This follows from the fact that {ti(P/N) 2': {ti(P)N/N.
q.e.d.
Turning to finite p-groups, we have seen that for given p, 0 and w the p-groups of width ~ wand obliquity::; 0 form a quotient-closed class of groups. Let r p,w,o be the graph whose vertices are the isomorphism classes of the p-groups of width ~ wand obliquity ~ o. Confusing a group with its isomorphism class, if G E rp,w,o is of class c, join G to G / IC( G). Then every just infinite pro-p-group of width at most wand obliquity at most 0 is the inverse limit of an infinite chain in rp,w,o. Thus we might hope to start a classification of finite p-groups by constructing all just infinite pro-p-groups of finite width, and taking the finite homomorphic images of these groups as our prime source of examples. Unfortunately, as we shall see, there are uncountably many pro-p-groups in this class. However, if we also bound the subgroup rank of the finite groups, we then find that there are only finitely many pro-p-groups to deal with (for fixed p, and with fixed bounds to the width, obliquity and subgroup rank of the finite p-groups in the graph). Moreover, we need to bound all three invariants, width, obliquity and subgroup rank, since if we do not bound the obliquity we will get pro-p-groups that are not just infinite as inverse limits of our finite p-groups, and if we remove the restriction on the width or the subgroup rank, we get infinitely many pro-p-groups. However, we do not wish to exclude groups of infinite subgroup rank completely from our considerations as they are of great interest.
c)
Four types of just infinite pro-p-groups
The insoluble pro-p-groups of finite width which are most accessible are p-adic analytic, cf. [LuM 87b], [DdMS 91]. For ease, we propose the following definition.
(I. 7) Definition. A p-adically simple group, for short a p-group, is a p-adic analytic just infinite pro-p-group. Every jJ-group is of finite width. One expects plenty of other just infinite pro-pgroups of infinite width. Determining the width might be rather difficult, e. g. it seems that the width of the p-completion of the Gupta-Sidki groups (cf. [Sid 84]) are not yet determined. Since these are p-analogs of the Grigorchuk group, the answer would be particularly interesting. In these notes we are mainly concerned with insoluble p-groups. One can distinguish four types of just infinite pro-p-groups of finite width: the soluble ones; the insoluble p-groups; those that are not p-adic analytic, but are linear over JFp((t)); and those that are non-linear, i.e. the rest, which are not linear over Qp or JFp(( t)). A comment on each type seems to be appropriate. The first two types are both jJ-groups, however the difference between the two is quite marked. The soluble ones are irreducible p-adic space groups. Essentially they can be investigated by the methods developed to study pro-p-groups of finite coclass. In fact, Conjecture C of [LeN 80], which has since been proved, cf. [Don 87], [Lee 94a], [Sha 94]' says: every pro-p-group of finite coclass is soluble. The insoluble p-groups
5
d) Non-soluble p-adically simple groups
are therefore a new class of groups to be investigated and they form the main topic of this work. The third type, e.g. Sylow pro-p-subgroups of Chevalley groups over JFp((t)), can often be treated at the same time as insoluble jJ-groups. For example when one computes lower central series. However, this class of groups has many unpleasant properties. They may be isomorphic to proper subgroups of themselves, they may have p-groups of outer automorphisms of arbitrarily big orders, they give rise to uncountably many just infinite pro-p-groups of finite width, cf. Chapter XIII. Also they are not easily investigated by computers, because it is not clear when one has investigated a sufficiently large finite factor group to predict properties of the infinite group e.g. the sections of the lower central series. The attitude taken in these notes is to treat them simultaneously with the insoluble f)-groups, whenever no extra effort is necessary. Finally, as for the fourth type, i.e. the non-linear groups, only a few classes of examples are known, namely the Nottingham groups and the p-completion of the Grigorchuk group, cf. [Gri 80], [Roz 96], and their open subgroups. A Nottingham group S(q) is the Sylow pro-p-subgroup of the group of JFq-algebra automorphisms of lFq [[t]], where q is a power of p, cf. Chapter XIII a). Our perception of the general structure of pro-p-groups of finite width has been radically changed by the recently discovered fact that S(p), for p any prime, contains as a closed subgroup an isomorphic copy of every countably based pro-p-group. See [Cam 97]. It seems far more hopeful to try to deSCribe all just infinite pro-p groups P with the property that every open subgroup of P is just infinite. Call these groups 'hereditary just infinite'. The known hereditary just infinite groups of finite width are the just infinite groups described above where the associated Lie algebras (for the second type cf. Chapter III) are simple, and just infinite groups commesurate with the Nottingham group. (Note that G and Hare commesurable if they have isomorphic open subgroups.)
d)
Non-soluble p-adically simple groups
Our main emphasis is on the groups of type 2; that is, on the insoluble just infinite p-adic analytic pro-p-groups of finite width, or insoluble jJ-groups for short. These H if occur in families, where a family is an equivalence class under the relation G G and H have open isomorphic subgroups. Then it turns out that the relation '~', defined by H ~ G ¢:} H is isomorphic to an open subgroup of G defines a partial order on each family, and that every family has a unique maximal group. It is these maximal groups that we exhibit, in various cases. I"V
The reason why insoluble jJ-groups P are accessible in a rather explicit way is that they are open subgroups of the group of Qp-rational points of certain semisimple algebraic groups G defined over the field Qp of p-adic numbers. In the p-adic topology P is necessarily compact and the topology induced from the Qp-topology of G(Qp) is the same as the original profinite topology of P. More precisely, it is possible to
I Introduction
6
attach a Qp-Lie algebra to P, which is the direct sum of pQ copies of a simple Qp-Lie algebra for some Q E Z>o (with Q = 0 as the most interesting case). The algebraic group G is the automorphism group of this Lie algebra. Therefore the classification of simple Qp-Lie algebras and simple algebraic groups over Qp, d. [Kne 65], [Sat 71], or [BrT 72], [BrT 84], [BrT 87] can be used. The last three references also cover the simple algebraic groups over local fields of positive characteristic which we sometimes also treat, cf. Chapters V and VI. Even if the simple Lie algebra is given, it is not always a trivial exercise to find the maximal p-group associated with it, which is a Sylow pro-p-subgroup of G(Qp). In practice, the problem of passing from a Sylow pro-p-subgroup of GO(Qp) to one of G(Qp) is not always routine. Here GO denotes the connected I-component of G in the sense of algebraic groups.
e)
Contents and organisation of these notes
Chapters II to VII and Chapter XIII are purely theoretical investigations into just infinite pro-p-groups of finite width. Chapters VIII to XII deal with algorithms, showing how to treat these groups with a computer. Where it was impossible to proceed theoretically, the tables of results of computer calculations have been given in Chapter XII. Finally, Chapter XIV lists some open problems. Chapter II concerns generalities on pro-p-groups of finite width, which are easy and do not use the distinction between the four types of groups discussed in c) above. Chapter III proves all the general results on p-groups indicated in d) above, in particular the interplay between simple Qp-Lie algebras and insoluble p-groups and the existence and uniqueness of maximal p~groups. It also discusses the obliquity of fr groups. Chapter IV deals with the question of which factor groups of a j)-group are sufficiently large to determine the pattern of the lower central series. The results are modelled after Blackburn's Theorems for the case U1,12, ... ) = (2,1,2,1, ...) where Ii = 10gp(lri(P)/'i+l(P) I), d. [Bla 61] reproduced in [Hup 67] p. 392. The next three Chapters, V to VII, deal with specific groups for which the lower central series is computed. Though the main intention is to deal with maximal fr groups, it turns out that the analogues in characteristic p can be treated at the same time. Chapter V deals with the Sylow pro-p-subgroups of (split) Chevalley groups over a local field. For most cases the lower central series is determined. Chapter VI deals with the Sylow pro-p-subgroups of the classical groups over a local field where the characteristic of the residue class field is not 2. It turns out that the Cayley map can be used to compute the lower central series thus avoiding the machinery developed in [BrT 72], [BrT 84], and [BrT 87]. Instead one has to deal with algebras with involution and one can assign a canonical hereditary order to the Sylow pro-p-subgroup, which is invariant under the involution. This order allows one to read off the lower central series. The method fails only for unitary groups of degree 3 over a local field whose residue class field has characteristic 3. As an interesting by-product one gets a formula similar to the Baker-Campbell-Hausdorff formula for classical groups which uses associative polynomials in the algebra with involution rather than Lie polynomials. It might be that this formula can also be used in the case when the characteristic of
e) Contents and organisation of these notes
7
the residue class field is 2. Finally, Chapter VII deals with the Sylow pro-p-subgroup of PC L(pQ, Qp), a case which is not covered by Chapter V on Chevalley groups, since so called rational automorphisms have to be taken into account. In any case, these rational automorphisms lead to groups with a rather small average width. Then Theorem (1.4) follows from the investigations of Chapter VII. The aim of Chapters VIII to XII is to investigate the lower central series and the obliquity of all insoluble, maximal p..groups whose associated Lie algebras have dimension ~ 14 mainly for primes p = 2,3. It is in these cases that it often becomes difficult to decide whether a given p-group is maximal or not, cf. Chapter XI. For p ~ 5 the lower central series can be investigated theoretically as described in Chapters V and VI, since up to dimension 14 there are no additional p-automorphisms on top of the Sylow pro-p-subgroups of CO(Qp) with C and Co as at the end of Section d) above. For p = 2,3 the groups considered in the theoretical chapters are often not maximal p..groups. The task of deciding maximality, respectively of finding the maximal p-groups, has obviously both a theoretical and a computational aspect. In making these groups explicit, we have faced a number of difficulties. First, we need to know the local fields of characteristic 0 and their automorphism groups in small cases. Although obtaining this information is a routine computation, we have not been able to find these fields in the literature, and have included them here in Chapter IX. Secondly, it is not clear how best to make these groups explicit. They are generated, topologically, by a finite number of matrices, together, perhaps, with some Galois automorphisms, etc. These generators are the starting point of our calculations. However, they require a fair amount of space on the page, and are not, in themselves, very informative. We could give a power-commutator presentation for a large quotient of each group. This gives a good insight into the structure of the groups, but is prohibitively expensive in space. A third possibility would be to make generally available the computational tools that we have used. Here we have two problems. First, the code has developed in an ad hoc way as the project has advanced. We are very grateful to Colin Murgatroyd and Matthias Zumbroich who wrote a great deal of it. The program, particularly the local field arithmetic, is subtle, complicated and extensive, but it works very efficiently, and appears to be completely reliable. However, it is not entirely portable, and experience has suggested improvements that could be made. We intend to get a publically usable version written; in the meantime, we will be glad to run any examples sent to us. In any case, the routines for doing the field arithmetic fast, which is absolutely necessary even for the examples with a Lie algebra of small dimension, are described in Chapter VIII. Possibly these algorithms are of independent interest. The algorithms which use the field arithmetic (characteristic o or p) to compute a power commutator presentation for suitable factor groups of a given group P are described in Chapter X. Chapter XI gives enough details to obtain generators for the maximal insoluble p.. groups. Applying the algorithms to a suitable factor group yields a power-commutator presentation of the factor group. Using these presentations for factor groups of a suf-
8
I Introduction
ficient size (such that the factor specifies the structure of the pro-p-group P) one obtains l by calculations in GAP [GAP 94L the lower central series and the obliquity of P. In particular l Chapter XII lists the tables of all insoluble maximal p..groups for p = 21 3 where the dimension of the associated Qp-Lie algebra is at most 14. The tables include the pattern of the lower central series and information about the obliquitYl i.e. the obliquity or at least the ultimate obliquity. The results show that the task is impossible to master without a computer. For instance the pattern of the lower central series often becomes periodic only after a long preperiod. We hope that these tables not only provide groups that are interesting in their own right but can also be used to investigate general insoluble p..groups for small dimensions.
II
Elementary properties of width
It is clear that the property of having finite width is closed under the formation of finite direct products, and finite p-extensions. For just infinite insoluble pro-p-groups it is probably also closed under passage to an open subgroup, however has not been proved. The best result we have for passage to open subgroups is the following.
(11.1) Proposition. Let P be a just infinite) insoluble pro-p-group of finite upper average width and let Q < P be open; then Q is of finite upper average width. Proof. Without loss of generality let IP : QI = p. Clearly li(Q) ::; li(P), Now Ii (Q) is characteristic in Q, and Q is normal in P, so Ii (Q) is normal in P. Therefore IP : 'i(Q)1 < 00 or li(Q) = (1). The last possibility is ruled out because Q is insoluble. Hence li(Q)/Ii+I(Q) is finite for each i. The exponent of li(Q)/li+I(Q) is bounded by the exponent pO: of QI12(Q). Let k be minimal with A k = 0 where A is the augmentation ideal of the group ring (ZlpO:Z)Cp • Then the k-fold commutator group of li(Q) with P satisfies bi(Q),k P] ~ Ii+I (Q) where, for any group G and H ~ G, we define [H'I G] := [H, G] and [H'i+I G] := [[H'i G], G] for i ;:: l. We now prove, by induction on i, that Iki(P) ~ li(Q)· Clearly Ik(P) ~ 12(P) ~ Q = 11 (Q) since PIQ is abelian and k ;:: 2. Assume the claim holds for i. Then li+I (Q) ;:: bi(Q),k P ];:: bki(P),kP] = Ik(i+I)(P), which proves the induction hypothesis. So
wa(Q)
= lim 10gp(IQ: :"i+I(Q) I) t~OO
::; lim 10gp(IP: l.k(i+O(P) I)
~
l~OO
= kwa(P).
~
q.e.d.
(11.2) Lemma. Let P be an infinite, finitely generated pro-p-group and let Q be an open subgroup of P. Then wa(Q) ;:: wa(P). Proof. Since Q ::; P, one has li(Q) logp IQ: Ii+I(Q)I
~
li(P) and therefore
+ logp IP: li+I(P) I + logp hi+I(P) : Ii+I(Q)1 > -logp IP : QI + logp IP : li+I(P)I· -logp IP: QI
Dividing by i and taking upper limits gives the result.
q.e.d.
There does not seem to be a version of (II. 1) for the width instead of the upper average width. However, the following holds.
(11.3) Lemma. Let P be an infinite pro-p-group of finite width w(P) and finite obliquity o( P). Then there is a kEN such that any central section MIN of P with M and N open normal subgroups of P and N ~ M satisfies
1M : NI
~ pk.
Proof. Let M, N ::] P be open with N ~ M and MIN centralised by P. Since o(P) is finite there is atE N independent of M such that the following holds. There
II Elementary properties of width
10
exists i E N with IHt(P) ::; M :S li(P), Since MIN is centralised by P, one has li+t+I(P) :S N::; IH1(P), Hence IMINI ::; p(t+l)W(P). q.e.d. (11.4) Corollary. Let P be an infinite, insoluble pro-p-group of finite width and finite obliquity. Then any open subgroup of P has finite width. Proof. By Lemma (11.3) the central sections MIN of P are of order bounded by pk for some k. Let Q ~ P be a normal open subgroup of P. Since P is of finite obliquity P is just infinite. Therefore because P is insoluble there is an 0: E N such that the exponent of QI12(Q) is i'Je. Let t be minimal with At = 0, where A is the augmentation ideal of the group ring of PIQ over 7!..lpo.7!... As in the proof of (11.1) one sees that the order of any central section of Q is bounded by pkt. Finally, assume that Q is any open subgroup of P. The core Ql of Q is also open in P. So one may assume that any central section of Ql has order bounded by pU for some u E N. Then any central section of Q is bounded by IQ : Qll pu. q.e.d. (11.5) Proposition. Let P be a finitely generated infinite pro-p-group. Then P maps onto a just infinite pro-p-group . Proof. If P is soluble then P maps onto 7!..p which is just infinite. Hence assume P is insoluble. Let (No.)o.EN be an ascending chain of closed normal subgroups of P of infinite index. The lower p-series of some group G is given by Aj+I(G) = [Aj(G),G]GP with Al(G) = G. Define Po. = PINo.. Since P is finitely generated the sequence (aj) jEN := IAi (Pj ) : Ai+ 1 (Pj ) I consists of fini te numbers and is constant for j ~ j(i) with certain j(l) :S ... j(i - 1) ::; j(i) :S ... for i E N. Therefore Pj(k)/ AHl(Pj(k)) ~ Pj(k)+nl Ak+l (Pj(k)+n) for all n ~ 0 and kEN. Since one has continuous epimorphisms Pj(i)/Ai+l(Pj(i)) ~ Pj(i-I)/Ai(Pj(i-l)) one can define H = lim Pj(i)/ AHI (Pj(i)) which is an infinite pro-p group. Since the epimorphisms
+--
11
case is ruled out since by an argument of Schur (d. [Hup 67] Kapitel V Hilfsatz 23.3 b)) Z(G) n [G, G] is a finite normal subgroup of G and hence is trivial. The commutator subgroup [G, G] is of finite index in G and therefore one has a contradiction to G being an infinite group. So [G, Hi] is open in G. Since N I is normal in G x ... x G, it follows that the subgroup of G x ... x G consisting of elements whose i th entry lies in [G, Hi], and whose other entries are the identity, lies in N. Since N is also normalised by a wreathing element, it follows that N contains the direct product of p copies of [G, Hi], and hence is open in G l Cpo q.e.d. (II. 7) Lemma. Let G be a pro-p-group with the property that G /i2( G) is of exponent p. Let i = ps + t where 1 ~ t ~ p. Let W = G l Cpo Then ii(W)/ii+I(W) :: is+I(G)/is+2(G) if i > 1, and W/i2(W) ~ (G/i2(G)) x Cpo Proof. Let B be the base of the wreath product, so that B is the direct product of p copies of G. For j ~ 0, ij(B) is the direct product of p copies of ij(G), and ij(B)/iHI(B) is naturally isomorphic to ij(G)/"iHI(G)0rlp Cpo Let B i be the inverse image in B of is+I (G) /is+2( G) 0lt-I, where I is the augmentation ideal of lFpCp. Note that lk / lk+ I is of order p for 0 ~ k ~ p - 1, and that lP = O. It is easy to see that, for i ~ 2, Bi = ii(W), For if i = 2 then s = 0 and t = 2, and the result holds in this case, and the general case follows by an easy induction on i. The lemma follows. q.e.d. If G/ i2( G) is of exponent greater than p the situation is somewhat more complicated. When we meet such cases, we work out the lower central series of the wreath product on an ad hoc basis. Note that the Lemma (II. 7) implies that the various widths of Ware essentially the same as for G.
III
p-adically simple groups (fi-groups)
The fundamental tool for studying these groups is their interconnection with Lie algebras over local fields of characteristic O.
a)
The Baker-Campbell-Hausdorff formula
We thank Bernd Souvignier, cf. [Sou 96], for having worked out the details of this section. To construct groups from Lie algebras, the Baker-Campbell-Hausdorff formula will be used. For some estimations on the p-adic value of its coefficients the following lemma is needed. (IIL1) Lemma. For m, n E N, n following hold
(i) lJp(m! . n!) (ii) lJp(n!)
~
= ;=~
= Ef=o aipi
with 0 ~ ai < P and a =
E:=o ai
the
lJp((m + n)!), (in particular lJp(n!) ~
;::::U.
(:t:t
Proof. (i) Since = (m~n) E N, one has lJp((m + n)!) - lJp(m! . n!) ~ O. (ii) The number of a E {l, ... , n} satisfying lJp(a) 2:: i is l np-i J. Therefore the number of a satisfying lJp(a) = i is lnp-iJ -lnp-i-IJ. Furthermore lnp-iJ = Ej:;;;tai+jpJ and it follows that lJp(n!) = Ef=l i( lnp-i J-l np-i-l J) = Ef=ll np-z J = Ef=l Ej:t ai+jpJ = Ek=l ak E7~l pI = E k= I ak (pk - 1)/ (p - 1) = Ek=o ak (pk - 1)/ (p - 1) = (n - a) / (p - 1). q.e.d. Let £ be a Lie algebra of finite dimension over a local field of characteristic O. For x, y E £ the Baker-Campbell-Hausdorff formula is formally defined as
(x, y) = log(exp(x) . exp(y)). This is a formal definition in the following sense. The definitions of log and exp are taken to be the usual power series definitions. For this to make any sense at all, one needs to work in an associative algebra; of course any Lie algebra can be embedded in an associative Lie algebra, where the Lie product is defined by [x, y] = xy - yx. There is also the question of convergence to be considered. By expressing exp and log as power series one has (cf. [Jac 62])
where [X pP yql, ... ,X pm ,Yqm] means rx, ... ,x,y, ... ,y, ... ,x, ... ,x,y, ... ,yl. In this \
.
PI
. ---....qj
---.....-... ~ Pm qm
expression one has eliminated the associative multiplication, but the problem of convergence remains. Writing (x, y) = En~l un(x, y), where un(x, y) is the sum of
13
a) The Baker-Campbell-Hausdorff formula
homogeneous terms of degree n in x, y, yields n
un(x,y) =
L m=l
In particular, UI(X,y)
Pt
lQ l
P, +q, >0 E(p,+q,l"'n
= x + y,
U2(X,y) = (1/2)[x,y], U3(X,y) (1/12)[x, y, y], U4(X, y) = - (1/48)[x, y, x, y} - (1/48) [x, y, y, x].
=
-(1/12)[x,y,xl
+
The following table gives, for small n, the least common multiple of the denominators appearing in un(x, y); this will be needed for the proof of the next lemma.
n
7
lem
2(i·3 3·5·7
8 29.33,5. 7
9
28.3 5,5 2.7
10 29.3 4.5 3.7
11 29.3 5.5 2.7,11
12 21:2.3 5. 52. 7·11
(IlL2) Lemma. Let L be a Lie lattice satisfying [L, L] C;;; pL if p =I: 2, and [L, L] ~ 4L if p = 2. Then un(x, y) E L for all x, y ELand all n 2: 1. The series (Ufl,(x, y)) converges to 0 E L as n ----7 00. Proof. From [L, L] ~ pL respectively [L, L] C;;; 4L it follows for the commutator of length s that [L, . .. ,L] ~ p.. -I L resp. S;;; 22. . - 2 L. So it is enough to prove for the coefficients cn == (_l)m~l(mn. PI!q1~" .Pm~qm!)-l in un(x,y) that lJp(C;;:-I) :::; n-1 respectively vAe;;:-l) ~ 2(n -1). Clearly by Lemma (IIL1), Vp(C;;I) :S lJp(n!) + lJp(n) + lJp(m) :::; (n ~ l)(p _1)-1 + 2logp (n). (i) p 2: 5: Since lJp(m) S; lJp(nt) for m:::; n, one has lJp(C~l) :::; 3lJp(n!) :S 3(p-l)-1(n1) :::; (p - 2)(p - l)-I(n - 1). Therefore un(x, y) E p(n-I)/(p-I} L. (ii) p = 3: For n ~ 13 it follows that n 5 :S :in-I, and therefore 10g3(n) :S 5- I (n - 1). One has lJ3(C~1) ~ lJ3(n~) + 2lJ3(n) :S (9/1O)(n - 1), hence u 71 (x, y) lies in 3(n~ll!10 L. For n ~ 12 one checks the claim by using the above table. (iii) p == 2: For n 2: 13 one has n 3 :S 2n- I . Then 10g2(n) :S 3- 1 (n - 1) and lJ2( c;;I) :S 10g2 (n!) + 2log2 ( n) S; (5/3)(n ~ 1). Hence one has Un (x, y) E 2(n- Il!3 L. For n :S 12 one checks the claim by using the above table. q.e.d. This lemma has the following consequence. (III.3) Proposition. For a Lie lattice L satisfying the above conditions (x , y) is well defined for all x, y ELand defining xy = (x, y) makes L into a group. Proof, The group axioms follow at once from the formal properties of log and expo q.e.d. Note that the zero element of L becomes the identity element in this group, and that the inverse of an element x is -x.
III p-adically simple groups (ij-groups)
14
To express group commutators in terms of commutators in the Lie lattice one uses the commutator Baker-Campbell-Hausdorff formula defined formally by
\l1(x, y)
= log(exp( ~x) exp( ~y) exp(x) exp(y)).
By the method used in [Jac 62] one obtains a formula for \l1(x, y), which is very similar to that of
Proof. Again this follows from the formal properties of log and expo
b)
q.e.d.
p..groups and their Lie algebras
A pro-p--group H is said to be uniformly powerful (or uniform) if it is powerful and satisfies the condition IAi(H) / Ai+l (H) I = IH/ A2(H) I < 00, where At(H) denotes the ith term of the lower p--series of P. Equivalently, H is uniform if H is finitely generated, powerful and torsion free (ef. [DdMS 91] Theorem 4.8.). Using this criterion it is easy to see that the Lie group constructed from a suitable Lie lattice, as in the previous section, is uniform. It is well known that any p--adic analytic group P has an open uniform pro-p-subgroup H, ef. [DdMS 91]. H can be assigned a Lie algebra L(H) over Zp in such a way that the assignment P to [,(P) := Qp @ L(H) defines a functor setting up an equivalence between the category of analytic groups with germs of analytic homomorphisms at 1 as morphisms and the category of Qp-Lie algebras. The Zp-Lie algebra L(H), henceforth referred to as the Lie lattice of H, has H itself as the underlying set and carries the following operations expressed in terms of the group operation of H (ef. [Laz 65]' or [DdMS 91] part 1). Denote the commutator in the group by [ , ]. Define an addition on H by
and a Lie bracket by
[g, h]L = n-too hm [gP ,hP ]P .
n
n
-2n
for all g, h E H. The action of Zp is obtained in an obvious way by the powering operation of Z. Note that the conjugation action of H on L(H) is an action by Lie algebra automorphisms. If H is a uniform pro-p--group, we have defined a Zp- Lie algebra L( H) with H as its underlying set, and it is easy to see that [L(H), L(H)] ~ pL(H) for p odd, and [L(H), L(H)] ~ 4L(H) for p = 2. It follows that we can then define a group operation on L(H) using the Baker-Campbell-Hausdorff formula. One can check that this gives the original group H, cf. [DdMS 91] Corollary 8.16. Similarly, one can start with a
b) p-groups and their Lie algebras
15
Lie lattice L, satisfying [L, L] ~ pL for p odd, or [L, L] ~ 4L for p = 2, form the corresponding group using the Baker-Campbell-Hausdorff formula, and then construct the corresponding Lie algebra. This gives back the original Lie algebra. Clearly the assignment H -----* L( H) defines an equivalence between the category of uniform pro-p-groups, and the category of finite dimensional Lie lattices over Zp. This in turn defines a functor £ from the category of finitely generated p-adic analytic groups to the category of finite dimensional Lie algebras over lQp taking P to £(P) = L(H) @ Qp where H is an open uniform subgroup of P. It is easy to see that this is a well defined functor, and that the image under the functor of a homomorphism between two finitely generated pro-p-groups depends only on its restriction to an open uniform subgroup. We now investigate the elementary properties of this functor. Clearly, if HI is a uniform subgroup of the uniform group H then L(Hd is a subalgebra of L(H), and if H 1 <J H then L(Hd <J L(H). In the reverse direction, the result is slightly weaker. (IlL5) Lemma. Let H be a uniform p-adic analytic group and X a Lie sublattice of L(H) satisfying [X, X]L ~ pX if p is odd, and [X, X]L ~ 4X if p = 2. Then X, regarded as a subset of H, is a subgroup of H. If, m addition, [L(H), X]L ~ pX ifp is odd, and [L(H),X]L ~ 4X ifp= 2, then X, regarded as a subset of H, is a normal subgroup of H. Proof. This follows from the Baker-Campbell-Hausdorff formula, and the commutator Baker-Campbell-Hausdorff formula. q.e.d. (III.6) Proposition. Let P be an insoluble p-group. Then
(i) £(P) is semisimple with isomorphic components) the number of which is a power of p and (ii) P acts faithfully on £(P).
Proof. (i) Let I be a characteristic ideal of £(P) and let H be an open uniform subgroup of P. Then I n L( H) is an ideal, J say, in L( H), and if X = p2 J, then X is a normal subgroup of H by Lemma (IlL5). Since X is P-invariant, it is a normal subgroup of P. But X is not open in P, since I is a proper ideal of £(P), so X = (1) and I = O. P permutes the simple components of £(P) transitively, since otherwise an orbit of the components under P gives rise to a normal closed subgroup of infinite index. As there is only a finite number of components it has to be a power of p. (ii) Assume the action of P has a non-trivial kernel. Since P is just infinite it follows that only a finite factor group of P acts non-trivially. Let U be a uniform normal subgroup contained in the kernel of this action. The group P acts trivially by conjugation on the Lie lattice L(U) and therefore it follows that any g E P and u E U commute. It follows that U is an abelian open subgroup of P, this contradicts P being insoluble. q.e.d.
16
III p-adically simple groups (p-groups)
Therefore an insoluble p-group P will be viewed as a subgroup of the automorphism group Aut(£(P)) of its Lie algebra £(P).
c)
j)-groups as open subgroups of groups of automorphisms of Lie algebras
Given a semisimple p-adic Lie algebra £ all derivations are inner, i.e. of the form adl : £ -+ £ : ,X r l [I, ,X] for I E £ , cf. [Zas 39] Satz 15 or 16, or [Jac 62] p. 73. Let L c £ be a Lie lattice satisfying [L, L] c p2 L. Define e(l) = exp(adl) for any ad-pronilpotent element IE L. Since (adltL C p21LL and /lp(n!) ~ (n -1)/(p -1) it follows that (l/n!)(adl)n L C L for alII ELand n E N. So, for I E L, e(l) lies in Aut(£). One uses the Baker-Campbell-Hausdorff formula to see that e(L) is a subgroup, cf. (IlI.3). Clearly e(L) is a pro-p-subgroup of Aut(£) (e(L) lies in the centraliser of L/pL) which is closed in Aut(£) because it is the image of a compactum under a continuous map. The aim is now to construct a jj-group for a given Lie algebra. This will be achieved in Proposition (III. 9). (III. 7) Lemma. Let £ be a simple finite dimensional Qp -Lie algebra. Any open pro-p-subgroup P of Aut(£) is a p-adic analytic group. Proof. Let n be the dimension of £. The automorphism group Aut(£) is a closed subgroup of GL(£) ~ GLn(Qp) which is p--adic analytic. Therefore, by [DdMS 91] Theorem 10.7, Aut(£) is p-adic analytic itself. Since P is open in Aut(£) and hence closed the assumption follows by the same argument. q.e.d. There is another way of constructing a Lie algebra of a p-adic analytic group P. If p is an odd prime, let H be an open uniform subgroup of P. If p is even, let H be the Frattini subgroup of au open uniform subgroup of P. The group algebra 7l.p H can be completed in such a way that a logarithm map log is defined, cf. [DdMS 91] part 2. Set A( H) := log( H). Then A( H) can be given the structure of a 7l.p -Lie algebra by defining a Lie bracket employing the product in the group algebra [x, y] := xy - yx and A(H) is closed under this operation. By using the Baker-Campbell-Hausdorff formula one can check that the map log : L(H) -+ A( H) : g 1---+ log(g) is a 7l. p - Lie algebra isomorphism. (IlI.8) Lemma. Let £ be a semisimple Qp-Lie algebra and L a julliattice in £ with [L, L] C p2 L. Then A(e(L)) is isomorphic to L via I rllogoe(l). Proof. Clearly L is an e(L) - module where the operation is defined by e(l)('x) = ead(!)(,X) = 2::~",=o ~!(adl)n(,X) for I E L. The lattice L is also a A(e(L))-module with the operation defined by log(e(l))('x) = 2:::'=1 ~l~n+J (1 - eadl)n(,X) = adl('x) = [I, ,X]. This is well defined because (1- ead!)n lies in the group algebra 7l. p[e(L)] and is interpreted as acting on the full p-adic lattice L. It follows that the map log oe is an isomorphism of Lie lattices because the action of L defined by this map is faithful. q.e.d. (III.9) Proposition. Let £ be a simple finite dimensional Qp-Lie algebra.
c) fj-groups as open subgroups of groups of automorphisms of Lie algebras
(i) Any open pro-p-subgroup P of Aut(£) is a fj-group and £(P)
'::::!
17
£.
(ii) Let W be an iterated wreath product of a copies of cyclic groups Cp of order p. An open pro-p-subgroup P of Aut(£) 1W is a fj-group, if and only if it permutes the pQ copies of £ transitively. Proof. (i) Applying Lemma (III.7) it remains to prove that there is no closed normal subgroup of infinite index in P. Choose a Lie lattice L c £ and a uniform open subgroup K of P such that e : L ~ K : I r l eadl is well defined and injective. Furthermore choose Land K in such a way that all open normal subgroups of K are uniform. Then e(L) is normal in K because ke(l)k- I (>.) = k(ead1k- I (>.)) = e(k(l))(>') for all >., I E L, k E K. The Lie pn lattice L(e(L)) is an ideal in L(K) because [e(l), k] = limn~(X)(e(l)pn, k ) E e(L). Hence the isomorphism of the two constructions of Lie algebras respects subgroups therefore L ~ A(e(L)) <J A(K) and consequently £ <J £(P). Since £ is simple there are no outer derivations. Thus £( P) ~ £ EB X decomposed as Lie algebras. It remains to show that X = {O}. Choose K and L such that A(K) = L EB X as Lie lattices. The action of K on L is faithful. The element e( x) E K for x E X acts trivially on L. Therefore X = {O}. (ii) To prove that P is just infinite under the assumed condition, see the proof of the first claim of Lemma (IL6). Conversely, let L(U) be a Lie lattice associated to P which decomposes into EBiCi . A non-transitive action of P on L(U) yields an orbit of infinite index in L(U) and therefore corresponds to a closed normal subgroup in P of q.e.d. infinite index which yields a contradiction to P being just infinite. (IlLI0) Corollary. An insolublefj-group P is an open subgroup of Aut(£)llV, where £ is a simple Lie algebra over Qp and W is an iterated wreath product of a copies of cyclic groups Cp of order p such that £(P) ~ EB P" £. Proof. It follows from (IlL6) that P is a closed subgroup of Aut(£(P)). Denote the dimension of £(P) by d. Let C be an open subgroup of Aut(£(P)) which is a fj-group. Take PI to be an open uniform subgroup of P and C I to be an open uniform subgroup of C. Then L(PI) and L( CI) are both full Lie lattices in £(P) of dimension d and hence L(PI) n L(CI) is a full Lie lattice in £(P). Therefore PI n C 1 is of finite index in C and P is open in Aut(£(P)). The rest of the claim follows from Proposition (IlL6). q.e.d. (IlLl1) Lemma. An insoluble fj-group P is not isomorphic to one of its proper subgroups. Proof. Assume PI to be a subgroup of P isomorphic to P. Let H be a uniform normal subgroup of P and HI the subgroup of PI corresponding to H under the isomorphism from P to Pl' The Lie lattices L(H) and L(HI ) are isomorphic and L(HI) is contained in L(H). The Killing form on L(H) and L(HI) is not degenerate and has the same discriminant on both lattices. Therefore these Lie lattices are equal. It follows that the subgroups H and HI are equal because they are equal as sets. The finite factor groups P/ H and PI! H are of the same order, so PI = P. q.e.d.
III p-adically simple groups (p-groups)
18
This result is in contrast to the situation oflinear groups of finite width over IFp [[t]]. For instance the Sylow pro-p-subgroup of S Ln (:IF'p [[ t]]) is isomorphic to the Sylow prop-subgroup S Ln(IFp [[t 2 ]]) which it contains properly as a subgroup of infinite index. The Nottingham group also contains proper subgroups isomorphic to itself. It may be that an insoluble just infinite pro-p-group P contains no proper isomorphic subgroup isomorphic to itself if and only if P is p-adic analytic.
d)
Obliquity and lattices of normal subgroups
The following theorem shows that there is a strong link between the uniform normal subgroups of an insoluble p-group P and the full Lie lattices in its Lie algebra £(P). This almost determines the structure of the lattice of normal subgroups of P since there exists an open uniform characteristic normal subgroup T such that all normal subgroups of P which are contained in T are uniform, ef. [DdMS 91] Corollary 4.5. (IILI2) Theorem. Let P be an insoluble p-group with Qp-Lie algebra £(P). Then there are finitely ma.ny P-invariant Lie lattices L l , ' .• ,Lk in £( P) such that the following holds.
(i) Every P-invariant Lie lattice in £(P) is of the form pC< Li for some i, a E N. (ii) There exist integers al,' .. ,ak such that there is a bijection between the sets {N <J PIN uniform in P} and {pCo, a ? ai, 1 ~ i ~ k} via N f--t L(N). Proof. (i) P acts irreducibly on £(P). Therefore, by the Jordan-ZassenhausTheorem, one has only finitely many isomorphism classes of P-invariant lattices in £(P). (ii) Define the set N := {N <J PIN uniform in P}. There are N 1 , ••• , N I EN which are maximal with respect to the condition that there exists no element MEN such that MP' = N for any i E N. It follows from (i) that l = k and L(Ni ) = PC no one has rn+AP) = Ff(rn(p)) = rn(p)pf, where F(X) = Fl(X) is the Fmttini subgroup of the group X and FHl(X) := F(Fi(X)). Call the lexicographic smallest such pair (z, 1) the periodicity of P and this f the defect of the periodicity. The existence of maximal p-groups will be established in the next section. The maximal jJ--groups which we have seen so far had 1 as their defect of periodicity. It is clear that this defect can become arbitrarily large, if the index
e) Uniqueness and existence of maximal p-groups
19
of P in its maximal j3-group gets large. Note, the average width is related to these parameters by
wa(P)
= d·-fz
where d is the dimension of the Lie algebra £(P). Theorem (III.12) indicates that the lattice of normal subgroups of P is essentially determined by a sufficiently big finite factor group. The simplest possibility is that each open normal subgroup of P lies between some "I~ (P) and "Ii +I (P). The deviation from this behaviour is measured by the obliquity. Recall its definition. (III.15) Definition. Let P be a pro-p-group of finite width. Define
/l-i(P) := (
n
N) n "Ii+l (P).
N :11',+1 (P),N
Then o(P) := max~EN 10gp(l!i+l(P) : /l-i(P)!) if it exists or otherwise o(P) := called the obhquity of P.
00 2S
If P is an insoluble p-group then by Theorem (III.12) and since P is just infinite o(P) is well defined and finite. Using the correspondence between the open normal subgroups and the Lie lattices one can determine the ultimate obliquity ou(P) := limi-tDO 10gp(l!i+l(P) : /l-i(P)I) from the Lie sublattices.
e)
Uniqueness and existence of maximal p..groups
Call a p-group maximal, if it is not properly contained in another p-group as an open subgroup. Let £ be a semisimple finite dimensional Qp-Lie algebra. One can view Aut(£) = A(Qp) as the group of Qp-rational points of a semisimple algebraic group A. Let Aut(£)O := AO(Qp) where AO is the connected component of 1 in A. One can apply to Aut(£t an analogue of the Sylow theorem by Matsumoto, cf. [Mat 66]. This proves that Aut(£)O contains an open maximal pro-p-group. Any two open maximal pro-p-groups are conjugate in Aut(£). Furthermore any pro-p-group is contained in an open maximal pro-p-group. It is well known that the index IA : AOI is finite. Furthermore AO is normal in A. Therefore, to get a Sylow theory for Aut(£), one has to deal with finite extensions of a connected semisimple algebraic group. This is done in the following lemma. (III.16) Lemma. Every pro-p-subgroup of Aut(£) is contained in an open maximal pro-p-subgroup. All open maximal pro-p-subgroups are conjugate in Aut(£). Proof. Let GO = AutO(£) and G = Aut(£). Since Co has finite index in G and is normal in G one can deduce a Sylow theorem for G as follows. For the first step let P be an open pro-p-subgroup in GO. From [Mat 66] Proposition 1, it follows that the normaliser Nco of an open pro-p-subgroup P is compact because P itself is compact. Therefore an open (Sylow) pro-p-subgroup P in GO is contained in its normaliser Nco (P) with finite (p-prime) index. For any x E G there exists
III p-adically simple groups (p-groups)
20
9 E GO such that px = pg. In particular xg- i lies in Nc(P). Therefore every element x in G can be written as a product of an element in Nc(P) and an element in GO. It follows that the factor N c( P) INCO (P) is isomorphic to the finite group GIGo. Applying Sylow's theorem to the finite quotiput Nc(P)IP the pre-image P of a Sylow p-subgroup of N c( P) I P is a maximal pro-p-subgroup of G. Clearly any two maximal pro-p-subgroups of G obtained in this way are conjugate. Let Q be an open maximal pro-p-subgroup of G. Claim, Q is conjugate to P under G. To prove this let Y = GO n Q. Then Nco (Y) : Y is finite and Q normalises Nco (Y). Hence Nco (Y)Q is an open subgroup of G with Y as an open normal subgroup. Since Q is a maximal pro-p-subgroup in G the prime pilNco (Y)Q : QI = INco (Y) : YI, hence Y is a maximal pro-p-subgroup of GO, proving the claim. It remains to deal with the case that the maximal pro-p-subgroup Q is not open in G. As proved in [Mat 66] one has that Y : = Q n GO lies in some open pro-p-Sylow subgroup P of GO. The intersection ngEQ p9 is of finite index in P and therefore open. Then Q lies in the open pro-p-subgroup (n9EQ P9)Q and by maximality it must be q.e.d. equal to it, contradicting the assumption that Q is not open. 1
1
(IIL1?) Corollary. An insoluble p-group P embeds into a maximal p-group Pmax with the same Lie algebra £(P) = £(Pmax ). Any two groups Pmax with this property are isomorphic. The gronp Pmax can be obtained by constructing a series of open subnormal subgroups Pi = P <J P2 <J ... <J Pmax in the following way: Pl is the Sylow pro-p-subgroup of Aut(Pi-r). Proof. Clearly P lies in a Sylow pro-p-subgroup of Aut(£(P)). Note, Aut(£(P)) acts faithfully on £(P) and the normaliser of a proper open subgroup X of a pro-pq,e.d. group contains X properly. The dimension of an analytic pro-p-group is the dimension of its Lie algebra. (IIL18) Proposition. There are only finitely many maximal p-groups of given dimension. Proof. Namely for a given finite extension K of Qp there are only finitely many absolutely simple Lie algebras over K of given dimension ([Kne 65], [Sat 71] p. 119). There are only a finite number of extensions KIQp of given degree, cr. [Nar 90] p. 216. q.e.d.
IV
Periodicity
As examples of the kind of result we have in mind, consider two of N. Blackburn's theorems. The first states that if P is a p-group such that P/rp+1(P) is of maximal class then so is P (cf. [Bla 58]). The second states that if P is a finite p-group with p > 3, lP/r3(P) I = p3, and PP = r3(P), defining Ii = logp hi(P)/ri+l(P)! then (iI, f2,' .. ) = (2,1,2,1, ... ,1,2,1,1), with 0 S; f S; 2 (cf. [Hup 67] p. 392, [Ela 61]). Also bi(P), rj(P)] = ri+j(P) if i or j is odd. The former result has been generalised in [Lee 94b] and [Sha 94] for sufficiently large p-groups of given coclass. The latter is much easier to generalise. We have not stated our generalisation in the strongest possible form, so it is easy, using the same idea, to obtain a stronger result in various special cases. In fact our 'generalisation' yields a weaker result, as it stands, when restricted to Blackburn's original situation, than that obtained by Blackburn. (IV.l) Definition. Let P be afinitep-group of class c, and let n 2': 1. Let N = rn(P) if p is odd, let N = rn(p? if p = 2. Denote by d(N) the number of generators of N. Then P is settled with respect to n if
(i) d(N) S; n, and (ii) NP 2': rc(P), We should observe that this is not the same concept as that used in [Lee 94b]. However the concepts are similar, in that in both cases, the structure of a pro-p-group is largely determined by its quotient by some term of the lower central series provided that the quotient is settled. (IV.2) Remark. By Proposition 1.13 and 4.1.13 of [LuM 87a], it follows that N is strongly hereditarily powerful in P. This means that if M <J P and M ~ N, then M is powerfully embedded in N; that is to say, [M, N] S; MP if p is odd, and [M, N] S; U2 (M) if p = 2, when U1 (H) := HP and Ui +1 (H) = U1 (Ui(H)) for i 2: 1. (IV.3) Lemma. Let P be a finite p-group of class c such that P/rc(P) is settled with respect to n. Then P is settled with respect to n. Proof. Assume that p is odd. Then rn(P/rc(P)) = rn(P)/rc(P), However (rn(P/rc(P)))P 2': rC-1(P)/rc(P), so d{rn(P/rc(P))) = d{rn(P)). The result follows. The proof for p = 2 is similar. q.e.d. (IVA) Proposition. Let N be a strongly hereditarily powerful subgroup of the pgroup P. Then 1r : M r l MP defines an isomorphism between the lattice £1 of normal subgroups K of P such that n(N) ~ K ~ N, and the lattice £2 of normal subgroups L of P satisfying (1) ~ L S; NP with n(N) = (n E Nln P = 1). Proof. Let p be odd. We first prove that 1r is onto. Let L E £2' By Proposition 1. 7 of [LuM 87a] every element of NT' can be written as a pth_power of an element in N since N is powerful. Hence L = {gP : g E N, gP E L}. Let K = {g EN: gP E L}.
IV Periodicity
22
Since L is powerfully embedded in N, it follows that K is a subgroup of N, and clearly K E £1, and 1f(K) = L. We now prove that 1f is injective. Let M E £1, and MP = L E £2' Let x P E L. We prove, by induction on ILl, that x E M. This is trivial if L = (1). By Proposition 1.7 of [LuM 87a], -3 y EM: yP = x p. Let x = yz, so zEN. Then zP E [M, N]P as M is powerfully embedded in N, and [M, N]P = [MP, N] by the remark to Proposition 1.6 of [LuM 87a]. So by induction, z E [M, N]n(N). Hence z E IvI and x E M, as required. It follows that M = {x EN: xP E L}, so M is determined by L. As 1f preserves inclusions, it follows that 1f is a lattice isomorphism. The proof for p = 2 is similar. q.e.d. (IV.5) Definition. Let P be an infinite pro-p-group, and let n ~ 1. Then P is settled with respect to n if, for some c, we have Plic+-1(P) is settled with respect to n. (IV.6) Proposition. Let P be an infinite pro-p-group that is settled with respect to n. Let N = in(P) if P is odd, and let N = in(P)2 if P = 2. Then N is a strongly hereditarily powerful subgroup of P, and 1f : M r l MP defines an isomorphism between the lattice £1 of closed normal subgroups K of P such that n(N) ~ K ~ N, and the lattice £2 of closed normal subgroups L of P satisfying (1) ~ L ~ NP with n(N) = (n E Nln P = 1). Proof. This follows from Lemma (IV.3) and Proposition (IV.4), by considering suitable quotients of P. q.e.d. (IV.7) Proposition. If P is an infinite pro-p-group that is settled with respect to n, then P is p-adic analytic. Also, if P is just infinite, then in(P) is uniform if p is odd; and in(P)2 is uniform if p = 2. Proof. The first statement holds because P contains a powerful subgroup N = in(P) if p is odd or N = in(P? if p is even. To prove that N is uniform it is sufficient to prove, by Theorem 4.8 [DdMS 91], that N is torsion free. If not then n(p) is a non- trivial closed normal subgroup of P, and hence is open. But this is impossible by Proposition (IV .6). q.e.d. (IV.8) Lemma. Let P be a p-adic analytic group, and let N be a uniform subgroup of P. Then L(MP) = pL(M). Proof. Trivial.
q.e.d.
(IV.9) Lemma. Let M :s N where N is a strongly hereditarily powerful subgroup of the p-adic analytic group P. Then the set of x, yEN, x y mod M has the same content if the congruence is interpreted group-theoretically or Lie-theoretically. Proof. If xy-1 E M then x - y = lim( x pi y-P' )P-' = lim Zi, where {zd is a convergent sequence of elements in M, since [M, N] is powerfully embedded in N. i Conversely, if x - y E M then since (x Pi y_pi)p- - xy-1 mod [M, N], it follows that xy-1 E M. q.e.d.
23
Before proving the next result about these subgroups, we need a commutator formula.
Y E G. Let p be a prime, and i > O. Then ypi X = xyp' [y, X]pi COCI ... Ci, where for all j, the term Cj is a product of pi- j -th powers of basic commutators in x and y involving at least y) copies of y for j > 0, and at least two copies for j = o. (IV.IO) Lemma. Let G be a group, and
X,
Proof. Collect the word ypi x by collection from the left. That is to say, always collect the left-most term that is not in place. To keep track of the collection process, label each of the original occurrences of y as Yl, Y2, ... ,ypi. That is to say, we proceed as if we had pi distinct commuting elements. Say that two basic commutators are equivalent if they become identical when suffices are ignored, and all equalities and inequalities between suffices in corresponding places are the same in both words. So, for example [[Y3' x, Yl, Y2, Y4], [Y2' x]] is equivalent to [[Y4' x, Y2, Y3, Y6], [Y3' x]]. Now if a basic commutator, with suffices, occurs in the collected word, it occurs only once, and every equivalent word also occurs. The number of commutators in this equivalence class is (p~), where k is the number of distinct suffices that occur in the original commutator expression. But (p~) is a multiple of pJ where p divides k exactly i - j times, for k S pi. This gives the required formula, once one has observed that [y, x] occurs exactly pi times in the collection. q.e.d.
(Iv.n) Lemma. Let P be a p-adic-analytic group, and N be a strongly hereditarily powerful uniform normal subgroup of P. Then n - n 9 == [n, g] mod [N, P]P. By Lemma (IV.9), this is unambiguous. Proof. By Lemma (IV.IO), n-n 9 is the limit of the convergent sequence {Yi} where Yi = ((npi(n9tPiy-i = (nP'gn-pig-l)P~i == ([n,g]pixr)p-i where Xi = COiCli···Cii. Here Cji E [N, P]P for all j, since N is a strongly hereditarily powerful subgroup of N. Since N is powerful, [n, g]p'Xfi = ([n, g]mii for some mi in N. Since mr E [N, p]p'+l, it follows that mi E [N, P]P since N is uniform. So n - n9 is the limit of the convergent sequence [n,g]mi, where mi E [N, P]P, and this completes the proof. q.e.d. (IV.12) Proposition. Let P be a p-adic analytic group, and let N be a strongly hereditarily powerful uniform normal subgroup of P. Then the Lie ideal [L(N), P] and the normal subgroup [N, P] are equal. Proof. This follows from the previous result.
q.e.d.
(IV.13) Proposition. Let M and N be strongly hereditarily powerful subgroups of
the p-adic analytic group P. If L(M) and L(N) are isomorphic as P-modules, then M /[ M, P] and N / [N, P] are isomorphic as groups. Proof. By the previous result, [M, P] and [N, P] may be interpreted in the Lie sense, so M /[M, P] and N /[N, P] are isomorphic, when both sides are interpreted as abelian groups with addition induced by the Lie addition. But as these quotients are
24
IV Periodici ty
abelian, this is simply the addition induced by the group multiplication.
q.e.d.
The three theorems with which we end this section give our 'generalisation' of the theorem of Blackburn with which the section began. (IV.14) Theorem. Let 'Yi(P) be a strongly hereditarily powerful subgroup of the padic analytic group P. Then there is at> 0 such that
for all j 2 i. Proof. The sequence {L("(j(P)) : j 2 i} contains only finitely many equivalence classes, where M and N are defined to be equivalent if M = pr N for some (not necT essarily positive) integer r. So for some j and t > 0 we have 'Yj+t(P) = 'Yj(P)P . Then T by Lemma (IV.8) and Proposition (IV.ll), 'Yj+t+I(P) = 'YHl(P)p . It follows that 'Yk+t(P) = 'Yk(p)pr for all k 2 j. As this can be interpreted as a statement about the Lie lattice, we can deduce that 'Yj+t~I(P) = 'Yj_I(P)pT provided that j > i. Iterating gives 'YHt(P) = 'Yi(p)pr. It follows that L("(i(P)) and L("(i+t(P)) are isomorphic as P-modules. The result now follows from Proposition (IV.13). q.e.d. (IV.lS) Theorem. Let P be a finite or infinite pro-p-group such that P!'YC+I(P) is settled with respect to n for some n 2 2. If P!'YC+I(P) :: Q!'Yc+I(Q) for some just infinite pro-p-group Q, and 'Yn(Q)P = 'Yn+k(Q) for some k > 0, then 'Yi(P)!'Yi+I(P) is a homomorphic image of 'Yi(Q) !'Yi+1 (Q) for all i 2: 1. Proof. Define a set-theoretic map () : Q ---t P as follows. Let a : Q! 'Yc+ I (Q) ---t P !'Yc+I(P) be an isomorphism. Fix a transversal {gIl"" gr} of Q!'Yc+I(Q) and define gi() to be a pre-image in P of a(gnC+I(Q)). For h E Q - 'Yn(Q)P with h'Yc+I(Q) = a gnc+I(Q) define h() := gi(). Now define inductively () : 'Yn(Q)p - 'Yn(Q)pa+l ---t P for a = 1,2, ... by g() = (gl/p())P. Clearly () maps 'Yi(Q) onto 'Yi(P) for all i, and induces a surjection of 'Yi(Q)!'Yi+I(Q) onto 'Yi(P)!'Yi+I(P) for all i. q.e.d. (IV.16) Remark. Although the condition that 'Yn(Q)P = 'Yn+k(Q) for some k > 0 is satisfied in most interesting cases, if it is not, we can replace it by an alternative condition that is always satisfied. We can simply assume that if N is any normal subgroup of Q such that N ~ 'Yn(Q) but N i 'Yn(Q)P then N 2 'Yc(Q). Note that this condition is satisfied for c big enough. In particular c depends on the obliquity of Q. (IV.17) Theorem. Let P be a pro-p-group. Let P!'YC+l(P) :: Q/'YC+I(Q) for some just infinite pro-p-group Q, where the length of the period of the sequence gi := 10gp(hi(Q) : 1~+I(Q)I) is k. Let P!'YC+I(P) be settled with respect to n for some n 2 2, and c sufficiently big. Define fi := 10gp(hi(P) : 'Yi+I(P)I). If fi < g~ for some i 2 c + 1 then P is finite. The size of c depends on the obliquity. Define mi := min{j E N I 'Ym) ~ !-ti(P)} - (i + 1) and m := max{mi liE N}, for the definition of !-ti(P) see (III. 15). If 'Yn(Q)P = 'Yn+k(Q) assume c + 1 2: n + m + k otherwise use an alternative condition as in Remark (Iv' 16).
25
Proof. Let N be the set of elements of Q mapped to the identity by (J, where (J is as in the proof of the last result. It is easy to see that N is a closed normal subgroup of Q, and that the image of ri( Q) jri +1 (Q) under (J is isomorphic to ri( Q)N jrHl (Q)N. The result follows. q.e.d.
V
Chevalley groups
To get a rough idea how the j)-groups and their lower central series look like, we investigate the Sylow pro-p-subgroups of Chevalley groups G = G(4), K) of adjoint type, where 4> is a simple root system and K is a local field with finite residue class field F of characteristic p. These groups are always just infinite of finite width. In the case that the characteristic of K is zero, they are j)-groups and often they are even maximal j)-groups.
Notation 4> a simple root system with base L}" 4>i := {,B = LQEll. a Qal,8 E 4>, a Q E Z, ht(,8) := LQEll. a Q = i} the set of roots of height i. So 4> I = L}, and l4>k I = 1 for the smallest k E Z wit h 4>k i= 0, since t here is a unique highest root. 4>+ the set of all positive roots and 4>- the set of all negative roots with respect to the base L}" Q the weight lattice of 4> (note, IQ : Z4>1 is equal to the determinant of the Cartan matrix), £ a complex Lie algebra with Chevalley basis (HQ) Xtli a E L}" ,8 E 4» cf. [Ste 68], (a,,B) (= 2( a,,B) / (,8,,8)) the Cartan numbers for roots a,,8 E 4>, £(Z) the Chevalley lattice, i.e. the Z-sublattice of £ spanned by the Chevalley basis, £(R) := R ® £(Z) for any commutative ring R, K a local field (i.e. a field which is complete with respect to a discrete valuation v : K ~ Z U {oo}) whose valuation ring 0 has finite residue class field F = 0/1r0 with 1r a uniformising element of 0, X Q and RQ the images of X Q and H Q in £(K) respectively £(0). As shown by Chevalley, cf. [Che 55] (see also [Car 72], [Ste 68]), there is a homomorphism . Write xQ(t)
:=
Then
X_Q(t)
t) )(=
01 1
=
-
exp(tXQ)).
~)).
Define G( 4>, K) := (xa(t) la E 4>, t E K) to be the Chevalley group of type £ over K. Set XQ,r = {xQ(t)1 t E O,v(t) 2 r} for a E 4> and r E Nand
1lr
=
{h(X) I X E Hom(Z4>, O;)}
with 0; = {x E 01 v(l - x) ~ r} where h(X)HQ = H and h(X)XQ = x(a)XQ for all a E 4>. Then XQ,r and 1lr are subgroups of Aut(£(K)), even of Aut(£(O)). We want Q
27
to study the lower central series of
P
= P(~,O):=
(Xa,o,
X,8,l,
HII a E ~+,,8 E ~-).
(V.1) Remark. The group P(~, 0) is a Sylow pro-p-subgroup of Aut(£(K)) for char(K) = 0 unless p I IQ : Z~I or p 1IAut(KIQp)1 or there is a diagram automorphism of the Dynkin diagram of ~ of order p. For a proof see [Don 87] p. 327-330 and [Iwa 66]. To see the lower central series of P it is easier to deal with its Lie algebra analogue first. Let j-l
E9( E9 1rOXa EB i=l aE'
E9
1r 2 0X,8) EB
,8E-k- 1 +i
k
E9( E9 OX, EB E9 i:::.j
,E'
1rOXd ) EB
dE-k-!+,
for j = 1, ... , k+1, where k is the maximal height of the roots in~. For j = (k+1)q+r with q 2 0 and 1 ~ r ~ k + 1 let £j = 1r QO£r. Then the £j(O) are full lattices in £( K) closed under the Lie bracket such that [£i, £j] ~ £Hj with equality if char( F) divides none of the m a,8 where [X a , X,8] = m a,8Xa+,8 and a,,8, a + ,8 E ~. To define a filtration for P let j
I E
To investigate when Pj
=
'J:',
(k
+ l)q + r
as above. Then define
1: L' U E .,y.,-k-l+i 'J:' lor 2. = ),• ... , k) .
Ij (P) we need two lemmas.
G IG i be a pro-p-group with G = G l 2 G 2 2 ... a central series of subgroups of finite index in G. If [G i , G]Gi+2 = GHI for all i 2 1 then li(G) = Gf l where li(G) is understood to be the closure of the i th term of the lower central series. (V.2) Lemma. Let G
=
.,y.,t
=
l~
Proof. One has Gi+l = [G i , G]Gi+2 = [G i , G][Gi+l' G]G i +3 = [G i , G]GH3 . An obvious induction yields Gi+ l = [Gi , G]Gi+n for any n ~ 2. Hence
Gi + l =
n[Gi ,G]Gi+n = [G
i,
G].
n>l
q.e.d.
V Chevalley groups
28
(V.3) Lemma. The following commutator relations hold in P. (Convention gh = hg[g, h].)
(i) Let a, {3 E
~
linearly independent. Then [xa(u), Xfj(t)]
X2a +jfj(Ci]afjU 2t j )
= II "YEN 'o<+JI3E
(ii)
where ha(u)Xfj = u(fj,a}Xfj and ha(u)Hfj = Hfj. (iii) [h(XI), h(X2)]
= 1 for
Xl, X2 E
Hom(Z~,
Oi).
(iv) [ha(u),Xfj(S)]:= xfj((l- U-(fj,a})S) fora,{3 E ~,U E O*,S E O. Proof. ii) can be checked by calculations carried out in S £2. The other equations follow from relations given by [Ste 68]. As an immediate consequence of (V.3) one sees that [P, P t ]
~
q.e.d. Pi +l .
As the proof of (VA) will show, there are three possible obstacles for Pi to being equal to li(P): (i) p 1 ma,fj for some roots a and {3 such that a mafj X a+fj,
+ {3
is a root, where [X a , X fj ] =
(ii) p 1 IQ : Z~I where Q is the weight lattice of ~, (iii) there is a root {3 E ~l U ~-k for which there is no a E ~ with gcd( ({3, a), p) = 1. Remark. Obstacle (i) occurs only for (p,~) E {(2, B n ), (2, Cn), (2, F4 ), (2, G2), (3, G2)ln 2:: 2} Obstacle (ii) occurs only for (p,~) E {(2, B n), (2, Cn), (2, D n ), (2, E 7 ), (3, E 6 )ln ~ 2} or p[(n + 1) and ~ is of type An. Obstacle (iii) occurs only for (p,~) E {(2, AI), (2, B 2 ), (2, Cn) In 2:: 2}. If obstacle (ii) does not occur also (iii) does not occur. On the other hand, if one extends the group by elements of H om(Z~, O~) (i.e. rational automorphisms) in the case of obstacle (ii) one possibly also avoids obstacle (iii). (VA) Theorem. Let d = IQ : Z~I as above. For P = P(~, 0) one has Pi = li(P) for all i E N, unless p 1 d, or p = 3 and ~ = G 2, or p = 2 and ~ E {G 2, F4 , B n , Cnln 2'
2}. Proof. Clearly, if one of (i)' (ii), or (iii) holds one has one of the exceptions listed in the theorem. Therefore we have to prove: if none of (i)) (ii), and (iii) is satisfied,
29
then [P, P~]Pi+2 = Pi + 1 for i = 1,2, ... and the result follows from (V.2). We use induction on i. The three cases considered yield the base and the induction step and the result follows by Lemma (V. 2), Case 1: i"t- O,k mod k+ l. In this case (V.3)(i) yields [P, P~]Pi+2 = P~+l since condition (i) above does not hold. k (mod k + 1) say i == q(k + 1) + k. Case 2: i In this case (V.3)(ii) shows that ha(x) lies in [P, PdP~+2 for all a E , since the X aand x_a-terms lie in P~+2' Therefore we have to prove (ha(x)la E ,x E 0;+1) = 1l q+ 1 . The definition of the ha(x) shows ha(x) = IT h(Xa)(a,~> with Xa E H om(Z, 0;+1) defined by Xa(f3) = 1 for (3 E .6., (3 i- a and Xa(a) = x. Since the h(Xa) with a E .6. obviously generate 1l Q+ 1 , this holds for the ha(x) with x E 0;+1 as well, if pJdet(((3,(l:))a,~E~ = IQ : ZI, which is the negation of (ii) above. Case 3: i - 0 (mod k + 1). Here relation (V.3)(iv) has to be applied. Since condition (iii) above is not satisfied one gets [P, Pi]P~+2 = Pi + 1 in this case. q.e.d. One should remark - especially important for the case = An - that Theorem (VA) remains valid if one modifies P = P(, 0), by working with the Sylow prop-subgroup of the Chevalley group rather than with the Sylow pro-p-subgroup of Aut(.L:(K)). (Of course the 1l i have to be modified accordingly. Note, however that the Sylow subgroup is no longer just infinite since it gets a finite centre and even after factoring out this centre it is not a maximal p-group in case of char(K) = 0.) Then condition (ii) becomes redundant and one gets Pi == 'Y~(P) except for the cases (p,
VI
Some classical groups
Most of the classical groups can be studied within the context of simple algebras with involution (cf. [Wei 61]' [Kne 69]), which at the same time provide the context for the Cayley maps (cf. [Wey 46]) connecting the unitary group with its Lie algebra if the characteristic is not 2. In the present situation also orders in the algebra which are invariant under the involution have to be considered. A general reference on orders is [Rei 75]. Apart from the general notation the first section can be skipped if one is only interested in the results, in particular in the orders of the factor groups of the lower central series of the classical groups. Example (VI.18) explains how to read Table (VI.17) which contains information about the generators of a Sylow pro-p-subgroup of the classical groups, about the lower central series, about the involution ° by which the groups are defined and about a minimal o-invariant order. The patterns of the lower central series are summarised in Table (VI.19) and can be understood without anything of the previous sections.
a)
Basic structures: orders and Cayley maps
Notation and assumption:
• K a local field with ring of integers 0 and a prime element Il. • p
-=/=
• - E
2 the characteristic of (0 jIl 0).
Aut(K) with
-2
• k the fixed field of
= id K .
~ 0
= k n 0)
1r
is a prime element of k.
Three situations are possible:
(i)
-= id)
then K
= k, 0 = 0
and let Il
= 1r.
(ii) - -=/= id) K jk unramified. In this case 0 (iii)
--=/=
id) K jk ramified. In this case 0
= 0 E9 oq
= 0 EB oIl,
with q2
E
assume Il2
0 -
1rO
and let Il
= 1r.
= 1r.
Note, in (ii) there is one and in (iii) there are two isomorphism types for K for a given k.
• (A, 0) a central simple K -algebra with involution ° such that ° induces - on K i. e. ° is a k-anti-automorphism on A fixing k and its square equals the identity. (Remember) if - is the identity on K then ° is called to be of fir8t kind otherwise ° is called to be of second kind.) • V a simple A-module. • A+:= {x E Alx o =x}. A- := {x E Alx o = -x}.
31
a) Basic structures: orders and Cayley maps
Note, A-is a k-Lie algebra with [x, y] := xy - yx for all x, yEA.
• U := U(A,O) := U(A)
:=
{x E Alxxo
unitary group of (A,O).
= lA}
• SU:= SU(A,O) := SU(A) := {x E Ulnr(x) of A over k.
= Id
where nr is the reduced norm
• SA:= {x E Altr(x) = O} where tr is the reduced trace of A over k. • SA- := {x E A-Itr(x) = O} where tr is the reduced trace of A over k. • X gen := {x E X •
1-1
is not an eigenvalue of x} for any subset X of A.
Ugen ----7 A;en : x t---+ (1 - x)(1 + xt 1 , CLP : A;en ----7 Ugen : x t---+ (1 - x)(1 + xt 1 are the Cayley maps, which are easily seen to be inverse to each other and compatible with conjugation by elements of u.
CPL
:
The aim of this chapter is to compute the lower central series of the maximal pro-psubgroup of SUo To this end these groups have to be described rather explicitly in terms of certain Lie lattices in A-which turn out to be of the form A-for certain carefully chosen a-invariant 0 -orders A in A. The powers of the radical of these orders turn out to be closely connected to the lower central series of the corresponding Sylow pro-p-subgroup of U(A). The link between the Lie lattices and the pro-p-groups is provided by the Cayley maps. The a-invariant orders are rather close to the 0span of the groups. There is an unpleasant but not so crucial side-effect in the case where the involution is of second kind and therefore the centre Z (U) is not finite. We are really interested in the maximal pro-p-subgroups of PU := U/ Z (U) with Z(U) = {x E K = Z(A)lxx = I} or PU extended by outer p-automorphisms. However, one shall investigate the maximal pro-p-subgroups of SU, which often amounts to the same. By the general theory, cr. Lemma (IILI6), SU has maximal pro-psubgroups and they are all conjugate under SUo It is worth while to set up correspondences between
• p
= {PIP
is an open pro-p-subgroup of U},
• D = {AlA is an o-order in A with AO = A}, and
• .c =
{LIL is a full o-Lie-lattice in A- with (... (ML)L ... ) o-lattice M in V}.
Note, for L E
.c every
~
1fM for some
element of L is ad-pro-nilpotent.
Clearly, one has the following maps: D D
----7
----7
P : A t---+ (1
.c : A t---+
+ (radA)) n U =: U(A),
(radAt := {x E radAlxo = -x}.
Neither map is surjective. There is an obvious modification of the maps, if one works with SU and SA-.
VI Some classical groups
32
Next there is an obvious map
£ ----+ V: L For L E £ let
H
(L)o-order = Ord(L).
L = rad(Ord(L))-. Clearly L
~
L E £ and L = L.
Next one defines a subclass l of £ for which the Cayley map is well defined. Define l = {L E £IL satisfies (*)} where (*) is the following condition.
(*) For all x, y ELand p(X, Y) E Z(X, Y) (= free associative algebra on X, Y) homogenous of degree k > 1, then p(x, y) E L provided p(x, y)O = ~p(x, y). Clearly L E £ implies
L E l. More generally one obviously has
(Vr.1) Remark. If A E V is an o-order with AO = A, and I <J A with I 1° = I then 1- E £.
~
radA and
Now use the second Cayley map to construct a map -
C LP : £ ----+ P: L
H
LCLP = {(1- x)(1
+ x)~ I Ix E L}
(VI.2) Lemma. The map C LP is well defined. Proof. Clearly L E £ =} L ~ A;en since all eigenvalues of x E L are congruent 0 (mod 1r a). Therefore xcLP is well defined and xcLP E U. Clearly (- x) CLP = (XCLP)-I. Let M be a full o-lattice in V with (... (M L) ... )L ~ M1r. Then clearly XCLP centralises an (full) o/1r-flag of M/1rM. Therefore LCLP is contained in an open pro-p-subgroup of U. To prove that LCLP is open, note that CLP : Ugen ----+ A;en is a homeomorphism and therefore maps the open subset L of A-onto the open set LCLP . That LC LP is a group requires L Eland follows now from the next lemma. q.e.d.
(Vr.3) Lemma. Let L E l. Then for any x, y E L there exists a Z E L with XCLPYCLP = ZCLP. More precisely, there exist Pk(X, Y) E Z(X, Y) homogeneous of degree k for k = 1,2, ..., such that Z = L~I Pk(X, y). E.g. PI = X + Y,P2 = -XY + Y X,P3 = -XY X - Y XY,P4 = (Xy)2 - (Y X)2,P5 = (Xy)2 X + (Y X)2y etc.. (The final formula for the Pk is established in [Sou 96].) Proof. XCLP = (1 - x) L~o( -X)i = 1 + 2 L~I (-X)i := 1 + 2£ and similarly yc LP = (1 - y) L~o (-y)j = 1 + 2Y. Hence XCLPyCLP = 1 + 2(i + fJ + 2£fJ). With i := £ + iJ + 2£y one gets 00
(XCLPyCLP)CPL
=
(1 - (1 + 2~))(1 + (1 + 2~)tl = -i L(-i)k =: z. k=O
Clearly everything converges, Z E L because of (*), and Z = L~=I Zk with a Z-combination of products of x and y with k factors.
zk = -Zk
33
a) Basic structures: orders and Cayley maps
Repeating the calculation in a suitable completion of Z(X, Y) and collecting the monomials of the same degree together, one gets the desired formula for Zl, Z2, . . . . q.e.d. Using the composite D ---* l ---* P one can exhibit a central series of the groups constructed this way, also this composite is equal to the map D ---* P introduced above. (VIA) Definition. Let A E D be an O-invariant o-Order in A. Ji- (A) := (radA)i n A- and Ui(A) := (1 + (radA)i) n U.
Define Ji- =
The two filtrations of (radAt and U(A) have the following properties.
(VL5) Proposition. (i) U(A) = U1(A) ~ U2(A) ~ '" with Ui(A) ~ U(A) open) Ui(A)/UH1(A) elementary abelian p-groups, [Ui(A), Uj(A)] ~ Ui+J(A) and niENUi(A) = {I} (ii) (radAt = J 1 ~ J 2- ?:" '" with J i- / Ji+ 1 finite o/1fo - modules such that [1i-' ~ Ji+jfori,j EN) andniENJi- = {O}. MoreoverJi- /Ji+1 r-v (Jt!Ji+1)-:: (-I)-eigenspace ufo acting on J 2 /Ji +1 with Ji = (radA)i .
In
(iii) Ji- Eland Ji-C LP
=
Ui(A) for i E N.
(iv) The second Cayley map CLP induces an isomorphism Ji- / Ji+ 1 ---* Ui (A)/U2+1(A) for each i E N. Proof. (ii) Jt! J H1 is a finite o/1fo -module; therefore Ji- / Ji+1 is a finite o/1fo-module, and since 2 =I- p = char(o/1f) one gets immediately (Jt! Ji+1t J i- /Ji 1· Since JiJj = Ji+j one gets [Ji-, ~ Ji+j . Finally niENJi- ~ nJi = {O}.
+
In
0l
(iii) By (VLl) Ji- Eland hence Ji-C LP E P is well defined. Since XCLP = 1- 2x+ 2x 2- 2x 3± ... for x E J2- , one has Ji-C LP ~ Ui(A). Since p =I- 2, Ui(A) ~ Ugen , hence Ui(A)cPL is defined. Let 9 E Ui(A). With y = (1/2)(g - 1) E Ji one gets
Since the Cayley maps CPL and CLP are inverse to each other, one has Ui(A)
=
Ji- CLP.
(i) By (VI.2) and (iii) Ui(A) is open in U, as also the definition shows directly. It remains to prove [Ui(A), Uj(A)] ~ Ui+j(A). For 9 E Ui(A) and h E Uj(A) write 9 = 1 + x and h = 1 + Y with x E J i and y E J j . Hence g-l = 1 - x + x 2 - " ' J h- 1 = 1 - y + y2 - y3 + ... yields the following expansion for the group commutator;
with Z E Ji+j +1. Hence [g, h] E Ui+j(A), more precisely [09, h] 1 + Ji+J+1.
1
+ xy - yx
mod
VI Some classical groups
34
(iv) This now follows immediately from the expansion of C LP in the proof of (iii). q.e.d. It will turn out that for suitable choices for A, cf. (VI.5) even gives the lower central series of the maximal pro-p-subgroup of U or SUO But first continue with the general discussion. Having constructed groups out of orders and Lie rings, now one goes in the other direction starting with groups.
P
~
D : PH oP :== (P)o
= o-span of the elements of P in A is clearly well defined. It gives rise to a saturation process: For P E P)et P = (l+rad(oP))nU. Clearly, P ~ PEP. A straightforward calculation shows P = P. Define 15 = {PIP E P}. The final map one has to discuss
is induced by the first Cayley map CPL.
CPL : 15 ~
l :P
H
PCPL == {gcPLlg E P}
(VI.6) Remark. The map CPL is well defined. Proof. As observed earlier, p =I- 2 implies that PEP lies in Ugen . Obviously PC PL ~ (rad(oP))-, cf. proof of (VI.5)(iii). Let x E (rad(oP)t. Then XCLP = (1 - x)(l + xt l E (1 + rad(oP)) n U, since both 1 + x, 1 - x E 1 + rad(oP) ~ (oP)*. Hence P <;: (rad(oPt)C PL ~ f\ but P = P. Hence PCPL :::: (rad(oP))- E l. q.e.d. Summarising one has : (VI.?) Proposition.
(i) The maps C LP : l ~
15
and CPL : 15 ~
l
induced by the Cayley maps are inverse to each other. ~V ~
(ii) C LP factors
l
(iii) CPL factors
15 ~ V
~
15: L Hord(L)
H
(1
+ rad(ord(L))) n U
l : P HOP H (rad(oP))-.
In the case of involutions of the second kind, i.e. k =I- K, one has a version for SU rather than U. We leave the necessary changes to the reader. Note, however, the Cayley maps do not necessarily map trace 0 elements to norm 1 elements and vice versa. However, if every eigenvalue a of x E A-occurs with the same multiplicity as -a then nr( aCLP) = 1. Our analysis, cf. in particular the proof of Lemma (VI.2), suggests that one may construct the maximal pro-p-subgroups P of U from certain o-invariant hereditary orders A. Note, for an hereditary o-order A in A the A-lattices M i in V with i E Z, which are ordered by inclusion the numbering can be chosen such that AIH1 ~ Mi and Mi+l = A1i radA. Moreover EndA(Mi ) is the maximal O-order in EndA(V) and therefore A contains O. Also A = {x E AIMix ~ M i for all i E Z} and radA = {x E AI.Nfix ~ Mi+1 for all i E Z}. There is an a E N with MJI = MHo: for
35
a) Basic structures: orders and Cayley maps
all i E Z, which at the same time is the number of simple Wedderburn components of A/radA, cf. [Rei 75] for details. Note, if Al , A2 are hereditary V-orders in A, then Al ~ A2 if and only if radA I ;:2 radA2. (VI.8) Corollary. Let P hold:
~
U be a maximal pro-p-subgroup of U. Then the following
(i) PEP. (ii) There is an hereditary V-order A with AO
=
A and (radA)-
= (rad(oP))-.
Proof. (i) P = P since P is maximal. (ii) oP is contained in an a-invariant hereditary order, cf. [Scha 74]. Among these choose one with (radAt maximaL By the maximality of P one gets (rad(oP))- = (radA) -. q.e.d. (VI.9) Definition. Let P and A be as in (VI.8)(ii). We call A semi-saturated if Mi(radA)- = Mi+l for all i E Z, where· .. Mi > Mi+l ." are the A-lattices in V. We call A saturated, if SAn((radA)i)- = ((radA)-)[iJ, where ((radA)-)[i] is the o-span of the [... [II, 12], 13"" Ii] with II, ... ,Ii E (radA)- and i E N. Clearly, A is saturated if and only if the defining condition is satisfied for i 1, ... ,ae - 1 where a is the number of components of A/radA and e is the ramification index of k in K. Also if A is saturated, it is semi-saturated. One can view the semi-saturated orders as those a-invariant hereditary orders which are maximal with respect to the property (radAt = (rad(oP))-. As a result of going through all possible K -algebras with involution, it will turn out that P can always be obtained from a semi-saturated hereditary order, which with one exception for p = 3 is already saturated and gives a description of the lower central series of P. We now derive a formula for Lie commutators, adapted to hereditary orders. Assume A = Dnxn where D is a division algebra with maximal order n and radical p = radn. An hereditary order A can be assumed to be of the form
where no, ... ,nQ-I E N with no + ... + nQ-I = n. With no, ... ,nQ-I fixed, (mij) E (N U {O})QXQ and po := n let A((mij)) 0
= {(aij)laij
E
(pmi;)n,xn; for 0
~
i,j
1 0
0
:s a
- I}.
0
1
Then A = A(
) , radA
= A(
), 0
1
1 0
1
1
VI Some classical groups
36
o
110 (radA)2 = A(
0 1
), ... , (radA)~ = A(Ei ),
1
211 where E i is obtained from E i - l by shifting all columns one position to the right and taking the last column of E i - l with entries increased by 1 as first column of E i . Call the matrix (mil) exponent matrix of the order or ideal. We introduce some notation for representatives of cosets in (radA)i/(radA)i+1 .
(VL10) Definition. Let 0 ~ i ~ a-I, ao E Dnoxn" al E Dnl xni +1 , ... , aa-l E Dn",-l xni+"'_l, where the indices s of n s are taken modulo a. Then
o
ao
o
o
o o
E Dnxn,
o o o o
0 aa-l where a/ is in the l-th block row and in the i-th block column. (In particular do(ao, ... , aa-l) = diag(ao, ... , aa-t).) If the entries of the ai lie in the appropriate powers of P (as indicated by the exponent matrix E i ), then ddao, ... , aa-l) E (radA)i, where i' i mod a. An elementary calculation yields the following formula for Lie brackets.
=
(VLll) Lemma. For 0 ~ i, j ~ a-I and ao E Dno xn" at E Dn",-l xn",+., bo E DnoxnJ, ... ,ba- l E Dn"'-l xn",+j, then
Dn 1 xn.+l , ... ,
aa-l E
[di(ao, ... , aa_I), dj(b o, ... , ba-l)] = di+j(co, ... , ca-d with i + j taken mod a to lie between 0 and a - I, and c/ :=
for 0
~
l
~ a-I)
a/b/+ i - b/a/+j
with all indices taken mod a.
Clearly, with this formula it becomes easy to check whether an hereditary 0_ invariant order A is saturated. We now translate Lie commutators into group commutators.
(VL12) Lemma. Let L E C. Then there are Pi(X, Y) E Z(X, Y) homogenous of degree i for i 2: 2 such that for any x, y E L the group commutator is given by [XCLP, YCLP] = 1 + p2(X, y)
+ P3(X, y) + ...
37
a) Basic structures: orders and Cayley maps
The first two Pi are P2(X, Y) = 4(XY - Y X) = 4[X, Y], P3(X, Y) = 4(X 2y - 2XYX - Xy 2 + YX 2 + 2YXY - y 2X) (The Lie expression of the Pi is established in [Sou 96].) Proof.
[XCLP, yCLP]
=
(XCLP )-1 (YCLP tIXCLPYCLP
= -4[[X, Y],X + Y].
= (-X)CLP( -Y)CLPXCLPyCLP
=
(1 + 2x + 2x 2 + 2x 3 + ...)(1+ 2y+2y 2 + 2y3+ .. ·)(1- 2x+ 2x 2 - 2x 3 + .. ·)(1- 2y+ 2y2 - 2y3 + .. -) = 1 + 4[x, y] + Z3 + Z4 ... and the claim follows similarly as in the proof of (VI.5).
q.e.d.
This was the last preparation for the computation of the lower central series of a maximal pro-p-subgroup of SUo
(VL13) Proposition. Let A = A0 be a saturated hereditary V-order in A. Then the maximal pro-p-subgroup P = SU n (radAtCLP of SU has the lower central series given by PI = P, Pi = SU n (1 + (radA)i) = SU n ((radA)i)-C LP . Proof. This follows immediately from (VL5), the definition of saturated orders, q.e.d.
(VL12), and (V.2).
Before going through the various cases, some general comments on the description of the involution ° of A = Dnxn leaving an hereditary order A invariant are in place.
(VL14) Remark. Let A = AO be an hereditary V-order in A. Then the permutation induced by the involution ° on the set of the Wedderburn components of A/radA has at most 2 fixed points. Proof. Let A/radA = X o EEl ... EEl XQ- I be the decomposition of A/radA into simple algebras and write Xl = X iL . The numbering can be chosen in such a way that radA/(radA)2 = Yo,! EEl YI,2 EEl .•• EEl YQ-I,o where Yi,i+I is a Xi-Xi+I-bimodule. Applying the involution ° yields r:~i+I = Y(i+I)L,u, and this is a X(i+I)t-XiL-bimodule. Assume ioL = i o, then (i o + j)L = i o ~ j (all indices taken modulo a). The claim follows. q.e.d. The decomposition of A/radA enables one to come up with a rather refined notion of the o/7ro-dimension of ((radA)i/(radA)i+It as follows: (ra dA) ij( ra dA)i+I
=
y;(i)
y(i) O,i EEl v(i) I I,i+I EEl .•• EEl Q-I,Q+i-I
with the convention that all indices are taken modulo a, where YR~j is a XrXi +r bimodule. Clearly }j:~~j° = Y(~~j)t,jL in the notation of the proof of the last remark.
(VI.15) Definition. For (r - l)a < i :::; ra, 0 :::; s, t < a, s + i == t mod a let
b~:i
dimo/7ro(Ys~~)t if (s, t) :=
{
dimo/7ro(Ys:~)) -
(tL,8L) if (s, t) is lexicographically earlier than (tL, SL) otherwise. =
B r = (b~r;)o<s , -, t
VI Some classical groups
38
One clearly has
(VL16) Remark. (i) B r
=
B r +e where e is the ramification index of Dover K.
(ii) For i E N let bi i
<
= L. b~:£
where s, t E {O, 1, ... ,a -I}) r defined by a(r -1) :S
t-S=l
ar with the convention that all indices are to be taken modulo a.
dim o / rro (radA)i / (radA)i+1
Then
= bi.
So in particular one can read off from the dimension matrices the orders of the sections of the lower central series of P = SU n (radA)-CLP , in the case the group is saturated. Note, however, if a I i this order might not be pf b; but p!(b;-I) where f =degree of o/1fO over IF'Pl depending on the difference between A-and SA n A-. If one has to take betr - 1 instead of betr, this is indicated by a • as an exponent to B r . The concrete way in which the involution ° : A = Dnxn ---1 A is given, is by
for a suitable F E A *, which can be thought of as a Gram matrix for some c:-Hermitian form on V with c: = ±l. In case D is a K -division algebra it is a non-split quaternion algebra Q and - is the standard involution. More precisely • Q has a unique maximal V-order • with
q2 E
n = (I, q, 1fQ, q1fQ)o
K = k a non-square unit, 1f~ =
• - maps 1, q, 1fQ and
q1fQ
to 1, -q,
-1fQ
1f, q1fQq-1
and
-q1fQ
= -1fQ'
respectively.
The special shape of F in the tables below is chosen in such a way that the components of A/radA and the permutations induced by ° on these components, ef. (VL14), becomes evident.
b)
Tables: Involutions, orders and lower central series
In the subsequent table one goes through the various simple algebras with involution (for characteristic 0 the list is complete) and the following information is given: (0) Dynkin diagram of the unitary group SU(A,O) as defined in [Sat 71] p. 119.
(1) Name of the unitary group, information on the division algebra D, (so that A = Dnx n is described, Q denotes a non-split quaternion algebra). (2) Matrix F (so that together with (1) (A,O) is described), (3) a = number of Wedderburn components of A/radA for the hereditary order A. In case a < n those terms of the sequence (no, . .. , net-I) which consists of degrees of the components of A/radA over their centres are given, which are not equal to 1. Then A is always of the form described before (VL1 0).
b) Tables: Involutions, orders and lower central series
39
(4) The dimension matrices B = B 1 or B 1 and B 2 respectively in case the ramification index of Dover K is 1 or 2. In case the involution is of the second kind one of the B/s gets a • as explained after (VI.16). Abbreviations:
denotes a matrix consisting of submatrices Ai) which have mi rows and mj columns. Such a submatrix is denoted by a, if all its entries are equal to a. It is denoted by (a b c) if it is square of the form a
a
b c
a
bee which degenerates into (b) if the degree is 1. The possible entries a, b, c can be natural numbers, 0, and -, with the meaning explained above for the dimension matrices B i and elements of D for the Gram matrices F. (5) The pattern of the lower central series is encoded in the following way. The number of copies of cyclic groups C p divided by the inertia degree f of k is given for each factor group 1d-'(i+l. A sequence of numbers enclosed by brackets and raised to some power x means the repetition of this sequence x-times. The whole sequence is overlined to indicate that it repeats itself. E.g. nn-l, n - 1 means: n n ,1n+kn/1n+l+kn ~ cpn-l) for k E , ... ,1n-l+kn/1n+kn ~ 1l+kn/12+kn r"V
N U {O}.
ct
ct
One finds an example of how to use the tables in (VI.18). If one is only interested in the orders of the factor groups 1i (P) /1i+l (P) one can use section VI c).
VI Some classical groups
40
(VI. 17) Table.
SUn(K, F), D
=K
wit.h K/k unramified, n:2: 3 odd 1 (2A n_l , quasi-split),
SUn(K,F), D ~ K with K/k ramified, n 2: 3 odd, (n,p)
=1=
(3,3), ( 2An_l
quasi-split) ,
(m + 1, m)m, m, (m, m + l)m, m for m = n~l
SUn(K, F), D ~ K with K /k unramified, n 2: 4 even, ( 2 An-I, quasi-split),
SUn(K, F), D ~ K with K/k ramified, n .?:: 4 even, ( 2 An-I, quasi-split),
,
41
b) Tables: Involutions, orders and lower central series
SUn(K,F), D = K with K/k unramified, n ~ 4 even, (2A n_d,
SUn(K, F), D ( 2
=K
~
with K /k ramified, n
4 even,
6
unit in k, -c ¢ nrK/k(K),
An-d, F:= (
~ ~ ~ (O~o) 1 (0 10) 0 0
0
,0:
2
I
1
= 2,
1'-2
11
n-2
= n-1,n1';:2
I
2
Bl=ClLJ ~ (1~-)L"I,.;,'B2=CIL ~ (l~_)).;"l";' G + 1, ¥)n-l
0----0--0- . ..
SOn(k, F), D = k with F:=
--0-0 ;::::} 0
n odd, (B1'-l, split), 2
((O~O) ~)
,0:
= n-1,n1';3 =
n-l,1
B= ,,;:3,1, ";:3,1 n+l n-l)~ ( 2 '2 2
2,
VI Some classical groups
42
0-0---0-
...
---0--0 ::::}.
SOn(k, F), D = k with n odd, c a unit in k, -c (j. k*2, (B"-l) 2
F'.-
0 0 0
0 0 1 0 0 c
,a = n - I, n,,-3 = 2, 2
(0 10 ) 0 0 0 0 0
,,;3,1,1, ";3 ,1
B=
n+1 n-1) ,,-1 (- - 2 2 '
2
0------0-0- . . .
Spn(k, F), D = k with n even,
(m + I, m)m for m =
~
---0--0 {:::: 0
(C~, split),
43
b) Tables: Involutions, orders and lower central series
..-0---+-
' ..
- 0 - - {=O
. "
--+--0 {=.
(n+ 1,n)n
..-0---+-
SUn(Q, F), D
= Q with
n odd, (en),
(( o10))n,a=n,
F:~
~
~
(2 -) ) -
, B2 ,,-1 2
(n+ 1,n)n
I
11
,,-1 2
=(
~ (2 1 _)
44
VI Some classical groups
SOn(k, F), D = k with n even, ( 1 D~, split ),
~
F:= ( (0 0)
0 0)
( 1f 0)
,ex
=n-
2, nn-4 = 2, nn-3
n-2,2
= 2,
2
B= n;4 ,1, n;4,1 m.-2
i
and m even
~
and m odd
m-2
(m+ 1,m)-2-,m+ 2, (m,m+ 1)-2-,m for m:= m-3
m.-l
(m + 1, m)-2-, (m + 1)2, (m + 1, m)-2- for m:=
SOn(k,F), D = k with n even, [a unit in k, -[ (j. k*2, (1D%: ),
F'.-
0
0
(0 10 )
0
0
(1 0 [)
0
0
(0 10 ) 0
0 0
0 0
0 (1f 0 1f[)
B=
1 2 (1 0 _)
,ex
=n-
2,nn-4 2
n;4 ,2, n;4,2
2 1
n;4 ,1, n;4 ,1 m.-2
(m + 1, m)-2-, m
771-2
+ 2, (m, m + 1)-2-, m for m:= ~ and m even (m + 1, m)-2-, (m + 1)2, (m + 1, m)-2- for m:= ~ and m odd m.-3
m-l
= 2,nn-3 = 2
45
b) Tables: Involutions, orders and lower central series
SUn(Q, F), D
(n
~"'-< n F
= Q with
even,
+ 1, n) ";2, n + 2, (n, n + 1)
n
skew Hermitian, ( I Dn ),
22, n
SUn(Q, F), D = Q with n odd, F skew Hermitian, ( I Dn ),
F'.-
0 0
q
0 ( ~1 0) 0
(0 10 ) 0
0 0
0 0
0
0 0 0
=n-
1, nn- 2
= 2,
(q7fQ 07fQ)
";3,1, ";3,1
(n
,a
+ 1, n) ";3, (n + 1)2, (n + 1, n) ";1
n;3,1,";3,1
VI Some classical groups
46
SOn(k, F), D = k with n even, ( 2 D!l, quasi-split, two possibilities: ramified 2 splitting field ),
( ~2' !l-I)~ 2
SOn(k, F), D field) ,
= k with n even, c a unit in k,
-c ¢ k*2, ( 2 D~ unramified splitting
·F .-
,a =
n - 2, nn-4 = 2, nn-3 = 2 2
n;4 ,2, n;4 ,2
B= n;4 ,I, n;4 ,1 Tn
2
Tn-2
+ 1, m ) -2-, m + 2, (m, m + 1)-2-, m for m := ~ and m even (m + 1, m) Tn;-3, (m + 1)2, (m + 1, m)"";' for m := ~ and m odd
(
m
47
b) Tables: Involutions, orders and lower central series
SUn (Q, F), D
= Q with
n-2
n even,
F skew Hermitian, ( 2 Dn ),
n-Z
(n+l,n)-2 ,n+2,(n,n+l)-z ,n
SUn(Q, F), D
= Q with
n even,
F skew Hermitian, (2D n , two groups),
B1 = n;2,1,n;2,1
(n, n - l)n
n;2 ,1, n;z ,1
VI Some classical groups
48
~
..
~)
SUn(Q, F), D = Q with n odd, F1 skew Hermitian, ( 2D n), 0 (0 -10) 0 q
0 0
F:=
2 2
B1 =
(n
(0 10 ) 0 0 0 0 0
(2 1 _)
2 0
(2 0 -)
0 0 0
~
0 0
0 0 0
0 0
7rQ
-7rQ
0
4 4 4 4
,a = n - 1, nn-2 "2 3 ,1,,,;3,1,1
,B2 = "~3 ,1'''23,1
+ 1, n) ";3, (n + 1)2, (n + 1, n) n;l
SUn(Q,F), D
= Q with n odd, o
( -10)
o o a= n,
(n,n-1)n
F skew Hermitian, (2D n1 two groups),
= 2,
b) Tables: Involutions, orders and lower central series
49
Here is an example of how to use the tables (VI.I8) Example. (i) One extracts information about topological generators and about the lower central series of a Sylow pro-p--subgroup P of SU5 (K, F) with K /k ramified of degree 2 (type 2A4 , quasi-split) from the table in the following way. The Gram-matrix F of the c-Hermitian form is given as
F=
00001 00010 0 0 1 0 0 o 1 000 1 0 0 0 0
An involution ° : K 5x5 ---7 K 5x5 : x ---7 FXtr F- I is defined where - denotes the Galois automorphism of Kover k. Then SU5(K, F) = {x E K 5X5 1xxo = 1, det(x) = I}. The ring of integers of K is denoted by 0 and the uniformising element is IT. The ring of integers of k is denoted by 0 and the uniformising element is 71". A minimal a-invariant o 0 0 0 0 10000 hereditary order A has exponent matrix 1 1 0 0 0 . This means that every 1 1 100 1 1 110 · 0 f th.r e1ement (aij) I:<.;i,j:S5 E A IS e torm {
aij alj
E 0 E ITO
for i S j . The radl'cal radA has for i > j
1 000 0 1 100 0 exponent matrix 1 1 1 0 0 1 1 1 1 0 1 111 1 The set {x E radAlxo = ~x} is denoted by (radAt. One observes from the table that a = 5. This means that A/radA is the direct product of a simple o/7ro-algebras; here A/radA = (O/ITO)5. It follows from (VLI3) that P = (radA)-CLP n SU where C LP is induced by the Cayley map. Therefore, to obtain generators for the group, one needs (Lie-) generators for (radA) - which are of the form dl (aI, ... ,an) with aI,"" an E K satisfying F(dI(ar, ... , an))tr F- I = -dI(aI,.'" an). This condition imposes linear equations on the entries aI, ... ,an- One applies the Cayley map CLP : x H (1 - x) (1 + x I to these elements. Specifying the field k to be
t
dl (1,0,0, -1,0) =
0 0 0 0 0
1 0 0 0 0
0 0 0 0 0
0 0 0 0 0 0 0 -1 0 0
, d I (O,I,-I,O,O), d1(0,0,0,0,IT)
Z;,
as Lie lattice over Zp, where IT 2 = p or IT 2 = cp for some non-square unit c E depending on the choice of K. Topological generators for the group are the images of
VI Some classical groups
50
these generators of (radA)- under the Cayley map 1 -2 o 1 o 0 o 0 o 0
0 0 1 0 0
0 0 0 1 0
CLp.
1 000 0 o 100 0 o 0 100 00010 -2Il 0 0 0 1
1 0 0 0 0 o 1 -2 -2 0 o 0 1 2 0 o 0 0 1 0 o 0 0 0 1
0 0 0 2 1
These are
The dimension matrices B r can be used to read off the orders of the factors of the lower central series of P since rad(A)- is saturated, cr. (VI.9). These matrices, given in the table, read as
B 1 -- (b(ll) s,t O'Ss,t'Sa-l -
1 1 1 1 1
1 1 1 1
1 1 1
1 0
0
• ,
B2
= (b(ll) s,t =
1 1 1 1 1 1 1 0 0
1 1 0
1 1
1
The dimension matrices B r = (b~ri)o<s t
_
1
_
bi == Et-s:i b~~i where s, t E {O, 1, ... , a - I} and a(r - 1) S i < ar and all indices taken modulo a. The sequence (b 1 , b2 , •• .) is thus given by 3,2,3,2,3,2,3,2,3,2, ... , in particular 3 == b5 == b15 == b25 = .... But this value must be reduced by 1 as it is indicated by • above the matrix B 1 • The reason for this is that, if i == 5 mod 10, then it(P)/ii+l(P) is generated by diagonal matrices, and the determinant condition reduces the dimension by 1. The orders of the sections of the lower central series of P for a general p-adic field k with inertia degree fare p3 f , p2 f p3 f ,p2 f , p2 f , p2 f , p3 f , p2 f , p3f, p2 f after which the pattern repeats. 1
(ii) Consider the Sylow pro-3-subgroup of S05('Cb, F) (type B2 , split) with
F=
0 0 0 1 0
0 0 1 0 0
0 1 0 0 0
1 0 0 0 0
0 0 0 0 3
The involution defining S05('Cb, F) is given by ~x5 ----'t ~X5 : x H Fx tr F- 1 . Let A be an associated c-invariant minimal hereditary order A. The number of simple components of the A/radA is a = 4 and nl ::: 2. Therefore (radAt is generated by C
0 0 0 0 0
1 0 0 0 0
:
0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0
d1 ( (0, 1), (-1, 0) tr , 0, 0) , d1 ( (0, 0), (0, 0) tr, 1, - 3) .
51
b) Tables: Involutions, orders and lower central series
Therefore topological generators for the Sylow pro-p-subgroup of S05((h, F) are 1 -2 0 0 0 0 1 0 0 0 0 0 1 2 0 0 0 0 1 0 0 0 0 0 1
1 0 0 0 0
0 -2 0 0 1 0 2 0 0 1 0 0 0 0 1 0 0 0 0 1
1 0 0 -6 6
0 1 0 0 0
0 0 1 0 0
0 0 0 0 0 0 1 -2 0 1
The lower central series has the pattern 33 , 32,3 3 ,3 2 after which the pattern repeats. Proof of the claims in the tables. (i) Completeness of enumeration of groups. Cf. [Sat 71] for the classification of simple algebraic groups over Qp up to isogeny. For each Dynkin diagram one is left with the problem of enumerating the Hermitian, bilinear or skew-Hermitian forms in each case, cf. [Tsu 61] for the skew-Hermitian case over quaternion algebras and [Scha 85] for the other cases. (ii) Maximality of the SU(A) as pro-p-subgroups of SUo By (VI.8) a maximal pro-p-subgroup of SU is of the form (1 + radA) n SU = (radAtCLP for some hereditary a-invariant order A in A. Since radA c radA I if and only if Al c A for hereditary orders, one is done in the case A is a minimal hereditary order. In the other cases one quickly convinces oneself that the a-invariant hereditary orders r contained in A yield (radr)- = (radAt. It is easy to see that our choices of A make (radA)- semi-saturated in all cases, even for SU3 (K), K/k ramified, p = 3. (iii) Lower central series of SU(A). We only have to prove that (radA)- is saturated in all cases except SU3 (K), K/k ramified, p = 3. Prior to the discussion of the general case, let us remark that with the help of Lemma (VI.l1) or with a symbolic manipulation package like MAPLE, one can check any of the cases of (VI.17) for a given degree n with K arbitrary. Let us do SU3 (K) with K / k ramified as an example. We arrange the elements of (radAt according to the layers ((radA)it /((radA)i+lt:
al = d l (l,-l,O), a2 = dl(O,O,II); bl :::: [aI, a2] = d2(0, -II, -II); CI = [aI, bl ] = do( -II, 2II, -II), ([a2' bd = 0); d l :::: [al,cI] = d l (3II,3II,0), ([a2,cI] = 0); el = [al,d l ] = d2(6II,0,0), e2 = [a2,d l ] = d2(0,-3II 2,3II 2); II = [aI, e2] = do( -3II 2, 0, 3II2), ([aI, ell = 0, [a2' ell = 2II, [a2' e2] = 0); [aI, II] = 3II2al' [a2' II] = - 6II2a 2. Note in the third layer there is just one element, since the trace must be zero, i.e. B I has a •. This calculation proves that (radAt is saturated if and only if p > 3. Passing to the general case, recall from (VI.14) that the involution ° induces a permutation i on the components X o, ... ,X n - l of A/radA with 0 or 1 or 2 fixed points.
VI Some classical groups
52
Accordingly the number of fixed points on the set of the components Yo(,:) , ... 'Y~~I,a-l+i follows the pattern 0,2,0,2, ... or 1,1,1, ... or 2,0,2,0, . ... In the setup of table (VLI7) the fixed points can be visualised in the same positions where the Gram matrix F has non zero entries if partioned according to A. For instance in the first case the permutation i is given by (0, a - 1)(1, a ~ 2) ... (i - 1, ~ + 1). Claim: The Lie lattices of table (VLI7) are saturated, if the first two or three in each series are saturated. Proof. To prove that [(radA)-, ((radA)Z-I)-] = ((radA)it, it suffices to see that any component of ((radA)it /((radA)z+I)- can be obtained from a component of ((radA)i-l)- /((radA)i)- and one of (radAt /((radA)2t by taking Lie commutators (of representatives). For involutions of the second kind, where the first dimension matrix B 1 gets a ., one has to allow for the fact that traces are zero, 1. e. which might force one to take two components rather than one in case i = a or i - I = a. The )- (= C(i) components of ((radA)i) - /((radA)i+l)- are of the form C(t) t~,s~ tL,SL ) s,t ;= (y(i) s,t EEl y:(t) with t - 8 :: i mod a and 8 #- ti or C~~l~ := (Ys(,~J- with 8i - 8 == i mod a. In the first case one would have to look at commutators of C~~t--=-l; with C~~L or of c~~-I~l . h C(I) . ht b e 0' h' h case one IS . WIt s-l,s' N ote, one 0 f t h e two C(i-l) s,t-l or C(i-l) s-l,t mIg ,Ill W lC forced to take the other one. But note also that none of the C~:~+1 is zero, which is very important and forces the choice of A as it stands. The other problem is that Lie commutators do not just induce o/1ro-bilinear maps (i-l) C(I) C(t) C st-l X t-lt ~ st I 1 j
or
C(I) C(t-l) C(i) s-ls X s-It ~ st , I 1
whose surjectivity one has to check, but that the image involves two rather than one component by chance. (This actually must happen on the diagonal because of the trace condition for involutions of the second kind.) The key idea is now that both issues on these bilinear maps can be checked in a smaller environment, i.e. for a Lie lattice of essentially the same type with smaller n, provided n is beyond a certain a-I size. To be more specific, write A = . EEl Aij , where Aoo , ... ,A a - 1,a-l are maximal
°
z,J=O
orders (covering the components of A/radA) and A ij are Aiz-A,wbimodules such that the involution of A permutes the AZJ according to i, i.e. Aij = AJi,n. That this is possible in all cases one immediately sees from the simple form of F in (VI.I7). To pass from A to a hopefully smaller A' to check the above statements on c~~l one chooses A' = . EEl Aij where S is chosen so that (x, y) E S =} (Yi, Xi), (x, x), (y, x) E S 0
zJES
to make A' an hereditary order with involution and (s, t) E S together with any of the above choices (8 -1, t) and (8 -1, 8) or (s, t -1) and (t -1, t) to have it represent the proper bilinear map. Applying this process once or twice respectively its version with a slightly more generous choice for S will reduce the n and if n 2: no + 1 where no is the smallest possible degree to yield a simple algebraic group. This proves the claim. To finish the proof one only has to investigate the first two Lie lattices of each series either by hand and (VI. 11 ) or by computer. q.e.d.
In conclusion of this chapter, remark that in most cases, where the characteristic of K is zero the groups SU(A)/Z(SU(A)) are maximal p-groups with the obvious exception where K /Qp has automorphisms of p- power order and the less obvious exceptions in the case of involutions of the second kind, where rational automorphisms
c) Table of patterns for lower central series
53
might turn up or if the group has a root system of type D 4 and a diagram automorphism might turn up.
c)
Table of patterns for lower central series
The pattern of the lower central series is encoded in the following way. For each factor group "'Ii/"'Ii+1 the number of copies of cyclic groups G p divided by f is given where f is the inertia degree of k. A sequence of numbers enclosed in brackets and raised to some power x means the repetition of this sequence x-times. The whole sequence is overlined to indicate that it repeats itself. E.g. nn-l, n - 1 means "'II / "'12 "" = GIn p " " ,"'In-l / "'In "" = GIn p ,"'I'f!, / "'In+l "" = GI(n-l) p an d so on. (VI.I9) Table.
lAn _ 1 split, n;::: 2, pfn, cf. Chapter VI 2 A n - 1 quasi-split, unramified splitting field 2 A n - 1 non-split, unramified splitting field, n even
(m + 1, m)m, m, (m, m
+ l)m, m
for m
= n21
2 An - 1
quasi-split, ramified splitting field, n odd
2 An - 1
quasi-split, ramified splitting field, n even non-split, ramified splitting field, n even
2 An - 1
IThe same holds probably for the non-split groups, but details have not yet been checked.
VI Some classical groups
54
(n+l,n)n
B n split B n non-split Cn split Cn non-split
(n
n-2 n-2 + 1, n)-2-, n + 2, (n, n + 1)-2-, n
1D n 1D n 1 Dn 2 Dn
2D n
split, n even non-split, n even, F orthogonal non-split, n even, F skew Hermitian form over quaternion algebra quasi-split, n even, unramified splitting field non-split, n even, F 1 skew Hermitian form over quaternion algebra
n
3
2
n
1
(n+l,n)-2 ,(n+l) ,(n+l,n)-2 1D n 1
Dn
1 Dn
2 Dn 2D n
split, n odd non-split, n odd, F orthogonal non-split, n odd, F skew Hermitian form over quaternion algebra quasi-split, n odd, unramified splitting field non-split, n odd, F 1 skew Hermitian form over quaternion algebra
(n,n-l)n 2D n
quasi-split, ramified splitting field 2 Dn non-split, F2 , F3 skew Hermitian form over quaternion algebra
VII
Some thin groups
In this section we exhibit a class of insoluble linear pro-p--groups that have average width arbitrarily close to 1. Let K be either Qp or lFp((t)), and let V be a vector space over K of dimension pO for some Q > O. Let VI, V2, .. . , v p" be a basis for V. For any integer i, define Vi = 1fk Vj where 1f is an element of K of value 1, and i = j + kpo, with 1 :S j :S pO. Let Vi denote the O-module generated by {Vj : j ?: i}. So Vi+1 is of index p in Vi for all i. Now let P be the subgroup of 5L(0, VI) that normalises Vi and centralises Vi/Vi+1 for all i. Then P is a Sylow pro-p--subgroup of 5L( 0, VI), and is generated by {91, 92, ... , 9p" }, where 9i maps Vi to Vi + Vi+1 and fixes Vj for 1 :S j :S pO, j =I- i. Let z be the K -automorphism of V that sends Vi to Vi+1 for all i. Then zP" is the scalar 1f I, where I is the identity. Now z normalises P, and we define G to be the subgroup of PGL(V) generated by z and P modulo scalars. Most of our calculations take place in P. Clearly P has a filtration P = P2 > "', where Pi = n~:ICp(Vj/Vj+d = n~ICp(Vj/Vj+I)'
H >
This filtration is in fact the lower central series of P, though we shall not need this fact. (VII.1) Lemma. Pz/ Pi+1 is an elementary abelian 9roup of rank pO if itO mod pO, of rank pO - 1 if i _ 0 mod pO. Proof. Let 9 E Pi, so Vj9 = Vj + ajvj+i mod Vi+j+I' Here the aj are integers in the range 0 to p - 1, which we also take to be elements of lFp . Now 9 t---+ (aI, a2, ... , ap") defines an injection (Ji of Pz/ Pi+1 into lFt. The only restriction on the aj arises from the fact that we are working in the special linear group, so the determinant of 9 must be 1. If itO mod pO then defining 9 by Vj9 = Vj + ajVj+i where aj is non-zero for just one value of j in the range [1, ... ,pO] implies that 9 is a transvection, and hence has determinant 1. So there is no restriction on the aj if itO mod pO. If i 0 mod pO then the condition that 9 should have determinant 1 implies at once that al + a2 + ... + ap" - 0 mod p. This is the only condition on the aj, as one can define 9 to have a diagonal matrix, with respect to the basis {VI, V2, ... , Vp"}, where one diagonal entry is b = 1 + a1f k , where a =I- 0, and i = kpo, another diagonal entry is b- I , and the other diagonal entries are 1. This completes the proof. q.e.d.
=
Note that P is a subgroup of 5L(0, VI), but we are really interested in P5L(0, vI). For odd p this will make no difference, as has no non-trivial units congruent to 1 modulo (1f) of order a power of p in this case. However, if p = 2 and we are in characteristic 0, then -1 is such a unit. This implies that if we replace P by its image in the projective linear group, the dimension of Pp" / Pp"+1 is one less in this case. A similar complication occurs if we allow K to be a totally ramified extension of Qp, as 1-units of order a power of p may also arise in this case.
°
VII Some thin groups
56
(VII.2) Lemma. Let A denote the group algebra IFpCp'" of the cyclic group generated by z, and let ¢ : IF;'" -t A be defined by (aI, a2, ... ,ap") H L ajzj-I. Then fJi¢ is a z-module monomorphism, where fJi is defined as in the proof of Lemma (VI!. i). Proof. It is easy to see that z acts in both cases by cyclic permutation of the aj' q.e.d. (Vl1.3) Lemma. The action of G on P~/ P~+I is uniserial. The image of fJ i¢ is the whole of A if i -¥. 0 mod pO, and is the augmentation ideal of A otherwise. Proof. The first statement follows from the fact that z acts uniserially on A, and the other statements are clear. q.e.d.
If 9 E Pi, we shall say that 9 lies in the i-th layer of Pi if the image of 9 modulo PHI under fJi ¢ lies in the i-th power of the augmentation ideal of A. For i = po - 1 we shall say that 9 lies in the bottom layer of Pi, and similarly for the next to bottom layer. We now consider the effect of commutation by the element y of P defined by VIY = VI + V2, and ViY = Vi for 1 < i ~ pO. Clearly y centralises Pd Pi+1 for all i. (VIlA) Lemma. Let 9 E Pi have image in Pd PHI corresponding under fJi ¢ to L gJ zj - l • Then [g, y] lies in P~+l, and has image in Pi+I/ Pi+2 corresponding under fJHI ¢ to gpo< + l-iZP'" -i - g2. Proof. This is a very simple calculation. Note that the first layer of Pi is not the top layer unless i also a zero-th layer.
q.e.d.
= 0 mod po as there is
(VII.5) Lemma. Let 9 lie in the bottom layer of Pi, where i =J- 0 mod pO. Then [g, z] lies in the first layer of PHI, and not in the second. Proof. To say that 9 lies in the bottom layer of Pi is to say that the gi are all equal, and this clearly implies that [g, y] defines an element in the augmentation ideal of A, but not in the square of the augmentation ideal, since i =J- 0 mod pO. Note that, if i = 0 mod po then 9 is congruent, modulo scalars, to an element of Il+I' q.e.d. (VII.6) Lemma. Let 9 lie in the next to bottom layer of Pi' but not in the bottom layer, where i = 0 mod pO. Then [g, y] lies in the zero-th layer of Pi+I , and not in the first layer. Proof. The image of 9 under fJi ¢ is c(l +2z+3z 2+ .. '+pOZP"'-I) for some non-zero constant c, which we may take to be 1. Then we see at once that the image of [g, y] modulo Pi+2 under fJi + 1 ¢ is 1, as required. q.e.d. To complete our analysis of the lower central series of G, we need to find how low down the series the top layers of the Pi come.
57
(VII.7) Lemma. Let 9 E Pi, where pO ii, and let v(i + 1) = fJ, where v denotes the
p-adic valuation of Z. Then [g, y] lies in the top layer of Pi+I , and not in the second layer, if 9 lies in the pO - p/3 - 1-th layer of Pi/Pi+I , and no lower. Proof. Let 9 lie in the pO - t- th layer of G, but no lower. Then we may take g(}i to be (b I ,b2 , ... ,bp<», where bj = (-l)j(P<>J-t) = (j~~~I). Then [g,Y](}i+l == (C I ,C 2, .. "ep.<>), where CI = -~, and Cp<>-i+I = bp <> -~+I' SO we need to find the least t for whIch
b2 i- bp<>-i+I. If t = 1 then bj == bk for all j, k. So t > 1. If t = 2 then bj = j + 1 for all j, so t = 2 unless i == -1 mod p. Now assume that i - 1 mod p. More precisely, let v(i + 1) == fJ. If m n = Lj njpl, where 0 :::; mj, nj :::; p - 1 for all j then
(:) "' If (:;)
modp.
Using this formula, if pO - k = 1 + k/3p/3 + k/3+I + ..., and t == to 0:::; kj , tj :::; P - 1 for all j, then b2 (t~l) modp, and
=
bp<>-k+I
- (to
=
= Lj mjpi, and
+ tIP + ..., where
+ 1 + tIP + ... + t/3_IpI3~1 + (t/3 + kfJ)p13 + ...) /3 I /3 to + tIP + ... + t/3-IP - + t/3P + ...
d mo p.
Assume now that p is odd. The above expression for bp<>-k+I is not in the form that allows us to apply our formula in all cases. For example, if to = 0 or to = P - 1. But in these cases b2 and bp<>-k+l are both zero modulo p. It is now easy to see that the least value of t that makes these two terms not equal modulo p is p/3 + 1, as claimed. For p = 2 the argument is essentially the same. q.e.d. We can now read off the structure of the lower central series of G.
(VII.8) Lemma. Let G i denote G / Pi, where i > I, and let p be odd. Let Ci be the nilpotency class of Gi . The last term of the lower central series of Gi is of order p, and corresponds to the bottom layer of Pi-I! Pi, and Ci+I == Ci + pO - 1. The last pO -1 terms of the lower central series of G i are of order p. For j :::; Ci the j -th lower central factors of G i and G i + I are equal, except that the Ci - p/3 + 1-st term of the lower central series of G i +1 is the direct product of the corresponding lower central factor for G i
with a p-cycle, where fJ = v(i
+ 2).
(VII.9) Theorem. The lower central series of G has the following structure if p is odd. G /"Y2 (G) 'V Cp X Cp<>, "Yi(G) /"Yi+I (G) ~ C p x C p if i = r(pO -1) - pI3 + 2 for some r E N such that v(r + 1) = fJ < a . Otherwise "Yi(G)/"Yi+I (G) 'V Cpo In particular the
average width is
.
Wa(G) = n-.+oo lim 10gp(IG: "Yn(G)I)/n = 1 + l/po.
Proof. The claims follow from Lemma (VII.8).
q.e.d.
VII Some thin groups
58
If p = 2 things are slightly different. Assume in this case that 0: ~ 2. Then, since one has to factor out the centre in characteristic zero, Pp'" / Pp"'+l is of dimension pO - 2, but is not trivial as 0: ~ 2. Otherwise the same analysis holds as above.
(VII.10) Theorem. The lower central series of G has the following structure if p = 2. G/'Y2(G) C 2 x C 2"" 'Yi(G)/'Yi+l(G) ~ C 2 x C 2 ifi = r(2° -1) - 2(1 + 2 for some r E N such that v(r + 1) = (3 < 0: and, in characteristic zero, r i- 2°, that is, i i- 2°(2° - 1) + 1 . Otherwise 'Yi(G)/'Yi+l(G) ~ C 2 . In particular the average width zs 'V
VIn a)
Algorithms on fields
Arithmetic in 0
We have four sets of algorithms, doing fast or slow arithmetic in characteristic 0 or p in a local ring 0 modulo some appropriate power of the (unique) maximal ideal with finite residue class field F. We can only use fast arithmetic if the ring is small enough for us to be able to pack each element into a machine word. In characteristic 0 we also need to be able to construct certain tables, which imposes a stricter condition on the size of the ring. Most examples constructed for these notes were computed using fast arithmetic taking up to 20 seconds; but some required higher accuracy, and needed to be done in slow arithmetic, taking about 8 minutes (on a HP 9000/730). The calculations needing slow arithmetic were those where the field was of relatively high degree over Qp. Arithmetic in the residue class field F is done in the standard way. That is to say, a primitive element a of F is found, and elements can be stored in additive form as polynomials of degree less than f in a, where F :::: GF(pf), or in multiplicative form as a power of a (or zero). Lookup tables are constructed to convert between these representations. Of course the additive form is packed into an integer. We also construct a table that gives, for each i, the integer j such that a i + 1 :::: aj (with suitable conventions for the zero element of the field) and add by using this formula. The characteristic p case is entirely straightforward. Calculating in F[[ t]]/ (t m ), in slow arithmetic, we store ring elements as arrays of length m, and we simply add, multiply, and invert units, using the naive algorithms. In the fast case, we encode E Citi as E Ciqi where the Ci lie in the range 0 ::; Ci < q with q :::: pl. It is a triviality to encode the algorithms for performing these operations without unpacking the elements into arrays. So we perform addition in O(m) integer operations, and multiplication in O(m 2 ) operations. Slow arithmetic in characteristic 0 is more delicate. The basic idea is to regard an element of 0 as a polynomial over Zp of degree f - 1 in a and degree e - 1 in 7r. So we store elements as double arrays of integers, where the entries are taken modulo some power of p. One problem is that we wish to compute modulo 7r m for some value of m, and this ideal need not be of the form pkO; so we work modulo pk for the least k such that 7r m ::; pkO, and then have a test for equality of ring elements that decides whether the value of the difference is at least m. A technical problem arises in that, in order to obtain answers that are correct modulo 7r m , we sometimes need, when considering the action of automorphisms, to work to a higher degree of accuracy. A more interesting problem arises with the element a. This is now a representative in 0 of a primitive element of F, and is required to be a precise root of unity. While this is not needed for us to be able to perform slow arithmetic, it is needed for various reasons of efficiency. We start from the minimum polynomial fo(x) of a modulo 7r, which is, in effect, the minimum polynomial of the primitive element used to define F. This polynomial is lifted to a polynomial !l (x) over 0 mod 7r m by interpreting the coefficients in (Z mod p) as elements of (Z mod pk). Let ao be a root of this polyi nomial. We then compute a as limi ab . This gives an exact root of unity, congruent
VIn Algorithms on fields
60
to ao modulo 1r. We then compute the minimal polynomial h(x) of a which is an irreducible f~ctor of the cyclotomic polynomial, and use h(x) instead of h(x) for further calculations in our ring.
b)
Calculating automorphisms in characteristic 0
Any automorphism of the ring 0 will take a to a where 0 ~ i < f, and 11" to some zero, in 0, of the image of the Eisenstein polynomial under the automorphism of the inertia field T that takes a to a i . The basic problem then is to find all zeros of a given polynomial over O. By Hensel's lemma, a sufficiently accurate approximation to a zero does lift to a unique precise zero, so we are justified in working in 0/ 1I" kO provided that k is big enough to ensure that the Hensel condition is satisfied, and that the accuracy is sufficient for our other calculations. Since the Hensel condition is liable to be the stronger of the two, we may have to compute the automorphisms using slow arithmetic even when our other calculations can be done using fast arithmetic. In practice, we always compute the automorphism using slow arithmetic, as this is fast enough for this purpose. If our matrices have entries in 0/1I" mO, we need to take k to be strictly greater than m, since we need on occasion to divide by 11", and this reduces the accuracy of our approximation by 1. We now work modulo successive powers of 11", finding allliftings of zeros accurate to one power of 11" to zeros accurate to the next power. Also for this procedure one might need a larger quotient of the ring than 0/ 1I" mO to ensure that the approximation of a zero coincides with a zero modulo 1I"m. The size of the ring depends on the value of the derivative of the Eisenstein polynomial. This naive approach is quite fast enough for polynomials of the small degrees needed. Each of these zeros, together with the corresponding image of a, defines an automorphism of 0, and we calculate the multiplication table, and store this and other basic information about the automorphisms. pi
c)
The group of units of 0
If 0 is of characteristic p, the group of units of 0 is easy to compute. In this case o is lFq [[t]] where q = pI, and the group of units is a free Zp-module with topological basis {I + tt} where i runs over all natural numbers prime to p. We now consider the more interesting case of characteristic O. There is a natural homomorphism of 0 onto the residue class field lFq , and this homomorphism induces a split homomorphism of the group U of units of 0 onto the cyclic group of order q - 1. The kernel of this homomorphism is the group U1 of I-units of O. As it is well known, U1 is the direct product of a free Zp-module of rank n = ef and a finite cyclic p-group occurring from roots of unity. The finite part, which may be trivial, is computed as follows. Let 1 + x be of order p, where x is of positive value. Then (1 + x)P = 1 + px + ... + x P, and for this to be 1, we need v(px) = v(x P), and hence, if the valuation is chosen so that v(1I") = 1, we need e + v(x) = pv(x), and hence e must be a multiple of p - 1. Taking this calculation to a greater degree of accuracy, let x = b1l"k modulo higher terms and b some unit. Then computing (1 + x)P = 1
d) Fast arithmetic in characteristic 0
61
gives pbJr k = bPJr kp modulo higher terms. So if p = C7r e modulo higher terms, we have bCJr e +k = IJPJr kp , so C must be a p - 1th power. If these conditions (which are vacuous for p = 2) are satisfied, it follows from Hensel's lemma that x as required will exist, and will have value (e - l)lp. If now e - 1 is a multiple of pI for some t > 1, then x will have a pt_Ith root, and so the torsion part of U1 will have order pt. A basis for a complement to the torsion part of UI can be constructed as follows. If UI is torsion-free we take the set of elements of the form 1 + aiJrJ, where 0 ::: i ::::; f and 1 S ) < pel (p - 1) and) is prime to p. If UI is not torsion-free, we make two changes. First we add a new generator 1 + afJrpe/(p-l) where a f is not in the image of the endomorphism J-t of the additive group of lFq defined by x H CX + x p . Secondly we remove a generator 1 + aZJrJ where) = e/(p - l)plJ if plJ is the highest power of p that divides e, and i is the least integer such that a~Pv J-t is an lFp-linear combination of the a UP" J-t for u < i. (Here we regard a as an element of lFq .) The above facts are easy to verify. For details see [Iws 86] or [Has 49].
d)
Fast arithmetic in characteristic 0
This is done using logarithm tables. Suppose that UI is torsion free. It is clear from the above that every element of 01 JrmO can be written uniquely in the form JrAak TI(l +azJrj)U,.] , where i and) are chosen as above so that the terms in the product are our chosen free generators for U I , and the exponents take values bounded by easily computed functions of m. Call this the multiplzcative form of the element. Similarly, any element can be written uniquely as a ~>linear combination of {azJrj} for 0 ::; i < f and 0 S ) < min(m, e), where the coefficients are non-negative, and bounded by a suitable function of m. Call this the additive form of the element. Now it is easy to add elements in additive form, and to multiply elements in multiplicative form. So when we are going to use fast arithmetic, we first go through all elements in multiplicative form, and compute, and store, their additive form, simultaneously constructing the inverse table. The problem is to construct these tables quickly. This is done by using a number of elementary tricks, for example x(l + aiJr) = x(l + aZ-IJr)a - xa + x. Now addition and multiplication by a are fast operations on elements in additive form, so if we need to compute an element y of the form akJrA TI(l +azJrJ)U,.] , and we have already computed akJrA TI(l + aiJrj)V,,] both in the case v10,Jo = Uio,jo - 1, Vzo-I,jo = Uio-l,jo + 1 for some fixed i o and )0, and U and v agree everywhere else, to give an element z, and also in the case Vzo,jo = U~o,jo - 1 for the same i o and )0, with U and v again agreeing elsewhere, to give an element x. Then by the above formula, y = Z - xa + x, and so is quickly computed. The case when U1 has torsion is harder, but similar principles are used. Computing these tables efficiently proved to be a very subtle exercise, and we congratulate Colin Murgatroyd and Matthias Zumbroich for the skill with which they wrote this code.
IX
a)
Fields of small degree
Extensions of t1h of degree 2, 3 and 4
Extensions with a soluble Galois group can be obtained by subsequent abelian extensions which are described by local class field theory. In particular, for every closed subgroup H of finite index in the multiplicative group of a local field k there exists a unique extension K of k in the algebraic closure k satisfying Gal(K, k) ~ k* / H where H = NK/k(K) is the norm group of Kover k, d. [Iws 86] p. 98-100. The number of different extensions of Qp in its algebraic closure Qp of given inertia degree and ramification index is also known from [Kra 62]. The (unique up to isomorphism) totally unramified extension of degree n over Qp is given by a primitive (pn - l)th root of unity. In the following an irreducible factor of degree n of the cyclotomic polynomial over Q is given as a minimal polynomial. The Galois group of this extension is cyclic of order n. The maximal abelian extension of exponent p of a local field K is denoted by Kab,p. Note, IKab,p : KI :::: IK* : (K*)PI is finite. (IX.l) Lemma. Let k be a normal extension field of finite degree over Qp. Then kab,p is a Galois extension. Proof. Let g E Aut(Qp, Qp). Then k~b,p and kab,p are both extensions of k because k is normal. Moreover the Galois groups over k of the two fields are isomorphic and Gal(kab,k) ~ k*/(k*)P which is the largest exponent p factor group of k*. Therefore k~b,p = kab,g and kab,p is normal. q.e.d. (IX.2) Lemma. There are 7 non-isomorphic extensions of «h of degree 2 in its algebraic closure ~. Minimal polynomials can he chosen as follows. Unramified extension: x 2 + x + 1. Ramified extension: x 2 + 2, x 2 + 6, x 2 - 2, x 2 - 6, x 2 + 2x + 2, x 2 + 2x + 6. Proof. The extensions correspond to the 7 different subgroups of order 2 in Constructing the GaI((Qp)ab,2,«h) ~ Q;/(Q;)2 ~ Z/2Z ffi (-1) ffi Zd2Z 2 rv extensions by roots of the equations x 2 - a one chooses a set of representatives of the 7 classes ofQ;/(Q2)2. By suitable substitutions one gets the enumerated polynomials as Eisenstein polynomials or, in the unramified case, as a factor of the cyclotomic polynomial. q.e.d.
Ci.
(IX.3) Lemma. There are 2 non-isomorphic extensions of Q2 of degree 3 in its algebraic closure ~. Minimal polynomials can be chosen as follows. Define a := (7 + (i + (i.e. a root of x 2 + X + 2) where (7 is a primitive 7th root of unity. Unramified extension: x 3 + (1 + a)x 2 + ax ~ 1 with Galois group C 3 · Ramified extension: x 3 - 2 with Galois group 53.
(i
63
a) Extensions of ((h of degree 2,3 and 4
Proof. Since Ql;/(Q2)3 ~ Z/3Z there is exactly one normal abelian extension with Galois group C 3 which is clearly the unramified one. To construct an extension with Galois group S3 one constructs an extension of degree 3 on top of an extension of degree 2 in a non abelian way. There is only one extension of ((h with Galois group S3 namely the above with intermediate field ((h[(3], where (3 is a primitive root of unity, since all other fields of degree 2 (cf. Lemma (IX.2)) yield abelian extensions. q.e.d. (IXA) Lemma. There are 59 non-isomorphic extensions of ((h of degree 4 in its algebraic closure ((h. Minimal polynomials can be chosen as follows.
Define a := (fs + (r~ + ({~ + (r~ for i = 1 or i = 7 where (15 is a primitive 15 th root of unity, i. e. a has mmimal polynomial y2 - Y + 4. Then, x 4 + (a - l)x 3 - 2x 2 - ax + 1 is a minimal polynomial for the un ramified extension with Galms group C 4 Totally ramified extensions no minimal polynomial 33 x 4 +4x 2 +2 34 x 4 + 12x2 + 18 35 x 4 + 20x 2 + 50 36 x 4 + 28x 2 + 98 37 x 4 + 4x 2 + 10 38 x 4 + 12x 2 + 90 39 x 4 + 20x 2 + 250 40 x 4 + 28x 2 + 490
with Galois group C 4 : factorisation over intermediate field (x 2 +2+ 2)(x 2 +2- 2) (x 2 + 6 + 3V2)(x 2 + 6 - 3V2) (x 2 + 10 + 5V2)(x 2 + 10 - 5V2) (x 2 + 14 + 7V2)(x 2 + 14 - 7V2) (x 2 + 2 + yC6)(x 2 + 2 - yC6) (x 2 + 6 + 3yC6)(x 2 + 6 - 3yC6) (x 2 + 10 + 5yC6)(x 2 + 10 - 5yC6) (x 2 + 14 + 7yC6)(x 2 + 14 - 7yC6)
Totally ramified extensions with no minimal polynomial 41 x 4 + 2x 2 - 4x + 2 42 x 4 + 14x 2 - 20x + 14 43 x 4 + 4x 3 + 6x 2 + 4x + 10 44 x 4 + 4x 3 + 2x 2 - 4x + 6 Totally ramified extensions no minimal polynomial 45 x 4 + 2x 3 + 2x 2 + 2 46 x 4 + 2x + 2 47 x 4 + 4x + 2 48 x 4 + 4x 2 + 4x + 2
Galois group V4 : roots to be adjoined to intermediate fields 2, yC2, y'=1 V2, J6,
v=s
y'=1,
J6,
v=o
V=S, JIG, yC2
not bi-quadratically constructible: Galois group A4 S4 S4 S4
64
IX Fields of small degree
Totally ramified extensions with Galois group D 8 : no minimal polynomial factorisation over intermediate field 4 1 x - 2 (x 2 + 2)(x 2 2) 4 2 2 2 x - 18 (x + 3V2)(x - 3V2) 4 3 2 3 x + 4x + 8x + 8x + 2 (x 2 + 2x + 2 + V2)(x 2 + 2x + 2 - V2) 4 3 4 x + 4x + 16x 2 + 24x - 14 (x 2 + 2x + 6 - 5V2)(x 2 + 2x + 6 + 5V2) 4 5 x - 6 (x 2 + J6)(x 2 - V6) 4 6 x - 54 (x 2 + 3J6)(x 2 - 3V6) 4 2 7 x + 4x - 2 (x 2 + 2 - V6)(x 2 + 2 + J6) 4 8 x + 12x 2 - 18 (x 2 + 6 - 3J6)(x 2 + 6 + 3J6) 4 9 x + 4x 3 + 16x 2 + 24x - 114 (x 2 + 2x + 6 - 5J6) (x 2 + 2x + 6 + 5)6) 2 (x + 2x + 2 + J6) (x 2 + 2x + 2 - V6) 4 11 x + 2 (x 2 + yC2)(x 2 _ yC2) 12 x 4 + 18 (x 2 + 3yC2)(x 2 - 3yC2) 4 2 13 x + 4x + 6 (x 2 + 2 + yC2)(x 2 + 2 - yC2) 4 2 14 x + 12x + 54 (x 2 + 6 + 3yC2)(x 2 + 6 - 3yC2) 4 15 x + 4x 3 + 8x 2 + 8x + 6 (x 2 + 2x + 2 + yC2) (x 2 + 2x + 2 - yC2) 4 3 2 16 x + 4x + 16x + 24x + 86 (x 2 + 2x + 6 + 5yC2) (x 2 + 2x + 6 - 5yC2) 17 x 4 + 6 (x 2 + y'=6)(x 2 - y'=6) 4 18 x + 54 (x 2 + 3y'=6)(x 2 - 3y'=6) 4 3 2 19 x + 4x + 8x + 8x + 10 (x 2 + 2x + 2 + y'=6) (x 2 + 2x + 2 - y'=6) 4 3 20 x + 4x + 16x 2 + 24x + 186 (x 2 + 2x + 6 + 5v!=6) (x 2 + 2x + 6 - 5y'=6) 4 2 21 x - 2x + 2 (x 2 - 1 + v=I)(x 2 - 1 - v=I) 22 x 4 - 6x 2 + 18 (x 2 - 3 - 3v=I)(x 2 - 3 + 3v=I) 4 2 2 23 x + 6x + 10 (x + 3 + v=I)(x 2 + 3 - v'=T) 4 2 2 24 x + 18x + 90 (x + 9 + 3v=I) (x 2 + 9 - 3v'=T) 4 3 2 25 x + 4x + 2x - 4x + 2 (x 2 + 2x - 1 + v=I) (x 2 + 2x - 1 - v=I) (x 2 + (-1 - v=I)x - 1 + v'=T) (x 2 + (~1 + v=I)x - 1 - v=I) 27 x 4 - 2x 2 + 6 (x 2 - 1 + ~)(X2 - 1 - ~) 28 x 4 + 2x 2 + 6 (x 2 + 1 + y'=5)(x 2 + 1 - yC5) 29 x 4 + 10x 2 + 30 (x 2 + 5 + yCS) (x 2 + 5 - ~) 4 2 30 x - lOx + 30 (x 2 - 5 + yCS)(x 2 - 5 - yCS) 31 x 4 + 4x 3 + 6x 2 + 4x + 6 (x 2 + 2x + 1 + yCS) (x 2 + 2x + 1 - yCS) 4 3 32 x - 2x + 8x 2 - 32x + 46 (x 2 + (-1 - ~)x + 1 + 3~) (x 2 + (-1 + ~)x + 1 - 3yC5)
a) Extensions of ((h of degree 2,3 and 4
65
Ramified extensions (f = 2) with Galois group D s: no minimal polynomial factorisation over intermediate field 49 x + 12 (x 2 + 4(3 + 2)(x - 4(3 - 2) 50 x 4 + 10x 2 + 28 (x 2 + 2(3 + 6)(x 2 - 2(3 + 4) 51 x 4 + 4x 3 + 6x 2 + 4x + 4 (x 2 + 2x + 2 + 2(3)(X 2 + 2x - 2(3) 52 x 4 + 4x 3 + 2x 2 - 4x + 4 (x 2 + 2x + 2(3)(X 2 + 2x - 2(3 - 2)
Ramified (/ = 2) extensions with Galois group C 4 : no minimal polynomial factorisation over intermediate field 53 x 4 - 4x + 52 (x - 8(3 - 6)(x 2 + 8(3 + 2) 54 x 4 + 4x 3 + 4x 2 + 12 (x 2 + 2x ~ 4(3 ~ 2)(x 2 + 2x + 4(3 + 2) 55 x 4 + 4x 2 + 52 (x 2 + 8(3 + 6)(x 2 - 8(3 - 2)
Ramified (/ = 2) extensions with Galois group V4 : no minimal polynomial roots to be adjoint to intermediate fields 2 4 56 x ~ 2x + 4 y!=3, H, 6 2 4 57 x ~ 6x + 36 y!=3, V2, yC6 2 3 4 58 x ~ 2x + 2x - 4x + 4 y!=3, v=T, J3 Proof. Those extensions which have a quadratic intermediate field K can be constructed by taking quadratic extensions of quadratic extensions each step according to Lemma (IX.2). The extensions might not be normal. Checking equality of the so constructed extensions is an easy task. The missing extensions have 54 or A 4 as Galois groups. First consider those with A 4 • According to the unique chief series of A 4 one has to construct first the unique (up to isomorphism) unramified extension by 0 = (7 + (i 1 which has C3 = (0) as Galois group. Now one constructs a Galois exten~ion K of k = ((h [0] with Gal(K, (b) ~ A4 such that the Galois group Gal(K, k) is "'4. Independent generators of the k* j(k*)2 are 1, 2, 5, 7, 1 + 0 and 1 + Since k*j(k*)2 obviously splits as a lF2 C3-module into a direct sum of 4 simple modules with trivial action of 0 and one 2-dimensional module. Adjoining V-I - 0 and VI - 02 to k, the fixed field of 0 is the required extension of degree 4 over (b.
oa.
To construct the fields with 54 as Galois group, one extends k = ((h [~, (3] (cf. Lemma (IX.3)) to K with G := Gal(K, (2) ~ 53 such that Gal(K, k) = V4 . Let N = k* j (k*)2 ~ C~ ~ Gal( k ab ,2, (b) and 0 be an element of order 3 in 53. Define Z := FixN(o) rv Ci. Therefore N, under the action of an element of order 3 in G, splits into N = Z EEl M, where M consists of 2 copies of the 2-dimensional irreducible lF2 C3-module. This structure is preserved when the representations of C3 on M is extended to 53. Since M ~ N j Z there are 3 normal subgroups in this factor with the pre-images M}, M 2 , M 3 such that NG j M i ~ 54' Therefore there are 3 extension fields of(b with Galois group 54' The fixed fields under G are the required fields. q.e.d.
66
b)
IX Fields of smaJl degree
Extensions of Q3 of degree 2, 3 and 4 (IX.5) Lemma.
(i) There are 3 non-isomorphic extensions of Q3 of degree 2 in its algebraic closure Q3. Minimal polynomials can be chosen as follows. Define a to be (8 + (~i where (s is a primitive 8th root of unity and i = 1 or 5 (i.e. a is a root of x 2 + 2). Unramified extension: x 2 + ax - 1. Ramified extensions: x 2 + 3, x 2 - 3. (ii) There are 10 non-isomorphic extensions ofQ3 of degree 3 in its algebraic closure Q3' Minimal polynomials can be chosen as follows. Define a to be a root of y4 + y3 + 2y2 - 4y + 3 i. e. any of (~3 + (f~ + (r3 for ~ = 1, 2, 4, 7 where (13 is a primitive 13 th root of unity. Unramified extension: x 3 + ax 2 + (1 - ~a - ~(3)x + l.
Ramified extensions: minimal polynomial x 3 + 3x 2 + 3 x3 + 3 x 3 + 12 x3 - 6 x 3 + 3x 2 + 3x + 3 x 3 + 3x 2 - 3x + 3 x 3 + 3x 2 - 3 x 3 + 3x 2 - 12 2 x 3 + 3x + 6
Galois group
53 53 53 53 53 53 C3
mtermediate field of degree 2 Q3[(S]
Q3[yC3] Q3[yC3] Q3[yC3] Q3[yC3] Q3[V3]
C3
C3
(iii) There are 5 non-isomorphic extensions of Q3 of degree 4 in its algebraic closure
Q3. Minimal polynomzals can be chosen as follows.
Define a := (80 + (§o + (io + (lZ to be a primit'lve 80 th root of unity i. e. a a root of x S + 60x 4 + 200x 2 + 225. Unramified extension: x 4 - ax 3+ 135- 1 ( a 6 +a 4+ 115a 2 + 180)x 2 + 135- 1 (_a 7 - a 5 -70a 3 - 225a)x-l. Define (s to be a primitive 8th root of unity. Ramified extension over the unramified field Q3 ((s) of degree 2: x 2 + 3 (Galois group 114), x 2 + 3(s (Galois group C 4). Totally ramified extension: x 4 + 3 (Galois group D s ), x 4 - 3 (Galois group D s).
Proof. (i) Straightforward. (ii) There are 4 normal extensions with Galois group C 3 corresponding to the different subgroups isomorphic to C 3 in 'Q;j('Q;)3. To construct the other extensions of degree 3 one has to consider normal extensions with Galois group 53 and the required fields are those corresponding to subgroups of index 3. On top of any of the 3 extensions k
b) Extensions ofQ3 of degree 2,3 and 4
67
in (i) one constructs extensions of degree 2 which have non-abelian Galois groups over Q3- This is only possible in one way when k = ~[J3] or k = ~[(8]- For k = Qp[y'=3] one has to consider the operation of C 2 on Gal(k ab ,3, k) ~ k* j(k*)3 rv C 3 X C 3 X C 3 X C 3 which splits into a two dimensional module on which C2 acts trivially and a two dimensional module on which C 2 acts non-trivially_ Hence one gets 4 different extensions corresponding to the I-dimensional non-trivial submodules. (iii) The given polynomialH are thoHe which have a quadratic intermediate field anu therefore can be constructed by taking quadratic extensions of quadratic extensions, d. Lemma (IX.4). Assume there are extensions K of Q3 with Galois group G isomorphic to A 4 or 54' Then G contains certain normal subgroups G I , ... ,G n for some n E N with G ~ Go ~ G I ~ ... ~ G n , called ramification groups in the upper numbering d. [Iws 86] p. 33-34. These subgroups satisfy IG : Gol = j, IGo : GIll (pi - 1) and IG I : (1)1 = pk for some kEN and j the inertia degree of K. Comparing these conditions with the (unique) chief series of G yields a contradiction. q.e.d.
X
Algorithm for finding a filtration and the obliquity
a)
The BASIS algorithm
Let P be a finite p-group, with a composition series
For 1 :s i S n, let ai E Pi - Pt +1 be fixed. So {ad forms a basis for P. It is useful to have (*) as a chief series for P, that is with Pt <J P for all i, but this condition will not always be satisfied.
at! ...
If 9 E P, then 9 can be written uniquely in the normal form 9 = a~n, where o::; U t < p. Assume that we have an algorithm for multiplying and inverting elements of P, and an algorithm for writing elements of P in normal form. Then we can compute in normal form the p-th powers of the basis elements, and the commutator of each pair of basis elements, thu:'3 producing a PC-presentation of P. A very wide range of algorithms are available, and have been implemented in the computer systems GAP [GAP 94] and MAGMA [MAG 95], for computing with p-groups (and soluble groups) defined in terms of PC-presentations. These algorithms, in most cases, simply require one to be able to multiply and invert elements, and to express the result in normal form, which we could do more efficiently without use of the PC-presentation. But constructing a PC-presentation for the groups in question, and sending the results to a file, enables us to analyse the groups we have constructed in a form that is independent of their provenance, and enables us to use the above packages.
The basic algorithm is performed in the following context. We have a finite p-group G, and a subset X of G. Let P = (X). We aim to construct a basis for P. When
the algorithm has been described, it will be clear how to write any element in normal form with respect to this basis, and hence how to construct the PC-presentation on this basis. We assume, of course, that we can multiply and invert in G, and can test for equality. There will be a fixed normal series G = G(l) > G(2) > ... > G(Wt+l) = (1) with elementary abelian sections, refined by the series (Gd. If 9 E G(i) - G(i+l) we say that 9 has weight i, and define the weight of 1 to be wt+1. We assume that we can calculate the weight of any element of G, and if g #- 1 has weight i, we assume we can compute the image of gin G(t)/G(i+l) as a vector with respect to some fixed basis of G(i)/G(t+l)' The algorithm to construct a basis for P goes as follows.
69
b) Split and Non-Split groups
BASIS(X, B, wt) 1* wt is as above; X is a finite sequence of elements of G. B is set to a sequence of elements of G that forms a basis for P = (X) . *I { t:= length(X); B :=0; for w := 1 to wt do {g := FIRST(X); repeat ifweight(g) = w; {FILTER(w, g, B); if weight(g) = w
{B:= B
U
{g};
X := X concatenate seq(gXt, ... , gXt, gP);
} else X
:=
X concatenate(g);
} } until NEXT(X, g) = FALSE;
} The procedure FILTER does the following: it takes as input an integer w, an element 9 of weight w, and a set B that is part of the basis being constructed. If the image of 9 in G(w)/G(W+l) is linearly deppndent on the images in G(w)/G(W+l) of the elements of B of weight w, then 9 is multiplied by elements of B of weight w to produce a new 9 of greater weight. This modified element is again called g. Otherwise, the procedure does nothing. FIRST(X) returns the first element of X, and NEXT(X, g) replaces 9 by the next element of X, or returns FALSE if 9 is the last element of X.
b)
Split and Non-Split groups
It is unfortunate that an algorithm that is so easy to describe should have required a C program of over 12000 lines. It should be admitted that the program as implemented does not quite fit the above description. We compute the PC-presentation at the same time as we compute the basis, and this enables us to use the power-commutator presentation to drive the construction of new elements of B. However, this is a minor variation. The main difficulty lies in computing with elements of G. The group G will have a somewhat more complicated structure than described above. It will in fact have a normal series, of length at most four whose factors will have the structure described; in particular, each factor will have a weight function. The weight functions of these factors define, in a natural way, a filtration of G, and the subgroups of G that arise in the filtration are required to be normal in G. It
X Algorithm for finding a filtration and the obliquity
70
is this normal series that is used in the procedure BASIS described above. We now distinguish two cases. In the simplest, or 'split' case, each extension splits, and we can express G as G[3] Xl G[2] Xl G[l] Xl G[O], where each G[i] is embedded as a subgroup of G. An element of G that lies in a G[i] will be called simple. It turns out that we can choose the embedding of each G[i] as a subgroup of G in such a way that the commutator of any two simple elements is again simple. This implies that, for any i < j, either G[j], as a subgroup of G, is normalised by G[i], or that G[j], as a section of G, is centralised by G[i]. As a consequence of the splittings and the commutator condition, if the generating set X of P consists entirely of simple elements, then only simple elements arise in the calculation. This will be assumed in the split case; so it is sufficient to write code to perform the basic group-theoretic operations in each G[i], to compute the commutator of two simple elements, and to compute the weight of a simple element. More precisely, the filter procedure requires us to be able to compute the image of an element of weight k in G[i](k)/G[i](k+l)'
c)
The groups G[i]
The group G[3], which is in general the largest of the eli], and the one in which most computation is performed, is defined as follows. Take a local field K with ring of integers O. Then the ring Od of d x d matrices over 0 has an ideal 1 (which is the radical of the standard minimal hereditary order in Od) consisting of those matrices in Od whose entries on and below the main diagonal lie in the maximal ideal of O. Since 1 is pro-nilpotent, the group H = H(d, K) of matrices of the form 1d +M, where 1d is the identity d x d matrix, and M runs through the elements of 1, is a pro-p-group. Defining H(i) to be the subgroup of H consisting of elements of the form 1d + M where M lies in Ii defines a filtration for H by normal subgroups with elementary abelian sections of rank d· j, where j is the degree of the inertia field; that is to say, the degree, over Qp, of the maximal unramified extension of Qp that lies in K. We take G[3] to be a subgroup of the quotient of H by H(w+l) for some w > 0, or, more often, of the quotient of H by the product of H(w+l) and the group of scalar matrices in H. To distinguish between these cases, we refer to the former as the linear and the latter as the projective case. The filtration {G[3](t)} of G[3] induced by {H(t)} is the filtration used in the BASIS algorithm. Since we work modulo G[3](w+l)' where w is fixed, it is sufficient to take the coefficients of our matrices to lie in the quotient of 0 by a suitable power of 1. However, matrices that represent the same element of G[3] may still have different coefficients modulo this power of 1. Before considering the details, let us agree to follow the inevitable practice of the program, and regard a matrix, with coefficients in a suitable quotient of 0, as an element of G[3]. This abuse of notation is not without dangers at the programming level.
71
c) The groups G[i]
Suppose that we are in the linear case. To prove that a matrix M = (mij) lies in G[3](t) one has simply to check that v(mij) 2: 1 + (t + i - j - l)/d for all i -I- j, and that v(mii - 1) 2: 1 + (t - l)/d for all i, as can be easily seen; so, given a basis for the residue class field of K over its prime field, it is easy to compute the image of M modulo G[3](t+1)' We shall illustrate the method by an example. Suppose then that 1 + 1Tall
M
=
(
1Ta21 1Ta31
a12 1 + 1Ta22 1Ta32
is an element of G[3]. Here, and throughout this section, 1T denotes a fixed uniformising element of K; that is to say, an element of 0 of value 1. To compute the image of M in G[3]/G[3](2)' form the image of (a12' a23, a3d in F3, where F is the residue class field. If this image is zero, to compute the image of M in G[3](2)/G[3](3) , form the image of (a13' a21, a32) in F 3. If this image is zero, to compute the image of M in G[3](3)/ G[3](4)' form the image of (all, a22, a33) in F3. If this is zero, to compute the image of M in G[3](4)/G[3](5)' form the image of (1T-IaI2, 1T-Ia23, 1T-1a32) in F 3, and so on. If we are in the projective case, everything is the same except on the diagonal. In this case, in order to check that M lies in G[3](t), we have to see that
v(aii - all) 2: 1 + (t - l)/d for all i > 1. Similarly, to determine the image of M as above in G[3](3)/G[3](4)' which is now of dimension 2 over F, form the image of (a22 - all, a33 - all)'
It remains to explain how arithmetic is performed in 0, or rather in quotients of this ring. This has been discussed in Chapter VIII. 'We now discuss the group G[2]. The group H is a maximal pro-p-subgroup of G L(d, 0), and can be described geometrically as follows. There is a chain of lattices L = L o > L 1 > L 2 > ... where L is the natural module for GL(d, 0), where L i +d = 1T L i for all i, and H is the group that stabilises this chain, and centralises the quotients. In fact we have taken £2 = (1Te1, e2, . .. , ed), and L 3 = (1Tel' 1Te2,"" ed), etc. Working over K we can extend this chain to negative exponents by defining L i - d = 1T- 1 L i . Then the normaliser of H in GL(d, K) is the subgroup that normalises this extended chain. If we work linearly, the quotient of this normaliser by H is a split extension N 2 ><J N I , where N 2 is the subgroup of the normaliser that normalises each L i , and N I is generated by an element that maps L i onto L i+ 1 for all i. Thus N 2 is the direct product of d copies of F*, and we can take N 1 to be generated by the matrix whose non-zero elements consist of 1 in each place immediately above the main diagonal, and 1To in the bottom left corner, where 1To is a uniformising element. If we work projectively, N I becomes of order d, as its d-th power is 1T 0 1d • Thus, if p divides d, we can extend H, and hence G[3], by a cyclic group of order pk, where k is the p-adic value of d. Note that N 2 ><J N 1 has several Sylow p-subgroups, and that different choices of 1To mod 1 2 will correspond to different Sylow subgroups. Because
72
X Algorithm for finding a filtration and the obliquity
of the commutator condition mentioned above (i.e. a commutator of two simple elements is simple), we will need to be careful of our choice of 1fo· Naturally we define G[2J to be this cyclic group of order pk, or to be trivial if we wish to omit this group. If G[2] is to be non-trivial, we must work projectively for invertibility. As G[2] is cyclic, the weight function on G[2] is forced, and does not need discussion. To compute the commutator of an element of G[3] with an element of G[2] involves dividing an element of our ring, of positive value, by 1fo. As we are only working in a quotient of 0, unfortunately 1fo is no longer a unit, so we have to be very careful at this point. Of course the answer mathematically speaking is to compute to a greater accuracy than is needed for the final result, but this causes problems of efficiency.
The group G[l] is a Sylow p-subgroup of the automorphism group of K if K is of characteristic 0, and is a Sylow p--subgroup of the group of automorphisms of K that fix a given uniform element if K is of characteristic p. The case when K is of characteristic p is the simpler, and we dispose of this case first. The group of automorphisms of K that act as the identity on the residue class field F of K is a so-called Nottingham group. It is known to contain a copy of every finite p--group, and Rachel Camina has recently proved that it even contains a copy of every finitely generated pro-p--group (cf. [Cam 97]). It is, however, difficult to find explicitly an example of any finite subgroup of order greater than p. We have therefore written our code to exclude subgroups of the Nottingham group acting on the matrix entries. Elements of the ring are represented as polynomials in some fixed indeterminate t, and we simply take the p--automorphisms of K that fix t, or we can take G[l] to be trivial. Thus G[l] is cyclic of order pk where k is the p--adic value of j, and F = GF(pl), or k = 0. Provided that we take the uniform element 1fo in the definition of G[2] to be t, elements of G[2] and G[l] commute, and there is no problem with the action of G[l] on G[3]. If K has characteristic 0, we compute its full automorphism group. The theory behind our calculations is well known; the original reference is [Has 49J. For a more recent reference, see for example [Iws 86]. We supply e, j and the Eisenstein polynomial, where e is the ramification index and j is the degree of inertia of Kover Qp. That is to say, K is obtained first by forming an unramified extension T of Qp, where T is of degree j over Qp, and is formed by adjoining the (pi - l?h roots of unity to Qp. Denote a primitive (pi - 1)th root of unity by a. The automorphism group of T does not act transitively on the set of primitive roots, so the question arises which primitive root has been chosen. The choice is made automatically, and is printed to the screen, the choice being determined by the minimum polynomial of a over Zp reduced modulo p. The Eisenstein polynomial is then provided as a monic polynomial of degree e over T, where the coefficients are given as polynomials in a over Qp. We shall take 1f to be some fixed zero in K of the given Eisenstein polynomial. Some details of the calculation of the automorphism group of K have been given in Chapter VIII b). Here we give an outline of the general theory. The automorphism group f of K has a filtration f 2: f o :;: f 1 :;: ... by normal subgroups, terminating in (1), where C is the subgroup that centralises 0/1f i +1 0. It is easy to see that the first two filtration quotients are cyclic of orders dividing j and pi - 1 respectively, and that the others are elementary abelian p--groups.
c) The groups C[i]
73
The first step is to compute f, finding the image of a and 11" under each element, as described later, and to construct the multiplication group of f. The next step is to find a Sylow p-subgroup of f. Clearly we may work modulo the normal p-subgroup f 1, and restrict ourselves to the subgroup consisting of elements whose image modulo f o is of p-power order. We now have a meta-cyclic group with a natural action on 11"0/(11"0)2. Clearly any p-subgroup must fix a non-zero coset of (011")2, and it is easy to see that a Sylow p-subgroup is the centraliser of some suitable coset. Thus we can find some a E F* such that the subgroup of f that acts on 0/11"0 as field automorphisms of prime power order, and fixes a1l" mod 11"2 is a Sylow p-subgroup of f. This Sylow subgroup is our C[l]. Having chosen C[l], we need to check the commutator condition. Clearly C[l] normalises C[3], and permutes the Sylow subgroups of the normaliser of H in PCL(d, K). One sees at once that, provided that 11"0 is chosen to be congruent to a1l" modulo (11"0)2, C[l] will centralise C[2] as a section of C, and all is well. Of course it is possible to choose the Eisenstein polynomial such that a may be taken to be 1. The filtration we use for C[l] is obtained from the filtration defined above (which is known as the upper numbering) by refining the cyclic quotient f /f o to have factors of order p, and deleting trivial quotients. It should be remarked that f /f 0 need not centralise fdfi+l for all i, so in general the filtration quotients that we obtain need not be central. Thus the procedure BASIS will not always produce an AG-system that corresponds to a chief series. Finally if p = 2 and d > 2, we may take C[O] to be the cyclic group of order 2 that acts on elements of C[3] by the inverse transpose automorphism. It commutes with C[l] and C[2]. If d = 2 this would simply act as conjugation by an element of the normaliser of H in CL(d, K); this corresponds to the fact that the Dynkin diagram for A( n) has no non-trivial automorphism if n = 1, as it consists of a single point. Unfortunately, we also have to consider non-split examples. It would be possible to deal with these by embedding the groups in larger split extensions, but this would have involved serious calculations to produce the required input. We therefore allow the program to run in non-split mode. In this case we represent an element of C as a linked list of length at most 4, whose entries are non-trivial elements of the C[i], arranged in strictly increasing order of i. This has the advantage that the identity ele~ent of C is represented by the null list. We now multiply and divide using collection. This enables us to remove the condition that the product and commutator of two simple elements should be simple. In practice we are only interested in the case in which this can apply to two field automorphisms. So in the non-split case, we have a different action of a field automorphism on a matrix, and the product (or commutator) of two simple elements in C[l] can be the product of an element in C[3] with the corresponding product (or commutator) in C[2]. These elements of C[3], which we call cocycles, are computed in advance. So in the non-split case, we have to supply an algorithm for computing these co cycles, and an algorithm for computing the action of C[l] on C[3]. There will also be a restriction on C[3]; but this will be supplied automatically by the input matrices.
74
d)
X Algorithm for finding a filtration and the obliquity
Calculating the obliquity
By the algorithm BASIS one can assume to have constructed a PC-presentation for some finite p-group P. Now an algorithm for calculating the obliquity is described. Define A := {N <J PIN 1:. 'Yi+l(P)}, Recall J-t~(P) := (nNEAi N) n 'Yi+l(P), The i th obliquity is defined to be Oi(P) := logp(I'Yi+1(P) : J-ti(P)I)· In particular Ai C A+l' Define ~ := {N E AilN C 'Yi(P)}. Therefore the first observation is that if one has determined J-ti(P) then J-ti+l (P)
= J-ti(P) n (
n
N) n 'Yi+2(P),
NEAi+l
Set M i := {N <J PIN E
J-ti+l(P)
Ai and
lJM
= J-ti(P) n (
E
At with M~N}. Clearly
n
N) n'Yi+2(p) for i EN.
NEM'+1
For determining M i one uses the following algorithm: L:= a set of candidates, K:= some set of normal subgroups, max (X) a function returning the set of all maximal subgroups of X which are normal in P
{ M z =0; L := maxbi(P)); for NE L do { K:= max(N); if M C 'Y~+l(P) for all M E K then M i :=MiUN; else { for M E K do { if M ~ 'Yi+l(P) then L :=LUM; K :=K-M;
} } L;= L - N;
} To prove that the algorithm finds all normal subgroups of M i one uses that P is a p-group and therefore for M E M i there exists a chain of normal subgroups such that 'Yi(P) = Xl > X 2 > ... > X s = M with IXi : X H1 ! = p for i = 1, ... 1 S - 1. The function max(N) returns the maximal normal subgroups of N which are normal in P. This is done by the following method:
75
e) Periodicity of the lower central series and obliquity
The group P acts on V := N/[N, N]NP which is an IFp-module. One determines the I-dimensional eigenspaces of V under the action of P. Since P is a p-group one has to consider only the eigenvalue 1. Call these eigenspaces 2 11 , ",25' Since the epimorphisms 1ri : V ---+ 2 i are P-admissible the kernels Ker(1ri) are invariant under the action of P. It follows that the set of pre-images {K er (1ri) [N, N]NPli = 1, ... ,s} is the set of all maximal subgroups of N which are normal in P.
e)
Periodicity of the lower central series and obliquity
Let G be an insoluble p-group. By Theorems (IV.6) and (IV.I4) the isomorphism types of the factors in the lower central series ri (G) / 'Yi+ 1 (G) and the sequence OJ of the obliquity repeat periodically for some i ~ io and some j ~ jo. To calculate the tables in Chapter XII it is necessary to determine io and jo and a c, such that the calculations can be carried out in a finite quotient G = G/'Yc+1(G) and repeat from io or respectively jo on. Denote the Qp-dimension of the corresponding Lie algebra by d. One can check that the groups investigated in Chapter XII are settled with respect to d. Therefore the periodicity of the lower central series will be guaranteed from 'Yd( G) onward or if p = 2 from the largest 'Ym (G) which is contained in 'Yd(G)2 onward (m ~ 2d). Unfortunately starting the investigation of the lower central series or obliquity from 'Yd(G), or 'Ym(G), requires a rather big quotient G of the group G. A more detailed analysis, as carried out below, can reduce the size of the quotient and therefore increase the number of groups which we actually can handle. We can observe that usually the lower central series shows already the pattern from some i1 < io. It can be proved by induction, that if one has an i 1 E N and there are r, kEN such that 'Yi(G)pT = 'Yi+k( G) and 'Yi( G) is powerful for i = i 1, ... ,i 1+ k-I and l'Yil (G) : !fIT (G) I is a multiple of d (all this can be checked in a suitable quotient G) then these conditions hold for all i ~ i1. To calculate the obliquity recall/li(G):= (nMI"Yi+l~G),M<1GM)n'Yi+1(G). The lattice of normal subgroups contained in 'Yd(G), or 'Yd(G) if p = 2, repeats periodically. The sequence (OJ) is periodic from)o onward, if all normal subgroups M which are to be considered for calculating J.1..jo (G) are contained in 'Yd( G) or 'Yd(G)2. A careful analysis also might give us a smaller index j1 from where on the obliquity repeats. Let N be a normal subgroup in G. Let i be the maximal index, such that N ~ 'Yi(G). Let s = s(N) be minimal, such that 'Yi+s(G) ~ N. To determine the period of the sequence (Iri+1(G) : J.1..i(G)I) , and to prove the repetition, one first estimates a E Nand j 2: i 1 such that max{ s (N) I N <J G, N ~ rj( Gn .s; a where i1 is where the period of the lower central series begins, cf. Lemma (X.I). In Lemma (X.2) one finds some z = z(a, k, j) such that all n~rmal subgroups of G contained in 'Yz(G) are uniform. It then remains to make sure that all normal subgroups which are necessary to calculate J.1..i(G) are contained in 'Yz(G). This will be the case for all J.1..i( G) with i 2: ()i where ()i is determined in Lemma (X.3). (X.I) Lemma. Assume that i 1 and the length of the period k are determined as
above such that 'Yn(G) is powerful and 'Yn(G)/'Yn+1(G)
'V
rn+k(G)/rn+k+1(G) for all
X Algorithm for finding a filtration and the obliquity
76
n ~ il. Let 1} then
0 :=
max{ s(N)IN
<J
G, N ::; It I (G), N 1. IZ1+k(G)}. Let j = max{il' 0
max{s(N)IN
<J
+
G with N ::; 1'j(G)} = o.
Proof. Claim: The maximal value of 0 is reached by normal subgroups generated by one element as a normal subgroup. One has (x, y)G ~ (x)G(y)G. Let Sl := s((x)G), S2 := s((y)G) und n, I such that 1'n+sl(G) ::; (x)G ~ 1'n(G) and 1'1+s2(G) S (y)G ~ 1'1 (G). Then 1'max{n+sl,I+S2}(G) ~ (x)G (y)G S (x, y)G ::; 1'max{n,l} (G). It follows that max{ n + Sl, I + S2} - max{ n, I} ::; max{ S11 S2}' Hence it holds that s( (x, y)G) ~ max{ s 1, S2} and therefore the claim, since all normal subgroups of G are finitely generated. Choose an x E N where N is a normal subgroup in G satisfying N ::; 1'n(G), for i ~ j + k . Then, n = j + f + kl with 0 ::; f < k. For any x E N there exits an x E 1'j+J(G) such that x PI = x. From the assumptions follows s( (x)G) So. The factor group (x)G /1'i+J+a+l(G) is abelian since 1'j+f+ a+l(G) > 1'2j+2f(G) because of j + f ~ 0+1. Clearly, (x)G S 1'Hj+lk(G). It remains to show that (x)G ~ 1'HJ+lk+d(G). Write Yl,'" Yo for the representatives of 1'J+J+a(G)/1'Hj+a+l(G). They are of the form IT~=1(Xam)9m, am E N, gm E G, some hEN. Applying the pi_power map ampl )9 m 1'H fHI+a+l (G). to Yn1'Hj+a+1 (G) yields (IT~= 1(x am )9 m )PI 1'j+J+kl+a+l(G) = IT (X These are generators of 1'J+J+IHa(G)/1'H/+lk+ a+1(G). The representatives of this factor group lie in (x)G. q.e.d.
z
(X.2) Lemma. Define G, j, k,o as above. Define z = max{j,o + k}, if P i- 2 or if p = 2. Then all N in G satisfying N S 1'z(G) are uniform.
= max{j,o + 2k},
Proof. Let N <J G, N S 1'z(G), s := s(N) S 0, choose n maximal such that 1'n+s(G) ~ N :::; 1'n(G) for n ~ z. Then [N, N] ~ bn(G), 1'n(G)] S 1'2n(G) and NP 2: 1'n+s(G)P = 1'n+,Hk(G) if p i- 2 or N 4 2:: 1'nH(G)4 = 1'n+S+2k(G) if p = 2. Since 1'2n(G) S 1'n+Hk(G) or 1'2n(G) S 1'n+H2k(G) the normal subgroup N is powerful. Since 1'n(G) is uniform, it is also torsions free. Therefore all N <J G, N S 1'z(G) are not only powerful but also torsion free and therefore uniform. q.e.d. (X.3) Lemma. With the above notation the set of normal subgroups {NIN ~ 1'nH(G) and N l 1'n+2d G )} equals the set {MPIM :::; 1'n(G) and M l 1'nH(G)} if n ~ z. In particular define m := min{n E NIN f. 1'z(G), N 2:: 1'n(G)}. Define a = max{ m ~ 1, z}. Then it follows that l1'n+k+ 1 (G) : f.lnH (G) I = l1'n+1 (G) : f.lTl (G) I if n ~ a. Proof. Let n ~ j. The pl-te power of N :S 1'n(G) is a normal subgroup of 1'n+lk(G). For every normal subgroup in 1'n+ldG) one gets by taking pi -th roots a normal subgroup in 1'n (G) . For the n - th obliquity one has to consider only minimal normal subgroups, which are not contained in 1'n+1(G). Therefore one only has to determine f.l1(G), ... , f.ll+k-l (G), then f.ln(G) for n > I + k - 1 are given by taking p-th powers, if all minimal normal subgroups which appear in the intersection for f.ll(G), .. . f.ll+k-l (G) are uniform. Therefore for n 2:: max{z, m-1} one has a bijection between the normal subgroups in 1'n(G) and 1'n+k(G). Every normal subgroup which is not contained in 1'z(G) contains
e) Periodicity of the lower central series and obliquity
77
ra+l(G) and hence need not appear in the intersection for J-Ln(G) with n ::::: a-l. q.e.d.
If one carries out the calculation of the obliquity for Q = G!'Yc+l(G) which is a finite group and if J-La+k-l(Q) contains a full set of representatives of 1c(G)/1c+l(G) then J-La(Q), ... , J-La+k-l(Q) determine the period of the sequence (J-Li(G)).
XI
The theory behind the tables
In this chapter we describe the technical details of how the input for the program described earlier is obtained in order to compute the tables. There are essentially two issues: enumerate all relevant Qp-Lie algebras up to dimension 14 for p E {2,3} and secondly obtain generators for the Sylow pro-p-subgroups of their automorphism groups. The Lie algebras which are relevant are the simple ones and the semisimple ones having pO isomorphic simple components for some a E N. The simple Lie algebras £ are absolutely simple over their centroid C = EndL(£). The possible centroids are extensions of ({h of degree 2, 3 and 4 and extensions of ~ of degree 2, 3 and 4 which are given in Chapter IX. We now enumerate the possible types of Lie algebras.
a)
The relevant Qp-Lie algebras up to dimension 14
It is well known that a simple Qp-Lie algebra is absol utely simple over its centroid C. The classification (cf. [Sat 71]) leaves the following possibilities: Notation: K r = Kr(K) is a central simple K-division algebra of index r for a p-adic field K. Let sln(Kr(K)) = {x E Kr(K)1 reduced trace (x) = O}.
dimension 3: Type AI: Sl2(Qp), Sll(K 2 (Qp)). dimension 6: Type AI: Sl2(K), Sll(K 2(K)) with IK: Qpl = 2. Type (A r)2: Sl2(({h)2, sir (K 2(({h))2. dimension 8: Type A 2: Sl3(Qp), Sll(K 3(Qp)), sU3(K, Qp) where IK : Qpl = 2. dimension 9: Type AI: Sl2(K), Sll(K 2(K)) with IK: Qpl Type (AI)3: Sl2(Q3)3~ Sll(K 2(Q3))3.
= 3.
dimension 10: Type B 2: S05(¢s,Qp), S05(¢ns,Qp) where ¢s and ¢ns can be chosen as the quadratic forms of the quadratic spaces (l)l..H l..H, N l..H where H is a hyberbolic plane over Qp, N the trace-O-subspace of K 2(Qp) with the quadratic form induced by the (reduced-) trace bilinear form of K 2 (Qp). dimension 12: Type AI: Sl2(K), Sll (K 2 (K)) with IK : Qp 1= 4. Type (Ar)2: Sl2(K)2, Sll(K 2(K))2 with IK: Qpl = 2. Type (Ar)4: Sl2(({h)4, Sll (K 2(({h) )4.
79
b) Genera tors for the maximal p-adically simple groups
b)
Generators for the maximal p-adically simple groups
Define 0; := {x E O*lv1r (x - 1) = i} for i E N. The following lemma is used at various places.
n
(XL1) Lemma. Let riJ E 0; such that (Ti) ·0;+11) = 1" .. , = 0;;0;+1' Then 0i is (topologically) generated by {r!) j = 1, ... , f, i = 1, ... , e + lp~1J}· 1
Proof. 0; ~ O;+e : 1 + x H (1 + x)P is a homomorphism of 0:+1 into 0;+e+1' Expanding (1 + x)P one gets terms of value 0, e + i, e + 2i, ... , e + (p - l)i, ip. Hence 0;+1 onto for p . i > e + i, i.e. i > P~ l' one obtains an epimorphism of 0;+el0;+e+1' The claim follows. q.e.d.
0;;
Note, the proof also shows that usually a smaller generating set can be found. Basically we have the following structure of AlltQp (£):
(XL2) Lemma. Let £ be a simple Qp-Lie algebra with automorphism group AutQp(£) and centroid C := End.c(£).
(i) AutQp(£) is an extension of the group of C-Lie algebra automorphisms of £ Autc(£) by a subgroup of Aut(C, Qp), the group of field automorphisms of C fixing Qp pointwise. If £
= £1 0Qp C
where £1 is a Qp-Lie algebra then
(ii) Let K be a (minimal) splitting field of £, i. e. £ := K 0c £ is a split K -Lie algebra of type cI> E {Ai, B i , ... } with Cart an subalgebra H. The following holds: (a) Every Q E Autc(£) extends uniquely to monomorphism,
n·
Q
E Aut K
(£) i.e.
Q
H
Q
is a
(b) AutK(£) = G(cI>, K) . 'D where G(cI>, K) is the Chevalley group with respect to the adjoint representation (cf. Chapter V) consists of representatives of Hom (ZcI>, K*)/Hom(Q, K*) and'D is the finite group of diagram automorphisms acting faithfully on the Dynkin diagram of cI>.
n
Proof. (i) and the first part of (ii) is clear, the rest follows from [Ste 61].
q.e.d.
Therefore there are various reasons why it is not always straightforward to find generators for maximal j)-groups. Here are names for the difficulties arising.
(i) the additional elements of'D, called diagram automorphisms, (ii) representatives of Hom (ZcI>, K*)/Hom(Q, K*) which are not contained in G(cI>, K) if IQ : ZcI>1 > 1 (Q denotes the weight lattice), called rational automorphisms (or d. [Car 72] diagonal automorphisms),
XI The theory behind the tables
80
(iii) suitable settings, e.g. if one has the choice of different representations, we prefer those of low dimension even if one has to factor out by the centre.
Let ll> be a root system of type A n- 1 and £.. = sln(K) with centroid C = K a finite field extension of Qp. The group G L n (K) acts on £.. by conjugation. Since the kernel of this action is K* In one has PGLn(K) acting on £... Denote by - : GLn(K) -t PGLn(K) the natural epimorphism. Use the notations for matrices di(al, ... , an) as defined in (VI.lO). (XL3) Lemma. The K -automorphism group AutK(sln(K)) is isomorphic to PGLn(K) XJ 'D where'D is the group of diagram automorphisms which is of order 2. Proof. The claim follows by (XI.2) (ii) and by the fact that PGLn(K) is a Chevalq.e.d. ley group with the rational automorphisms on top. A set of generators of the maximal j)--group acting on sln(K) (= Sylow pro-psubgroup of AutiQlp (sln(K))) can be obtained as follows. (XI.4) Lemma. Let Eij(a) E GLn(K) be the matrix with i on the diagonal, a in position (i, j) and 0 everywhere else. Define W to be a Zp-basis of OK and B a generating set of OJ (cf. XI. i). Define £ := {Ei,Hl(a),En,l(na)!i {Ei,Hl(a),En,l(na)li = 1,. ",n1 and a E W}. Define rn(n):= d1 (1, ... ,1,n). Denote by n the set of rational automorphisms {rn(n)l, do(b, 1, ... , l)lb E B} if n = pkl and pfl for some kEN. Define F :=
(1 ... 1).
The action of the diagram automorphism on the group
generated by £ is given by - : X
rl
F X-tr F.
a) A Sylow pro-p-subgroup of AutK(£..)
rv
PGLn(K)XJ 'D is generated by
(i) £ ifpfn andp odd, (ii) £
U
C}
if pfn and p even,
(iii) £ un if n = pkl for some k, lEN, pfl and p odd. (iv) £ un u {-} if n = pkl for some k,l E N, pfl and p even. b) Let r be the Sylow pro-p-subgroup of Aut( K, Qp) such that every (J E r satisfies n(J n(mod n 2 0). The Sylow pro-p-subgroup P of AutK(£..) constructed in a) can be extended in a split way by r such that the elements of r act on the entries of the matrices.
=
Proof. a) Clearly, £ generates a Sylow pro-p-subgroup of SL n (K). The factor group GLn(K)j SLn(K) ~ K* and A := GLn(K)j(SLn(K) . K* In) ~ K* j(K*)n. Hence if pfn one has that (£) is a Sylow pro-p-subgroup of PGLn(K). Otherwise, observe that the elements of n act on (£). The natural image in A of the elements in n generate, via the above isomorphism, a Sylow pro-p-subgroup of
81
K* /(K*)n. Therefore the set {£, R} generates a Sylow pro-p--subgroup of PGLn(K). If p = 2 additionally the diagram automorphism acts on the Lie algebra by D : £ --} £ : l H -Fltr F and induces - on the group generated by £ or {£, R} preserving the chosen Sylow pro-p--subgroup. b) W.l.o.g. choose r in the described way. The condition 7[0 == 7[ (mod 7[ 2 0) guarantees that P is mapped to itself by r. q.e.d.
(XI.5) Lemma. Let K/Qp be a quadratic extension with Qp -automorphism order 2 and F = ( :
...
: ) E K nxn .
of
The special unitary Lie algebra I:- =
sun(K,Qp):= {X E KnxnlXF+FXtr = O,trace(X) = O} has prUn(K, Qp):xl C) as its Qp-automorphism group, where rUn(K, Qp) := {g E GL(n, K)lgFgtr = AgF for some Ag E Q;}. The factor group prUn(K,Qp) = rUn(K, Qp)K*In/K*In acts by conjugation on £, i.e. gK*In : x E £ H gxg- 1 and - acts by entry-wise application. Proof. Clearly £ := K 0iQlp £ ~ sln(K). By identifying £ with Qp 0iQlp £ C £, a Qp-basis of £ yields a K-basis of the Lie algebra of trace zero matrices sln(K). Let D : £ --} £ : X H ~ F xtr F denote the diagram automorphism on the Lie algebra. By (XI.2) each K-automorphism of £ (viewed as a Qp-subalgebra of £) lies in PGLn(K):xl (D). The natural K - £- (or £-) module K 1xn is absolutely simple. -tr Therefore by Schur's Lemma {F1 E KnxnlXF1 = F1(-X ) for all X E £} = KF. Hence any 9 E GLn(K) conjugating £ into itself, must map F onto a multiple, Le. gFgtr = AgF or 9 E rUn(K, Qp). Conversely every 9 E rUn(K, Qp) induces an automorphism on £. The diagram automorphism D maps £ into itself and it restricts to the Galois automorphism - on £ acting entry-wise on the matrices. q.e.d.
(XI.6) Lemma. (i) If in the last lemma n is odd, then prUn(K, Qp) is isomorphic to PUn(K, Qp), i.e. AutiQlp(sun(K, Qp)) '" PUn(K,Qp)XJ (-).
(ii) Moreover, for arbitrary n, PUn(K, Qp)/ PSUn(K, Qp) where N : K* --} Q; : X H xx is the norm map.
~
Ker(N)/(Ker(N))n
Proof. (i) Let r = rUn (K, Qp) and A : r -t Q; : 9 H Ag where gFgtr = AgF. Then N(K*) ~ rA .:; Q;, because A maps the scalar matrices in r onto N(K*). Taking determinants one gets A; = N(det(g)), i.e. (rAt S N(K*). But n is odd and by local class field theory (cf. [Neu 86] p. 42) Q;/N(K*) ~ C2 ('" Gal(K/Qp)), hence rA = N(K*). Since Ker A = Un(K, Qp) one obtains rUn(K, Qp) = Un(K, Qp) . K*In, thus proving the first claim. (ii) The map det : Un(K, Qp) --} Ker(N) is surjective, cf. [Tay 92] p. 115, and has SUn(K,Qp) as its kernel. Since the determinant of a1n E Un(K,Qp) is an the last claim follows. q.e.d.
XI The theory behind the tables
82
We proceed to determine generators for the Sylow pro~2-subgroup of SU3 (K, ((b). We want to describe the group SU3 (K, ((b) in terms of algebras with involutions, cf. Chapter VI. To fix some notation let K be a quadratic extension of Q2 with maximal order 0 := Z2[W]. Let A = K 3x3 and (0) == Gal(K, ((b). Denote the involution
a faa Ca ) H e b on A by The unitary group U3(K, ((b) is given ( 9 h i a ga d aa in the following by {g E Algo = g-I}. A a-invariant minimal hereditary order is given
(ihaa
be) f
a d e
by
r
:=
0.
(~ ~ ~). Its radical equals J = J(f) = ( : ~ ~). ~
~
Let (JijJi+l)-
~
0
= {x
E PjJi+l
~
~
I XO = -x}.
(XI.7) Lemma. Define G := U3 (K, ((b) n (1 + J) and G i := G n (1 + JZ) for Then G has a filtration G = G 1 > G 2 > ... G n > .,. with monomorphisms
i E N.
(G zj Gi+l,') -t ((JijJi+l)-,+). Proof. The monomorphisms are induced by the isomorphisms (1+Ji )j(1+J1+ 1) ~ Ji j Ji+ 1 via 1 + x H x. For any 9 = 1 + x E G z one ha:::; g-l - 1 - x mod (1
Since (1
+ xt = 1 + XO
+ Ji+l).
it follows that XO == -x (mod Ji+l) for 1 + x E G z.
q.e.d.
The notation for matrices dz(ao, aI, a2) given in (VI.10) will be used. The matrices are to be read as coset representatives in Jl j J1+1. First consider the case of a ramified splitting field K of SU3 (K, ((b). Let w = ~ where ~ is a uniformising element of K. One wants to find generators of GdGi+1 that are mapped into the 2-dimensionallF2 -module (Ji j Ji+l) -. One observes that (Ji j p+l)+ = (Ji j Ji+l because the characteristic of the residue class field is 2.
t
(J3i+l j J31+ 2)- = (dl(~i, _(~a)i, 0), d l (0, O,1I"i+l)), (J3i+2jJ3i+3)- = (d2(~i,O,O),d2(0,~i+l,_(~a)i+l)), (J3(i+l) j J 3(i+l)+I)- = (dO(~i+t, I, _(~a)i+l), do(O, ~i+t, 0)). The map Gd G i +l -t (Ji j P+ 1 1 + x H X is not surjective for every i. This
t :
is due to obstacles which can be seen by carrying out calculations of the following kind: Let x E (Jij Ji+l and y E p+l. Assume 9 = 1 + x + Y E G t • Then 1 = gg-1 = ggO = (1 + x + y)(l + XO + yO) = 1 + x + XO + XXO + zx(Y), where x + XO E Ji+l and XXO E J2i and zx(Y) is a term in y and x. But it may happen that there is no solution y E Ji+l such that x + .,£0 + XXO + zx(Y) = 0 as explained in detail for every case below. In such a case we have a contradiction to the above assumption 9 E Gi · The series of GdG i +1 for i E N have the following dimensions over lF2 C marking the repetition) :
t
Case 1: 1, I, 1, 1, 1, 2, 2, 1, 2, 1,2, 1 for the extensions with minimal polynomials x 2 x 2 + 6, x 2 - 2 or x 2 - 6. Let i ;::: 1. Consider G6i +2/G6i+3'
+ 2,
83
°
If 9 = 1 + x + y with x = d 2(2\ 0, 0) E (J6H2 / J6i+3t. Then = 99° - 1 = x + XO + Y + yO = d 2(2HI, 0, 0) + y + yO (mod J 6(i+I)+3) because all other summands lie in higher powers of J. Since d 2(2 i+1, 0, 0) E J 6(H1)+2 it is necessary for the existence of a solution to the equation that there is ayE J6i+3 - J 6(1+1)+3 satisfying y + yO == d2 (2 i + 1 , 0, 0) (mod J 6(Hl)+3). But no such solution exists. Consider G6i+4iG61+5' For x = d 1(0,0, 2i ) one finds that there is no solution for 9 = 1 + x (mod J6i+5) in an analogous way. Consider G 6(Hl)/G6(HI)+I' i.e. x = d o(0,2 i+l,0) E (J 6(i+1)/J 6(i+ 1)+1)-. If = 99° - 1 x + XO + Y + yO (mod J 6(H2)+1) it follows that x + XO E J 6(i+2). But since there exists no solution y E J 6(i+1)+1 - J 6(1+2)+1 to the congruence x + XO + Y + yO == (mod J 6(i+3)) it follows that 9 = 1 + x tj. G6i+4/G6i+5' The arguments are very similar for i = but one also has to consider the term XXO which causes a slightly different behaviour. For all 9 - 1 + x (mod p+l) with x E (J i / JHl)- and x not as above the congruences are solvable. This can be checked by calculations or follows from Lemma (XI.8) below.
°
°
°
Case 2: 1,1, 2, 2, 1, 2, 1, 2, 1, 2, 1,2 for the extensions with minimal polynomials x 2 + 2x + 2, x 2 + 2x + 6. Let i 2:: 1. Consider G 6z +1 / G 6i + 2 Assume 9 = 1 + x + Y E G 6i +1 with x = d 1(0,0, 21 n) E (J6i+l / J6i+2)-. Then = 99° - 1 x + XO + Y + yO d 1 (2 i +1 ,0,0) + Y + yO (mod J6i+5). Hence x + XO E J6i+4 there exists no solution to the equation. For G6i+5/G6H6, in particular x = d 2(2 I n, 0, 0) E (J6i+5 / J6i+6t the calculation is done in an analogue way and yields no solution for 9 = 1 + x (mod j3i+6). Now consider G6H3/G6H41 i.e. x = do(O, 2i n, 0) E J 6i+ 3. If = 99° - 1 == x + XO + Y + yO (mod J 6(Hl)+1) hence x + XO = do(O, 2i +1, 0) E J 6(H1). But the equation is not solvable. The arguments are very similar for i = but one also has to consider the term
°
=
=
°
°
XXO.
For all 9 == 1 + x with x E (P / Ji+l) - and x not as above the congruences are solvable. This can be checked by calculations or follows from Lemma (XI.8). Now consider the case of an unramified splitting field. Let w be a primitive 3rd root of unity. The lF2 -modules (Ji/Ji+1)~ are 4-dimensional. (j3Hl / j3 H2 = (d 1(2 1 , - 21 ,0), d 1(2 iw, -2 iwl7 , 0), d1 (0,0, 2H1 ), d l (0,0, 2i + 1w)) H (j3 2/ j3i+3t = (d 2(2\ 0, 0), d 2(2 1w, 0, 0), d 2(0, 2i + 1, - 2i + 1 ), d 2(O, 2i+ 1 w, - 2i + 1W (j3i+3 / j3i+4)~ = (d 3(2i+ 1, 0, _2 i+ 1), d3(2 1 +1W, 0, _2 i + 1W d 3(O, 2i+1, 0), d 3(0, 2i+1W, 0))
t
(7
))
(7
),
But due to obstacles as in the ramified case one gets smaller dimensions for
G I /G 1+1
:
Case 3: 3,3,3 for the unramified extension (x 2 + x + 1). For G 3i+J / G 31+J+1l 1 .:S j .:S 3 and i ~ 1 consider 9 = 1 + x + y with x E (j3Hi / j3i+J+1)~ then = 99° - 1 == x + XO + Y + yO (mod j3i+i+4) If x fixed under ° it follows that x + XO E j3i+H3. In particular these elements are x = d 1(0,0, 2t x = d 2(2i, 0, 0), x = d 3(0, 2\ 0). But there exists no y in j3i+Hl _ j3i+H4 satisfying x + XO (mod j3Hi+4) Y + yO (mod J3i+ i +4). The
°
=
XI The theory behind the tables
84
case i = o works out similarily. Forallg == l+x (mod Ji+1) with x E (Ji/Ji+ 1)and x not as above the congruences are solvable. This can be checked by calculations or follows from Lemma (XI.8).
(XI.8) Lemma. Using the notation for matrices di(ao, aI, a2) explained in (VI.10) the Sylow pro-2~subgroup G of U3 (K j 02) is generated by
Case L extensions x 2 + 2, x 2 + 6, x 2 - 2, x 2 - 6 h + d1 (OJ 0, n), do (1 + n, 1, (1 + n a)~ 1 ), h + d1 ( n, n, 0) + d2(- N (n) . 2- 1 , 0, 0), do ( -1, 1, - 1), do (1, - 1, 1), h + d1 (0, 0, - 2) + d2(0, 2, - 2), do (1, c, 1) for cc a = 1 and c - 1 == 2n(mod 4)' Case 2: extensions x 2 + 2x + 2, x 2 + 2x + 6 /3 + d 1(0, 0, c) + d2(Ojn, -n a ) with c + ca = -N(n) and v(c) = 1, h + d2(c, 0, 0) with trace(c) = 0, v(c) = 0 do(l+n, 1, (1 +na)-l), do(l, c, 1) for c-1 = n(mod 2) and N( c) = 1, do( -1, 1, -1), do (1, -1, 1), Case 3: unramified extension x 2 + x + 1 /3 + d1(1, -1,0) + d2(w, 0, 0), h + d1(0,0, -2 - 4w), do(l h + dl (w, -w a j 0) + d2(w, 0, 0), do(3 + 2w 1, (3 + 2w a)).
+ 2w, 1, (1 + 2W a)-1)j
j
The Sylow pro-2-subgroup of AutQ2(£) is isomorphic to G/Z(G) Xl (ij) where Z(G) denotes the centre of G and ij is induced by the Galois automorphism (J acting entrywise on the matrices. Proof. Let H be topologically generated by the above elements. In all three cases the calculation of the lattice of H-invariant sublattices of a 1x3 (by a p-adic version of the sublattice algorithm d. [PIP 77]) shows that there is a chain of distinguished sublattices through 0 1 X3. The Sylow pro-2-subgroup of the normaliser of the group acts on this chain. Define Hi := HnG i . If the splitting field is ramified (e = 2) let io = 12, otherwise (e = 1) i o = 6. By computation one checks that H z /HH1 rv GdG i+1 for i S i o. For i > i o one uses the map x H x 2 to prove the isomorphism by induction since for 9 = l+x+y E Hi with x E Ji_Ji+1 follows g2 = (1+x)2 1+2x(mod j3e+i+I) and 2x E j3e+i. Therefore all elements of the normaliser which fix this chain and lie in U3 (K, Qp) have already been found. The rest follows from (XI.5) and (XI.6). q.e.d.
=
(XI.9) Remark. See also [Tit 79]. Apart from the residue class field being of characteristic 2 part of the difficulties here might result from the fact that the root system of SU3 (K, Qp) is not reduced in the sense that some roots allow multiples which are also roots. Now we determine the generators for SU3 (K, (3) for p = 3 and a ramified extension of Q3 as splitting field K. Since (radA)- is only semi-saturated (cf. VI.18) one has to choose a generator additionally to those which are images of ((radA)l)- - ((radA)2)under the Cayley map CLP : X H ~~~ for constructing SU3 (Q!s [n]) n (1 + J) (n 2 = 3 or n 2 = -3). From (XI.6) it follows that for generating PU3 one also has to calculate extra diagonal elements which have a non-trivial entry in KerN/(KerN)3. There is
85
one element in this factor if 11"2 = 3 or if the splitting field is unramified. There are two elements if 11"2 = - 3. Hence we have the following lemma. (XI.10) Lemma. The Sylow pro-3-subgroup of AutQ3 (.C) for a ramified splitting field is generated modulo central elements by the following matrices. If 11"2 = 3 : (d I (11",11", O))CLP, (d I (1, -1, O))CLP, (d I (0,0, 1I"))CLP and do(l, aI, 1) where 0 "th - 21093(1+3) al = (1 + 3)(1 + 11" ) w~ 0: = 1093(1-3) . If1l"2 = -3: (d I (1I",1I",0))CLP, (d I (l,-l,O))CLP, (d I (0,0,1I"))CLP, do(l,aI,l) and do(l, a2, 1) where aI = (1- 3)2(1 +11")0 with 0: = 2 -12;;:N1~)3) and a2 = (1- 311" )2(1 +11")# "th (3 = 2 - 1093(1+27) w~
10 93(1+3)
.
(XI.H) Lemma. The Sylow pro-3-subgroup of AutQ3 (1:) with an unramified splitting field (e.g. K = (Ch [(g]) is generated modulo central elements by the following matrices. (d I (l, -1, O))CLP, (dI(w, -w a , O))CLP, (dI(O, 0, C))CLP with trace(c) = and I/(c) = 1, do(l, -1 + (g - (l, 1).
°
Proof. The generators follow straightforward from (VI.17) by adding do(l, -1 + (g - (i, 1) as an extra generator to pass from SU3 to PU3 hence z = -1 + (g - (i generates Ker(N)j(Ker(N))3. This last claim holds because the factor group Ker( N) modulo the torsion elements is isomorphic to Z3 since the index of N(K*) in Q3 is finite. Furthermore there is no 3-torsion in Q, and no third root of z lies in Z3[(g]. q.e.d.
Assume a division algebra D of dimension d 2 over its centre K is given in the following way, cf. [Rei 75]: Let w be a (qd - l)th root of unity and W = K[w]. By 1I"K E K denote an element of valuation 1 in K" An automorphism () generating Gal(W, K) qT maps w t---+ w for some r, r fn. The elements of D are given as K-linear combinations of (w*r and (11" D)j, 1 < i, j ::; n, where 0:* := do(0:,0: 0 , ... ,0: 0,,-1) for any 0: E Wand 1I"D := d 1 (1, ... ,1, 1I"K) which is an element of valuation 1 in D. Consider the Lie algebra 1: = Sll (D) consisting of those elements of D which have trace 0. Tensoring by W yields W ®K D = Wdxd and W ®K 1: rv sld(W). Therefore we view D as K-subalgebra of W dxd . The automorphism () E Gal(W, K) applied entry-wise to the matrices induces an automorphism on D. Now we want to exhibit how to extend the Qp-automorphisms of the centre K to an action on D. (XI.12) Lemma. Let K be an extension ofQp with IK : Qpl ::; 4 and W an unramified extension of K of degree d. Define F := FixAutQp(K)(K) a fixed subfield in K. Then) W is a Galois extension over F with an abelian Galois group. Proof. Define Fu < K to be the maximal unramified subfield over F of the extension Kover F. Since K is a Galois extension over F with abelian Galois group
XI The theory behind the tables
86
Gal(K, F) it follows that W = K F where F is the unramified extension of degree d of Fu . The Galois group of F over F is cyclic and the Galois group of Kover F is abelian of degree less or equal to 4. The claim follows since in the Galois group G = Gal(W, F) the subgroups corresponding to K and F are normal with trivial intersection and have abelian factor groups in G. q.e.d.
(XI.13) Lemma. With the above notation and IK : Qp I ::; 4 it follows that AutQp(D) = AutF(D)r>AutK(D) and AutQp(D)/AutK(D) ~ AutQp(K) ~ Gal(K, F). In particular, every Qp -automorphism of the centre K of D can be extended to a Qpautomorphism of D. Proof. Since the automorphism group of D acts on K it only remains to show how the extension of the automorphisms Aut(K, Qp) works out. Let [i· d-Ij be the Hasse invariant of Dover K and (j E Gal(W; K) the i-th power of the Frobenius automorphism. Applying Lemma (XI.12) one gets that every 7/J E Aut(K, Qp) 'V Gal(K, F) lifts to d elements 7/JI, ... ,7/Jd E Gal(W, F) with 7/Ji = (ji-\7/J\ for i = 2, ... ,d. Applying 7/Ji to D yields D1/Ji which has the same Hasse invariant as D since 7/JI and (j commute. Hence there exists a matrix in W dxd which conjugates D1/Ji to D. A matrix for this purpose is given by M ;= do(1, n1/J' ... , n~+o+._+Od-2) where n1/J E W is a solution of ,p
N w / Kn1/J = ~. The conjugation operation by M is denoted by KM. Then, 7/JIKM is a Qp-automorphism of D. q.e.d. (XI.14) Lemma. Let K be the centre of the division algebra D and let £., be sll(D). Then AutK(£") = PGLI(D). Proof. If d > 2 then any automorphism of £., is induced by conjugation of an element of PGL I (D) since the two epimorphic images of the universal enveloping algebra U(£.,) namely D and DOP which come from the two representations of sld(W) of lowest degree are not isomorphic as algebras. If d = 2 the two division algebras of this degree and the two representations of the universal enveloping algebra U (£.,) become isomorphic such that one also gets all automorphisms by conjugation of PGLI(D). The automorphism (j E Gal(W, K) applied entry-wise on PGL l (D) acts in the same way as 1fD E PGL I (D) by conjugation. q.e.d.
(XI.15) Lemma.
a) Denote by - : GL I (D) -t PGL\ (D) the natural epimorphism. Let 1f be a uniformising element of OK. Let e be the ramification index of the centre K of D and W a Zp-basis of Ow. Define U(s) = {1 + 1fbn*ln E W, j = 1, ... , s}. The Sylow pro-p-subgroup of PGL\(D) is generated by (i) U(de
+ lp~\J)
if pfd,
(ii) U(de+ lp~IJ)U{1fb} ifd=pkl andpfl.
b) Let IK ; Qpl ::; 4. Define r to be the Sylow p-subgroup of AutQp(K) such that for 7/J E r holds 1f1/J 1f(mod 1f2). The Sylow pro-p-subgroup AutQp (£.,) zs isomorphic to PGLI(D) extended by the r described in Lemma (XI. 13).
=
f) sos(Qp), split case
87
Proof. a) Let A be the maximal order of D. Let w be a primitive (pf d - 1)th root of unity where f is the inertia degree of K. It follows that A = 2:t':-J R[w]7rb with valuation ring R of K ([Rei 75] p. 146). The Jacobson radical is given by J(A) = 2:t=l R[w]7rb· The Sylow pro-p-subgroup G of GL1(D) is l+J(A). There is a filtration of the group G by normal subgroups Gt = l+(J(A))i such that there is an embedding Gi!G t +1 y. J(A)i j J(A)i+l via 1 + x 1---7 x. Let H be generated by one of the sets U according to the specific case. Define H1 by Hi := H n (1 + J(A)i). For i S de + lpC:~\ J Hi! H i +1 maps surjective to J (A) i j J (A) i+ 1. The map y 1---7 yP has the effect that for an element 1 + x E Hi with x E Ji - p+l the p-th power (1 + x)P == 1 + px(mod p+de+l) with px E Ji+de and px(mod Ji+de+1) is non-trivial. Therefore it follows by induction that for every i > de + lp~lJ every quotient HdHH1 ~ J(A)ijJ(A)i+l. This proves that H is a full Sylow pro-p-subgroup of GL1(D). Now consider PGL 1(D). If d is divisible by p, the matrix 7rD is the only missing representative for a generating set. For the Sylow pro-p-subgroup of PGL 1(D) one has to take the l-th power of this matrix where n = I . pU with I not divisible by p. b) The choice of r makes sure that r normalises the chosen Sylow pro-p-subgroup of PGL 1(D). The claim follows from Lemma (XI.13). q.e.d. Remark. The given number of generators is far too big. In every specific case one easily reduces this number by also taking into account what one gets by commutators.
f)
305 (Qp),
split case
(XI.16) Lemma. The Qp-Lie algebra I: = sos(Qp) is generated by X a = d1(0, 2, -1 1 01 0), Xa = d1(1, 0, 0, -1 1 0), and X- 2a -{3 = d2(0, 01 0,1, -1) as a Lie subalgebra of sls(Qp)· For p =J. 2 the generators for the maximal Sylow pro-p-subgroup of AutQp(l:) are x a (l) = exp(Xa ), x{3(l) = exp(X{3) and X-2a-{3(P) = exp(pX_ 2a -{3)' Proof. This is an immediate consequence of (V.3).
q.e.d.
(XI.17) Remark. For p = 2 the situation is more complicated because in the first instance one has to deal with obstacle (i) (see after (V.3)) since one of the Cartan numbers is 2. To overcome this one has to add an extra generator for example x2a+{3(1) = exp(X2a +{3)' But also obstacle (ii) occurs that is IQ : Z
XI The theory behind the tables
88
the exterior square tensor module /\2V. Consider S05((h) in the 4-dimensional projective representation in terms of algebras with involution (see Chapter VI). Let A =
=
(~~~a31 ~~~a32 ~~:a33 ~~:) a34
1----7
Fa tr
-a41
An a-invariant minimal hereditary order denoted by
where SP4((h) = {a E Alao
~ ~1! ~).
° °° F-I = (~:: ~:: =~~: =~~:) on A by -a42 -a32 a22 al2
a41 a42 a43 a44
Its radical is of the form J
and F = (
-1
Denote the involution a
Q~X4
= J(r) = =
-a31
r
a21
all
is of the form
(H ~ ~).
0.
(~ ~ ~ ~). 222
One has S05((h)
~
0
SP4((h),
2 2 2 2 -a}. Define the symplectic group
Sp4((h) := {x E Alx- 1 = XO}. (XI.I8) Lemma. With the above notation on has
a) (JI/J2)~ = (dl (I,0,-I,0),d l (0,I,0,0),d l (0,0,0,2)), (p / J3)- = (d 2 (I, 1, 0, 0), d2 (0, 0,2,2)), (J3/J4)- = (d 3(I,0,0,0),d 3(0,2,0,-2),d 3(0,0,2,0)), (J4/ J5)- = (d o(2, 0,0, -2), do(O, 2, -2, 0)), (J4k+I/J4k+2)- = 2k(Jl/P)~, dimension 3, (J4k+2/J4k+3)- = 2k(P/J3)-, dimension 2, (J4k+3 / J4k+4)- = 2k( j3 / J4) -, dimension 3, (J4k+4/J4k+5)- = 2k(J4/J5)-, dimension 2. b) G:= Sp4((h) n (1 + J) has a filtration by normal subgroups G i := G n (1 + Ji) G 1 = G > G 2 > '" where (Gi/G i + l ,') is embedded into ((Ji/J i+1 )-, +) via 1 +x
1----7
X.
c) The group G is generated by 'H:= {14 + dl(I, 0, -1, 0), 14 + dl(O, 1,0,0), 14 + dl(O, 0,0,2),14 + d3(I, 0, 0, 0), 14 + d3(0, 0, 2, On. The group G and representatives of the rational automorphisms 'R.. = {do(I, 1, -1, -1), do(I, 1,3,3), d2(I, 1, 2, 2n act by conjugation on L. The kernel of this action consists of scalar matrices. The Sylow pro-2-subgroup of AutQ2 (£) is generated by 'H and'R.. modulo scalar matrices. Proof. a) and b) can be checked by straightforward calculations analogous to Lemma (XI. 7) and the calculation which follows Lemma (XI.7). The maps from Gi/Gi + l ---1' (Ji/Ji+l)- are surjective in this case. c) Let H be generated by 'H. Define Hi := H n Gi . For i S 8 it is checked by
89
g) S05('Op), non-split case
calculations that HdH i + 1 ~ (Ji / ji+l) ~. For i > 8 these isomorphisms follow by induction applying the map x 1----+ x 2 to the elements hence for g == 1 + x + y E Hz with x E Ji - Ji+l follows g2 == (1 + x + y)2 = 1 + 2x (mod Ji+5) and 2x E Ji+4 is not trivial modulo Ji+5. All other terms lie in higher powers of J. Any tlh-automorphism of I: is induced by conjugation of an element ¢ E PGL 4(tlh) acting on 'O~ x4 since there is only one faithful irreducible 4-dimensional representation for the Lie algebra B 2 • The lattice of H-invariant lattices of olx4 consists of a chain. The Sylow pro-2-subgroup of the normaliser of G in PGL 4(tlh) acts on this chain. Its elements either lie in SP4 (tlh) n (1 + J) or act non-trivially on the form F. Any automorphism ¢ of I: acts on {A E 'O~x4llA = -Altr for alll E I:} = tlhF. The ones acting non-trivially on the form F map F to a multiple and hence are similitudes. The factor group Sim(tlh1° )/(Sp4(tlh) . tlh14) embeds into '0;/('02)4 by mapping the elements to their scaling factor. The group of similitudes Sim(tlh 1°) contains G = G· '0;14. Let N be the the normaliser N S1m(fb,O)(G) and ~ := Sp4(Q2) . tlh14. Then Z := N/(N n Sp4(tlh)tlhI4) embeds into '02/('0;)4. Since G is of finite p-prime index in Nu(G) = NSim(fb,O)(G) n U it follows that G can be extended by Z. Such an element has determinant y2k for kEN and y E 'Op since det(aFa tr ) = det(yF) = y4F and therefore det(a)2 = y4. Hence as representatives modulo scalar matrices one has do(1 111 -1, -1), do(1 1, 31 3) and d2(1111 2 2) as generators. q.e.d. j
j
Let K be an unramified extension of tlh of degree 21 w a 3rd root of unity Gal(K 1 tlh) = (0). Denote by Q := JC 2(tlh) the quaternions over tlh given as Q =
{(
2~a
(I =
The map - : Q -> Q :
:a )la,b
(~
n,
(2~'
i
E
tlh[w]}
~ (~ ~. )
:. ) -> (
,j
~ (~ ~), k ~ ij)
_~. -:zb)
defines an involntion on Q.
The Z2-maximal order of Q is called D and its radical J(D). On
Q2x2
we have the involution" :
hermitian form
(~ ~)
0-+
(g 1)
given by the --
(~ ~) and a minimal "-invariant hereditary order r:= (J~)
The radical J(f) is given by
g).
(~~g~ J~))'
It follows that the factors J(f)i/ J(f)i+l ~ lFf. Note, because the characteristic of the residue field is 2 it follows that (J(f)i / J(f)i+l)(J(f)i / J(f)i+l)-. (XI.19) Lemma. With the above notation and I: = {x E Q2X2 1xo = -x} which is
of type B 2 the following holds.
=
XI The theory behind the tables
90
a)
(J(r)IJ(rf)-=((~ ~) (~ ~) (~ ~)), (J(rfIJ(r)3)- = (( j
o
(J(f)'/J(f)')- = ((
0 -, ) -J
(kO-k 0 - )),
~ ~) (~ ~) (~l ~ )),
(J(r) 4I J (r)5)-=((2/0 ) (2i 0 -,)), o -21 0 -21, k (J(f)4k+1 I J(r)4k+2)- = 2 (J(f) II J(r)2)-, dimension (J(f)4k+2IJ(r)4k+3)- = 2k(J(r)2IJ(f)3)-, dimension (J(r)4k+3 I J(r)4k+4)- = 2k(J(r)3 I J(r)4)-, dimension (J(r)4k+4IJ(f)4k+5)- = 2k(J(r) 4IJ(f)5)-, dimension
3,
2, 3, 2.
b) The subgroup G:= U2(Q, F)n(l+J(r)) has ajiltration Gi := Gn(l+J(f)i) with G I = G > G2 > ... > Gi > ... where GdGi+1 embeds into (J(f)iIJ(f)i+I)via 1 + x I-t x.
c) The set 1-£1
:= {
0) ' (I0 1j) ' (I0 1k) } and the j 1 ' (Ik 1 ( 0I 1I) ' (/0)
rational automorphisms 1-£2 := {
0 31 ' (j0 0) j } ( 0/0- I ) ' (/0)
_
generate a maxtmal Sylow pro-2-
subgroup of AutQ2 (£). Proof. a) and b) can be checked by straightforward calculations analogous to Lemma (XI. 7). c) Any ~ -automorphism of £ is induced by conjugation of an element ¢ E G L 2 ( Q) since the associative span (£)asso = Q2X2 is the only epimorphic image of the universal enveloping algebra U (£) in the requested dimension as seen from representation theory of B 2 • Note, by the theorem of Skolem and Noether it follows that the ~-automorphisms of Q2x2 are inner and therefore embed into PGL 2 ( Q). Any automorphism ¢ maps {A E Q2x21lA = -Aztr for alll E £}
= ~ (~ ~)
to itself.
Henceforth AutQ2(£) maps into the similitudes modulo their centre PSim(Q,O). The set 1-£1 is a generating set for the Sylow pro-2-subgroup G of U2 (Q, F). This can be proved by considering H := (1-£1)' Hi := H n Gi . For i ~ 8 one checks that the factors Hi/HHI are isomorphic to (J(f)iIJ(r)i+I)-. The same follows by induction for i > 8 using (1 + X)2 _ 1 + 2x (mod J(f)i+5) and 2x E J(r)i+4 is non-trivial modulo J(f)i+ 5 if x E J(f)i I J(f)i+l. One finds that G fixes a chain of D-lattices. Define U := U2(Q)· ~/2' The factor group Sim(Q,o )IU embeds into Q2/(Q2)2 ~ Cl by mapping the elements to their scaling factor. The group of similitudes Sim( Q,o ) contains G = G· Q2/2. Let N be the normaliser NSim(Q,O)(G). Then Z := NI(N n U) embeds into ~/(~)2. Since G is of finite p-prime index in Nu(G) = NSim(Q,O)(G)nU it follows that G can be extended by Z. In particular, the elements of 1-£2 are representatives of this factor where the first 2 elements act trivially on the chain of sublattices and the third one non-trivially. q.e.d.
91
(XI.20) Lemma.
(i) The Qp-Lie algebra L of type G 2 is generated by
Xet = d1(0, -1, 0, 0,1,0,0),
X{3 = d 1(1,0,-1,2,0,-1,0) andX_ 2et - 3 {3 = d2 (0,0,0,0,0,1,-1) as a Lie subalgebra of sl7(Qp)·
(ii) For p i- 2,3 the generators for the maximal Sylow pro-p-subgroup of AutQp (L) are given by x et (1) = exp(Xet ), x{3(l) = exp(X{3) and X-2et-3{3(P) = exp(pX- 2et - 3{3). For p = 2 one needs additionally
For p = 3 one needs additionally
Proof. This is an immediate consequence of (V.3) and dealing with obstacle (i), i.e. that certain Cartan numbers are divisible by 2 or 3. q.e.d.
XII
Tables
The following tables give computer generated information about the lower central series and normal subgroups of the maximal insoluble p-adically simple groups for p = 2 and p = 3 up to dimension 14. For comparison it also includes theoretically generated information on the corresponding groups, in case they exist, for primes p ~ 5. The groups are Sylow pro-p-subgroups of the automorphism groups of certain semisimple Qp-Lie algebras. Details about the groups, e.g. generators of a Sylow pro-p subgroup, are given in Chapter XI. The tables are organised as follows. 1.) Dimension of the Lie algebra 2.) Name for the Lie algebra (usually the classical name e.g. Lie algebra of type G2 , which is denoted by 92)
Sl2 (Qp)
except for the
3.) Primes and information on fields, e.g. for ramified extensions the minimal polynomial of a generator where numbers refer to chapter IX or for unramified extensions the degree since the extensions are (up to isomorphism) unique (In the general case for p 2: 5 ramified means any ramified extension of the appropriate degree.) 4.) Structural invariants of the groups (i) The isomorphism types of the factors of the lower central series. Define the following sequence 9i' If 'Yi(G)/'Yi+1(G) is elementary abelian i.e. 'Yi(G)/'Yi+1(G) ~ define 9i := n. Otherwise if 'Yi(G)/'Yi+I(G) 9:: C;11 x ... x C;c>c> where q1"'" qa are distinct powers of p then define 9i:= q~1 . .... q~c>. For example if 'Y1(G)/'Y2(G) ~ C? then 91 = 2. If 'Y1(G)/'Y2(G) ~ Ci x C4 then 91 = 23 .4 1. The - indicates a period of the sequence. If p = 2 or p = 3 this is proved by the p-th powering map. For p ~ 5 the results follow from Chapter V and VI. The semicolon before the i-th term indicates that 'YJ (G) is powerful for j > i.
C;
(ii) The obliquity, if it was calculated. Namely the sequence
where J.Li(G) is defined to be the intersection of 'Yi+1 (G) with the intersection of all normal subgroups N of P with N 1:. 'Yi+ 1(G). The - indicates that the sequence repeats periodically which is proved by calculating in a sufficiently big quotient of the group, see Chapter X e). If this was too expensive, e.g. for groups with rather 'oblique' normal subgroups and big central sections it happens that the sequence obviously runs into some pattern but it has not yet been proved. We believe that the last k numbers will repeat where k is the length of the period of the lower central sequence marked by The groups of dimension 12 are rather wide in the first sections. For these examples we give the ultimate obliquity
93
a) Dimension 3
where
for a chosen kEN. Unfortunately, it was not possible to prove by calculations in a finite quotient that the ultimate obliquity will repeat periodically like the given patterns because it is not clear whether one is deep enough down in the group. Choosing k = 5 makes the computation practically manageable. We give Ui only for those i E N where we have the moral certainty that Ui = 0i' There are a few remaining groups where we could not give any information about the obliquity since the sections 'Yi(G)/'Yi+l(G) are too wide. (iii) Define, if
or if
'1L 2
0i
is calculated
is calculated
mi
:=
min{jl'Yj(G)
s: Pi(G)} - (i + 1)
i.e. 'Yi+ Hm, (G) is the largest term of the lower central series contained in Pi (G). Periodicity is treated as in (ii).
a)
Dimension 3
Sl2(Qp), cf. XLc)
p=2 9j O2
mz
p=3 4,2; 2,1 5,3, 3,2 3,2, 3, 2
p=2
p~5
9i
2;1,2
OJ
0, 0, 0, 0,
m2
°°
p=3 4,2;2,1
9i
2; 1,2
Oi
5,3, 3, 2 3,2,3,2
OJ
0,0,0 0,0,0
mi
2; 1,2
p~5
9i
m
9i
9i
2;1,2
94
b)
XII Tables
Dimension 6
sI2(K) with IK: Qpl
= 2,
cf. X1.c)
p=2
9i Oi
Tnt
x2
4,4,2,1,2,1;2,1,2,1 7,11,9,8,9,8,9,8,9,8 3, 7,6,5,6,5,6,5,6,5
+ 2x + 2 , x 2 + 2x + 6
9i 0t
Tn i
4, 3, 2, 1, 1, 1; 2, 1,2, 1 9,8,6,5,4,3,5,4,4,3 6,6,5,4,3,2,4,3,3,2
9i 0i
Tni
5, 3,2,1,2,1;2,1,2,1 14,11,9,8,9,8,9,8,9,8 8, 7:6,5,6,5,6,5,6,5
unramified of degree 2
9i 0i
Tni
4,3,2,3;2,2,2 10,7,5,4,6,6,4 5,4,3,3,4,4,3
p=3
9t 0t
Tni
2,1,2;1,2,1,2 0,0,0,0,0,0,0 0,0,0,0,0,0,0
9i Oi
Tni
2, 1, 2; 1, 2, 1,2 0,0,0,0,0,0,0 0,0,0,0,0,0,0
unramified of degree 2 9t 0t
Tni
4;2,4 0,0,0 0,0,0
p?:5 ramified of degree 2 9t
2, 1,2, 1
sI2(~)2, cf.
(11.6)
p=2 9i 0i
Tni
5, 4,2,2; 2, 2, 1, 1 14,10,8,6,8,6,5,4 7, 6,5,4,7,6,5,4
unramified of degree 2
9i
4,2
95
b) Dimension 6
Sll (Kd K )) with IK : Qp I = 2, cf. XLe)
p=2
gi 0i
Tni
x
2
4, 4,2,1,2,1;2,1,2,1 7,11,9,8,9,8,9,8,9,8 3, 7,6,5,6,5,6,5,6,5
+ 2x + 2 , x 2 + 2x + 6
9t O~
Tni
unramified of degree 2
4,3,2,1,1,1;2,1,2,1 9,8,6,5,4,3,5,4,4,3 6,6,5,4,3,2,4,3,3,2
Tni
gi
2, 1, 2; I, 2, 1, 2
g~
Oi
0,0,0,0,0,0,0 0,0,0,0,0,0,0
Tni
gi o~
Tni
23 ·4\ 3,2,1,2,1; 2,1,2,1 14,11,9,8,9,8,9,8,9,8 8, 7,6,5,6,5,6,5,6,5
gi Oi
23 .4 1 ,2,2,1,2; 1,2,1,2 10,8,6,5,5,4,6,6,5 7,6,5,4,4,3,5,5,4
p=3
Tni
Oi
2,1,2;1,2,1,2 0,0,0,0,0,0,0 0,0,0,0,0,0,0
unramified of degree 2
gi
4; 2, 4
0i
0, 0,
Tni
0,0,0
°
p~5
ramified of degree 2
gi
2,1,2,1
Sll(Kd«:b))2, cf. (11.6)
p=2 g~ 0i
Tni
5, 4,2,2;2,2,1,1 14,10,8,6,8,6,5,4 7, 6,5,4,7,6,5,4
unramified of degree 2
96
c)
XII Tables
Dimension 8
sI3(Qp), ef. XLc)
p=3
p=2
9i
3,1,2,1;1,1,2,1,2,1
Oi
3,2,2,1,2,1,2,2;1,2,1,2,2 3,1,3,2,4,3,1,3,2,4,3,1,3
0z
5,4,2,1,0,0,5,4,2,1
mz
2, 1,2,1,3,2,1,2,1,3,2,1,2
m~
4,3,2,1,0,0,4,3,2,1
9i
p'25
sU3(K, Qp) with IK : Qp I = 2, cf. XLd)
p=2 x 2 + 2, x 2 + 6, x 2 - 2, x 2 + 6 9i 0i
mi x2
5, 4, 4, 2, 3, 2; 3, 2, 2, 2, 2 13,15,15,13,14,16,14,19,17,16,14 5, 6, 8, 7, 7, 9, 9,11,10,10, 9
+ 2x + 2, x 2 + 2x + 6
9i ~
mi
4, 4, 4, 2, 2, 2; 2, 2, 2, 2 9,10, 9,14,12,11, 9,14,12,11 3, 5, 5, 7, 6, 6, 5, 7, 6, 6
unramified of degree 2
OJ
3, 2, 2, 1, 2, 1, 2, 2; 1, 2, 1, 2, 2 3,1,3,2,4,3,1,3,2,4,3,1,3
mz
2,1,2,1,3,2,1,2,1,3,2,1,2
9i
p=3
9i
4,2,2; 1,2,2,3
9i
Oi
10,8,6,5,4,2,6 6,5,4,3,2,1,4
0z
3,2,3; 1,2,2,3 4,2,6,5,4,2,6
mi
2,1,4,3,2,1,4
mi
unramified of degree 2 9i
4,3;2,3,3
Oi
5, 3, 8, 5, 3
mi
3,2,4,3,2
unramified of degree 2
ramified of degree 2 9i
2, 1, 1, 1, 2, 1
97
d) Dimension 9
p=2 9i 0t
mt
p=3 3,3,2,3,3;2,3,3 0,0,0,3,0, 0,3, 0,0,0,1,0,0,1,0
°
9i 0i
mt
3,1,2,1;1,1,2,1,2,1 5,4,2,1,0,0,5,4,2,1 4,3,2,1,0,0,4,3,2,1
p~5
d)
Dimension 9
p=3 9t 0i
mi
3,2,2;1,1,1,2,2,2 4,2,0,0,0,0,4,2,0 2,1,0,0,0,0,2,1,0
p=3 9t 0i
mi
3,2,2;1,1,1,2,2,2 4,2,0,0,0,0,4,2,0 2,1,0,0,0,0,2,1,0
p=2
6, 2, 2, 2, 1, 1, 2, 1, 2, 1; 2, 1, 2, 1, 2, 1 19,17,15,13,12,11,12,11,12,11,12,11,12,11,12,11 12,11,10, 9, 8, 7, 8, 7, 8, 7, 8, 7, 8, 7, 8, 7 unramified 9i 8,3,5;3,6 0i and mi unknown
XII Tables
98
p=3 x 3 + 3, x 3 + 12 , x 3 - 6 , x 3 + 3x 2 + 3, x 3 - 3x + 3, x 3 + 3x 2 + 3x + 3 (non Galois extensions) 9z
x 3 + 3.7: - 3, x 3 + 3x - 12, x 3 + 3x + 6 (Galois extensions) 9i
0z
2,1,2,1,2; 1,2,1,2,1,2 0,0,0,0,0,0,0,0,0,0,0
0t
3,1,2,1,2; 1,2,1,2, 1,2 3,0,0,0,0,0,0,0,0,0,0
rni
0,0,0,0,0,0,0,0,0,0,0
rni
1,0,0,0,0,0,0,0,0,0,0
unramified of degree 3 9i 0i
3,2,2; 1, 1, 1,2,2,2 4,2,0,0,0,0,4,2,0
rni
2,1,0,0,0,0,2,1,0
p~5
unramified of degree 3
ramified of degree 3 91
sIt (K: 2 (Qp)) with
IK : Qp I =
2,1,2,1,2,1
3, cf. XI.e)
p=2
9i 0i
rni
6, 2, 2, 2, 1, 1, 2, 1, 2, 1; 2, 1, 2, 1, 2, 1 19, 17,15, 13, 12, 11, 12, 11, 12, 11, 12, 11, 12, 11, 12,11 12,11,10, 9, 8, 7, 8, 7, 8, 7, 8, 7, 8, 7, 8, 7
unramified of degree 3 9i 8, 3, 5; 3, 6 0i and rni unknown
p=3 x 3 + 3,x 3 + 12,:r 3 - 6, x 3 + 3x 2 + 3, x 3 - 3x + 3, x 3 + 3x 2 + 3x + 3 (not Galois extension) 9i
x 3 + 3x - 3, x 3 + 3x - 12, x 3 + 3x + 6 (Galois extension)
2,1,2,1,2; 1,2,1,2,1,2 0,0,0,0,0,0,0,0,0,0,0
9t
0i
0i
3,1,2,1,2;1,2,1,2,1,2 3,0,0,0,0,0,0,0,0,0,0
rni
0,0,0,0,0,0,0,0,0,0,0
rni
1,0,0,0,0,0,0,0,0,0,0
99
e) Dimension 10
unramified of degree 3 9j
3,2,2; 1, 1, 1,2,2,2 4,2,0,0,0,0,4,2,0 2,1,0,0,0,0,2,1,0
OJ
mi
unramified of degree 3
ramified of degree 3 9j
e)
2,1,2,1,2,1
Dimension 10
S05(Qp)
~
SP4(Qp) (split), cf. XLf)
p=2 9i OJ
mt
6, 4, 3, 3, 3; 2, 4, 3, 2, 3, 3, 3, 3, 2, 3, 2 23,19,17,14,13,11,13,13,12,11,12,12,12,12,14,12 9, 8, 7, 6, 5, 4, 7, 6, 5, 6, 7, 6, 7, 6, 7, 6 p~5
p=3 9i
3,2,3;2,3,2,3
0i
4,2,4,3,4,2,4
mj
2,1,2,1,2,1,2
9i
3,2,3,2
S05(Qp) ~ SP2(JC 2(Qp)) (non-split), d. XLg)
p=2 9i OJ
mi
6, 4, 3, 3, 3, 2, 4, 3, 2, 3, 3, 3·, 3, 2, 3, 2 23,19,16,13,12,10,12,12,11, 9,12,11,10,12,13,11 9, 8, 7, 6, 5, 4, 7, 6, 5, 5, 7, 6, 6, 6, 7, 6
p=3 9i OJ
m1
f)
p~5
3, 2, 3; 2, 3, 2, 3 4,2,1,1,4,2,1 2, 1, 1~ 1,2,1,1
Dimension 12
sh(K) with IK : Qp I = 4, cf. XLc)
9i
3,2,3,2
XII Tables
100
p=2
polynomials: 1-4,7-10,13-20,27-30 9i 5,5,2,2,2,1,1,1,2,1,2,1,2,1; 2,1,2,1,2,1,2,1 Ui 15, 14,15,14, 15, 14, 15, 14, 15, 14, 15, 14 ni j 10, 9,10, 9,10, 9,10, 9,10, 9,10, 9 polynomials: 5,6,11,12,21-25,31 9t 6,4,2,2,2,1,1,1,2,1,2,1,2,1; 2,1,2,1,2,1,2,1 Ui 15,14,15,14,15,14,15,14,15,14,15,14 nit 10, 9, 10, 9, 10, 9, 10, 9, 10, 9, 10, 9 polynomials: 26,32 9i 6,3,2,1,2,1,2,1,1,1,2, 1, 2, 1; 2, 1, 2, 1, 2, 1, 2, 1 Ut 11, 10,11, 10,11, 10,11,10,11,10,11,10 Tni 8, 7, 8, 7, 8, 7, 8, 7, 8, 7, 8, 7 polynomials: 33,35 9i 23 .4 1,4,3,3,2,1,1,1,2,1,2,1,2,1; 2, 1,2,1,2,1,2,1 Ui 15,14,15,14,15,14,15,14,15,14,15,14 Tnt 10, 9, 10, 9,10, 9,10, 9, 10, 9, 10, 9 polynomials: 34,36,37,39 9i 24 .4 1,3,3,3,2,1,1,1,2,1,2,1,2,1; 2,1,2,1,2,1,2,1 'Ui 15,14,15,14,15,14,15,14,15,14,15, 14 Tni 10, 9,10, 9,10, 9,10, 9,10, 9,10, 9 polynomials: 38,40 9i 23 .4 1 ,4,3,3,2,1,1,1,2,1,2,1,2,1; 2,1,2,1,2,1,2,1 Ui 15,14,15,14,15,14,15,14,15,14,15,14 nit 10, 9, 10, 9, 10, 9, 10, 9, 10, 9, la, 9 polynomials: 41,43 6,4,2,1,2,1,2,1,1,1, 2, 1, 2, l', 2, 1, 2, 1, 2, 1, 2, 1 9i 10, 9,11,10,10, g, 11, 10, 10, 9,11,10 Ui 8, 7, 8, 7, 8, 7, 8, 7, 8, 7, 8, 7 Tnt polynomials: 42,44 6,3,3,1,2,1,2,1,1,1, 2, 1, 2, 1-, 2, 1, 2, 1, 2, 1, 2, 1 9i 10, 9,11,10,10, 9,11,10,10, 9,11,10 Ui Tn, 8, 7, 8, 7, 8, 7, 8, 7, 8, 7, 8, I polynomials: 45,46,47,48 9i 7,2,2,2,2,1,1,1,2,1,2,1,2,1; 2,1,2,1,2,1,2,1 Ut 15,14,15,14,15,14,15,14,15,14,15,14 Tni 10, 9, 10, 9,10, 9,10, 9,10, 9, 10, 9 polynomials: 49,50 ~=----= 9i 6,6,4,2,2,2;4,2,4,2 Ui and ni i unknown '7
polynomials: 51,52 ~=----= 9i 6,5,4,2,3,2;4,2,4,2 U t and ni i unknown
f) Dimension 12
101
polynomial: 53 9i 2 3 . 4 1 ,4,3,4,2, 2, 2, 2, 2, 2; 2, 2, 2, 2, 2, 2 Ui 12,10,12,12,10,12,12,10,12,12,10 rni 7, 6, 7, 7, 6, 7, 7, 6, 7, 7, 6 polynomial: 54 9i 2 3 .4 1 ,4,2,3,3,1,2,2; 1, 2, 3, 1, 2, 3 Ui 10, 9,10,10, 9,10,10 n1t 6, 5, 6, 6, 5, 5, 6 polynomial: 55 9t 24 . 4 1 ,3,3,4,2, 2, 2, 2, 2, 2; 2, 2, 2, 2, 2, 2 Ui 10,12,12,10,12,12,10,12,12,10,12 Tnt 7, 6, 7, 7, 6, 7, 7, 6, 7, 7, 6 polynomials: 56,57 9i 6,4,3,3,2, 2, 2, 2, 2, 2; 2, 2, 2, 2, 2, 2 Ui 12,10,12,12,10,12,12,10,12,12,10 n1i 7, 6, 7, 7, 6, 7, 7, 6, 7, 7, 6 polynomial: 58 9t 6,5,2,3,2, 2, 2, 2, 2, 2; 2, 2, 2, 2, 2, 2 Ui 12,10,11,12,10,11,12,10,11,12,10 n1t 7, 6, 7, 7, 6, 6, 7, 6, 7, 7, 6 unramified of degree 4 9i 2 3 . 4 1 ,3,3,3,2,3,3, 3; 2, 2, 3, 3, 2 8,12,11, 9, 10, 8 5, 6, 5, 5, 6, 5 ~
p=3
9i
2,1,2,1,2,1,2;1,2,1,2,1,2,1,2
Oi
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0 x + 3, x 2 - 3 E K[x] with K/Qp unramified of degree 2 n1z 2
9i
4,2,4;2,4,2,4
Oi
0,0,0,0,0,0,0,0 0,0,0,0,0,0,0,0
n1i
totally ramified of degree 4 9i
2,1,2,1,2,1,2,1
unramified of degree 4
unramified of degree 4 9i 8;4,8 and n1i unknown
0i
degree 4, ramified of degree 2
9i
4,2,4,2
XII Tables
102
p=2
polynomials: 1-4,7-10,13-20,27-30 9i 5,5,2,2,2,1,1,1,2,1,2,1,2,1; 2,1,2,1,2, 1,2,1 Ui 15,14,15,14,15,14,15,14,15,14,15,14 mi 10, 9,10, 9,10, 9, 10, 9, 10, 9, 10, 9 polynomials: 5,6,11,12,21-25,31 9i 24 .41,4,2,2,2,1,1,1,2,1,2,1,2,1; 2,1,2,1,2,1,2,1 Ui 15,14,15,14,15,14,15,14,15,14,15,14 mi 10, 9,10, 9,10, 9,10, 9,10, 9,10, 9 polynomials: 26,32 9i 24 .41, 3,2, 1,2,1,2,1,1,1, 2, 1, 2, 1; 2, 1, 2, 1, 2, 1, 2, 1 Ui 11,10,11,10,11,10,11,10,11,10,11,10 mi 8, 7, 8, 7, 8, 7, 8, 7, 8, 7, 8, 7 polynomial: 33 9i 2 3 .41,4,3,3,2,1,1,1,2,1,2,1,2,1; 2,1,2,1,2,1,2,1 Ui 15,14,15,14,15,14,15,14,15, 14, 15, 14 rni 10, 9,10, 9,10, 9,10, 9,10, 9,10, 9 polynomial: 34 9i 23 ·81, 3, 3, 3, 2,1,1,1,2, 1, 2, 1, 2, 1; 2, 1, 2, 1, 2, I, 2, 1 Ui 15, 14, 15, 14,15,14,15,14,15,14,15,14 mi 10, 9,10, 9,10, 9,10, 9,10, 9,10, 9 polynomials: 35,38,40 9i 23 .4 1 ,4,3,3,2,1,1,1,2,1,2,1,2,1; 2,1,2,1,2,1,2,1 Ui 15,14,15,14,15, 14, 15, 14,15,14,15,14 mi 10, 9,10, 9,10, 9,10, 9,10, 9,10, 9 polynomials: 36,37,39 9i 23 .8 1 ,3,3,3,2,1,1,1,2,1, 2, 1, 2, 1; 2, 1, 2, 1, 2, 1, 2, 1 Ui 15,14,15,14,15,14,15,14,15,14,15,14 rni 10, 9, 10, 9,10, 9, 10, 9, 10, 9,10, 9 polynomials: 41-44 9i 23 '4 1 ,4,3,1,2,1,2,1,1,1, 2, 1, 2, 1; 2, 1, 2, 1, 2, 1, 2, 1 Ui 10, 9,11,10,10, 9,11,10,10, 9, 11,10 mi 8, 7, 8, 7, 8, 7, 8, 7, 8, 7, 8, 7 polynomials: 45,46,47,48 9i 7,2,2,2,2,1,1,1,2,1,2,1,2,1; 2,1,2,1,2,1,2,1 Ui 15,14,15, 14,15,14,15,14,15,14, 15,14 mi 10, 9,10, 9,10, 9,10, 9,10, 9,10, 9 polynomials: 49,50 9i 6,6,4,2,2,2;4,2,4,2 Ui and rni unknown
--,-=--~
polynomials: 51,52 9i 6,5,4,2,3,2;4,2,4,2 Ui and mi unknown
--,--~:----=
103
f) Dimension 12
polynomial: 53 9i 23 .4 1 ,3,3,2,2,2,2, 1, 1, 1, 2, 2, 1; 1, 2, 2, 1, 1, 2, 2, 1 Ui 12,11,10,12,12,11,10,12,12,11,10,12,12,11 mi 9, 8, 7, 9, 9, 8, 7, 9, 9, 8, 7, 9, 9, 8 polynomial: 54 9t 23 .4 1 ,3,2,2,2,2,1,1,2,1,1,1; 2, 2,1,1,2,2,1,1 Ui 11,10, 9,10,11,10, 9,10,11,10, 9 mi 8, 7, 6, 8, 8, 7, 6, 7, 8, 7, 6 polynomial: 55 9i 23 .8 1 ,2,3,2,2,2,2,1,1,1,2,2,1; 1,2,2,1,1,2,2,1 Ui 12,11,10,12,12,11,10,12,12,11,10,12,12,11 mi 9, 8, 7, 9, 9, 8, 7, 9, 9, 8, 7, 9, 9, 8 polynomial: 56 9i 23 .4 1 ,4,3,1,2,2,2,1,1,1,2,2,1; 1,2,2,1,1,2,2,1 Ui 12,11,10,12,12,11,10,12,12,11,10,12,12,11 rni 9, 8, 7, 9, 9, 8, 7, 9, 9, 8, 7, 9, 9, 8 polynomial: 57 9i 22 '4 2 ,3,3,1,2,2,2,1,1,1,2,2,1; 1,2,2,1,1,2,2,1 Ui 12,11,10,12,12,11,10,12,12,11,10,12,12,11 mi 9, 8, 7, 9, 9, 8, 7, 9, 9, 8, 7, 9, 9, 8 polynomial: 58 9i 23 '4 1 ,4,2,2,2,2,2,1,1,1,2,2,1; 1,2,2,1,1,2,2,1 Ui 12,11,10,11,12,11,10,11,12,11,10,11,12,11 mi 9, 8, 7, 9, 9, 8, 7, 8, 9, 8, 7, 9, 9, 8 unramified of degree 4 23 .8 1 ,2,2,2,1,1,2,1,2,2, 2', 1, 1, 1, 2, 2, 2, 2, 1 9i 9, 8, 7, 6,12,11, 9,10, 9 Ui 8, 7, 6, 5, 9, 8, 7, 9, 8 rni
p=3 x 4 + 3, x 4 - 3 9i 2,1,2,1,2,1,2; 1,2,1,2,1,2,1,2 Oi 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0 mi 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0 x 2 + 3, x 2 ~ 3 E K[x] with K /Qp unramified of degree 2 9i Oi
mi
4,2,4;2,4,2,4 0,0,0,0,0,0,0 0,0,0,0,0,0,0
unramified of degree 4 9i 8; 4,8 and mi unknown
Oi
p?5
totally ramified of degree 4 9i
2,1,2,1,2,1,2,1
degree 4, ramified of degree 2 9i
4,2,4,2
XII Tables
104
unramified of degree 4
p=2
5,4,4,4,2,2,1,1, 2, 2, 1, 1; 2, 2, 1, 1, 2, 2, 1, 1 20,18,17,16,20,18,17,16,20,18,17,16 13,12,11,10,13,12,11,10,13,12,11,10
9i Uj
mi x2
-
2, x 2 + 6
9i Ui
mi
6,5,3,3,2,2,1,1,2,2,1,1; 2,2,1,1,2,2,1,1 20,18,17,16,20,18,17: 16 13,12,11,10,13,12,11,10
x 2 + 2x + 2, x 2 + 2x 9i Ui
m·
~
+6
5,4,3,3,2,2,1,1,1,1,1,1; 2, 2, 1, 1, 2, 2, 1, 1 12,10, g, 8,10, 8, 7, 6 g, 8, 7, 6, 7, 6, 5, 4
unramified of degree 2 9t Uj
mi
6,5,3,3,2,2, 3, 3; 2, 2, 2, 2, 2, 2 14,12,14,12,10, 8 10, g,lO' g, 7, 6
Sll(K 2 (K)? with IK: Qpl = 2, cf. (11.6) p=2
9j Ui
mi x2
-
5,4,4,4,2,2, 1, 1,2,2,1,1; 2, 2, 1, 1, 2, 2, 1, 1 20,18,17,16,20,18,17,16 13,12,11,10,13,12,11,10 2, x 2
+6
9j
24 '4 1 ,4,4,3,2,2,1,1,2,2,1,1; 2, 2,1,1,2,2,1,1
Ui
20,18,17,16,20,18,17,16
mi
x
2
13,12,11,10,13,12,11,10
+ 2x + 2, x + 2x + 6
9i Ui
mi
2
5,4,3,3,2,2,1,1,1,1,1,1; 2, 2, 1, 1, 2, 2, 1, 1 12,10, 9, 8,10, 8, 7, 6 g, 8, 7, 6, 7, 6, 5, 4
g) Dimensi on 14
unramified of degree 2
9i Ui
mi
24 .4 1 ,4,3,2,2,2,1,1,2,2,1,1; 2,2,2,2,1,1,1,1 14,12,14,12,11,10, 9, 8 11,10,11,10, 9, 8, 7, 6
Sl2(~)4, cf.
(II.6)
p=2 9i Ui
mi
6,5,4,4,2,2,2,2; 2, 2, 2, 2, 1, 1, 1, 1 18,16,14,12,11,10, 9, 8 15,14,13,12,11,10, 9, 8
Sll (K:2(~))4, cf. (11.6)
p=2 9i Ui
mi
g)
6,5,4,4,2,2,2,2; 2, 2, 2, 2, 1, 1, 1, 1 18,16,14,12,11,10, 9, 8 15,14,13,12,11,10, 9, 8
Dimension 14
92(!Qp) (simple Lie algebra of type G2 ), cf. XI.h) p=2 9i 0i
mi
4,4,3,2,3,2,4,3;2,3,2,4,3 8,6,7,5,9,8,6,7,5,9,8,6,7 3,4,4,3,4,3,4,4,3,4,3,4,4
p=3 9i Oi
mi
4, 3, 3, 2; 3, 3, 3, 2, 4, 3, 3, 2, 3, 2, 2, 2 11,12,10, 9,11,11,11, 9,11,10,12,10,14,12,10,10 4, 5, 4, 3, 4, 5, 6, 5, 6, 7, 8, 7, 8, 7, 6, 7
p'25 9i
3,2,2,2,3,2
105
XIII Uncountably many just infinite pro-p-groups of finite width
a)
The Nottingham group
Let G q be the group of continuous automorphisms of the ring IFq [[t]] where q is a prime power. Clearly G q can also be regarded as the group of continuous automorphisms of IFq (( t)). In fact every automorphism of IFq [[ t]] is continuous. The simplest proof is to observe that an element b of IFq [[t]] is of the form a q - a if and only if b has positive valuation. Thus every automorphism preserves the valuation on IFq [[t]l, and hence is continuous. Similarly one can prove that every automorphism of IFq (( t)) is continuous, but we do not need these facts, since we are in any case only concerned with continuous automorphisms. Clearly any element of Gq is determined by its action on t, and this can be any element L~l ai ti , where the at lie in IFq, and al =J. o.
(XII!.l) Definition. Let G q be the group of continuous automorphisms of the ring IFq [[t]]. The Nottingham group Sq is defined to be the Sylow pro-p-subgroup of Gq; that is, the subgroup of G q = {L~l aiti I ai E IFq, al =J. O} defined by the condition al = 1. Suppose now that q is a prime p > 2. If we define ei in Sp to be given by t ~ t(l + t i ) then it is easy to see that (el' e2, . ..) forms a base for Sp; that is, every k i < p for all i. If element of Sp can be written uniquely as e~l e~2 .. " where 0 Sp( i) is the subgroup of Sp that centralises t mod (tHI) then Sp(i) is generated topologically by {ei,ei+l""}' and it is easy to see that [ei,ej] e~~~ mod Sp(k) where k = min(2i + j, i + 2j). From this it follows at once that Sp is just infinite, with width 2, and ii(Sp)liHl(Sp) has order p2 if i = 1 mod (p - 1), and has order p otherwise. In fact ii(Sp) = Sp(r) where r = i + i(i - l)/(p - l)l. Similarly, it is easy to see that Sp(i)P < Sp(2i), so Sp is not p-adic analytic. These results are easily generalised to q being a prime power. If q is an odd prime power, the i-th lower central factor of Sq may be naturally regarded as a vector space of dimension 1 or 2 over IFq , the dimension being 2 exactly when i _ 1 mod (p - 1). For q = 2, SplSp(2) rv O2 X 0 4 , but the other terms of the lower central series are of dimension 1 or 2 over 1F2 . The case when q is a power of 2 is similar. In all cases, Sq is a just infinite non-p-adic analytic group of finite width. For details see [Joh 88].
s:
=
b)
Construction of uncountably many groups
Let p be a prime. Note that the group of units V of IFp [[t]] is the direct produc.t of a cyclic group of order p - 1 with a free Zp-module VI on {li I pfi}, where li = 1 + t i . That is to say, given any Zp-module M containing a sequence {fi I p fi} that converges to 0, there is a unique homomorphism of VI into M mapping li to Ii for all i (with pfi). It follows that V := VI/Vi is a compact vector space over IFp, with topological basis {Vi I Pfi} with Vi = viti. The Nottingham group Sp acts faithfully on this
107
b) Construction of uncountably many grollps
vector space. The stabiliser of Vi = Ufli is a closed subgroup C(i) of infinite order and index in Sp. Now V has a filtration V = VI ~ V2 ~ .. " where Vi = (Vj I j ~ i, p fj). If V E Vi - Vi+l, define wt(v) = i. If W is a finite dimensional subspace of V, then W has a basis (b I , ... , bd) with wt(b i ) < Wt(bi+I) for 1 ~ i < d. Then (wt(b l ),' ", wt(b d)) is an invariant for the orbit of Wunder Sp. (XIII.2) Lemma. If d(l - lip) > 1 (i.e. d > 1 and p > 2 or d > 2 and p = 2) and U I is as defined above, then the Nottingham group Sp acts on the set of finite dimensional subspaces of UI!Uf, of given weight invariant (WI, ... , Wd), with uncountably many orbits. Proof. Let W be such a subspace. Now W has a unique reduced echelonised basis the above basis {Vi I P)'i}, and the ith basis element can be written as L ajvj, with aj in lFp , where j runs through the positive integers prime to p, aj = 0 if j < Wi, aj = 1 if j = Wi, and for every j > i aj = 0 if j = Wk for some i < k ~ d. There are no other restrictions on these basis vectors, so the set of subspaces W with the given invariant weights is in natural 1-1 correspondence with the set S of ordered d-tuples of sequences (aj) satisfying these conditions. Now define an equivalence relation rv n on S for each n > 0 so that two elements of S are n-equivalent if and only if the corresponding sequences agree on terms aj for j < n. Since j cannot be a multiple of p, rv and rv are the same for k > O. Now let Sp(n) be the subgroup of Sp that
W.r. t.
kp
kp+l
fixes t modulo t n +I . It is clear that Sp permutes these n-equivalence classes, and that Sp( n) fixes them. But Spl Sp( n) is of order pn-l, and the number of equivalence classes under rv is at least pdn(l-I/p)-c for some easily calculated constant c. It follows that n the number of orbits of Sp on the n-equivalence classes increases exponentially with n since d > 1 and p > 2. q.e.d. Define (PSLp(lFp[[t]]))p to be a Sylow pro-p-subgroup of PSLp(lFp[[t]]). Now let W be a d-dimensional subspace of V, and let UI, •.• ,Ud E U1 define a basis for W. Let Pw < PGLp(lFp[[t]]) be generated by
(PSLp(lFp[[t]]))p and Now
Af
= uJp '
{Ai = (~i p-I
(
u iO
0 -:-11
Uz
p-I
Ip~l) 11 ~ i ~ d} )
E
modulo scalars.
(PSLp(lFp[[t]]))p modulo scalars, so Pw is an
extension of (PSLp(lFp[[t]]))p by an elementary abelian group of order pd, and clearly Pw is determined by W. Conversely, det maps PGLp(lFp[[t]]) homomorphicallyonto UIUP, and maps Pw onto W, so Pw determines W. Now assume that p > 2. Let WI and W 2 be finite dimensional subspaces of V. It is clear from [HOM 89] Proposition 3.2.7 that PW1 and PW2 are full groups (cf. [HOM 89] p. 106) since p > 2, so by [HOM 89] Theorem 3.2.29, any isomorphism between these groups is the restriction of an automorphism of PGLp(lFp[[t]]), and this automorphism group is ge~erated by inner automorphisms, the inverse transpose automorphism, and field automorphisms. Only the latter affects the determinant, and hence the isomorphism exists if and only if WI and W2 are in the same orbit of the Nottingham group. This gives us uncountably many finitely generated non-isomorphic groups of finite width. It is clear that, since UI contains no elements of order p, these groups are all just infinite.
XV
References
Bla 58 N. Blackburn, On a special class of p-groups. Acta Math. 100 (1958), 45-92. Bla 61 N. Blackburn, Generalizatiun uf certain elementary theorems on p-groups. Proc. London Math. Soc. 11 (1961), 1-22. BrT 72 F. Bruhat, J. Tits, Groupes reductives sur un corps local 1. Donnee radicielles valuees. Pub!. Math. 1. H. E. S. 41 (1972), 5-251. BrT 84 F. Bruhat, J. Tits, Groupes niductives sur un corps local II. Schema en groupes. Existence d'une donnie radicille valuee. Pub!. Math. L H. E. S. 60 (1984), 197-376. BrT 87 F. Bruhat, J. Tits, Groupes algebriques sur un corps local III. Complements et applications a cohomologie galoisienne. J. Fac. Sci. Univ. Tokyo Sec. 1A 34 (1984),671-688. Cam 97 R. Camina, Subgroups of the Nottingham group. appear.
Journal of Algebra, to
Car 72 R. W. Carter, Simple groups of Lie type. Wiley, London 1972. Che 55 C. ChevalIey, Sur certains groupes szmples. Tohoku Math. J. (2) 7, (1955), 14-66. DdMS 91 J. D. Dixon, M. P. F. du Sautoy, A. Mann, D. Segal, Analytic pro-p Groups. LMS Lecture Note Series 157, 1991. Don 87 S. Donkin, Space Groups and Groups of Prime-Power Order VIII. Pro-pGroups of Finite Coclass and p-Adic Lie Algebras. J. Alg. 111 no. 2 (1987), 316-342. GAP 94 M. Schonert (ed.), Groups, Algorithms, and Programming. GAP-3.4 Manual, Lehrstuhl D fur Mathematik, RvVTH Aachen. Gri 80 R. 1. Grigorchuk, On the Burnside problem for periodic groups. FunktsionaL Anal. i Prilozhen. 14 (1980), 53 - 54, [Russian]. Eng!. trans!.: Functional AnaL App!. 14 (1980), 41-43. Has 49 H. Hasse, Zahlentheorie. Akademie-Verlag 1949, 3rd ed. 1969. HOM 89 A. J. Hahn, O. T. O'Meara, The Classical Groups and K -Theory. Springer Berlin 1989. Hup 67 B. Huppert, Endliche Gruppen I. Springer Berlin 1967. Iwa 66 N. Iwahori, Generalized Tits system (Bruhat decomposition) onp-adic semisimple groups in Algebraic groups and discontinuous Subgroups. Proc. Sympos. Pure Math., vol. IX, Part 1, 71-83, Providence RI 1966.
110
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Authors' addresses and e-mail: Queen Mary and Westfield College, University of London School of Mathematical Sciences Mile End Road London El 4NS England [email protected],uk
RWTH Aachen Lehrstuhl B fUr Mathematik Templergraben 64 52062 Aachen Germany plesken@willi,math.rwth-aachen.de gz@willi,math.rwth-aachen,de
XVI
Notation
l aJ ial lim :xl
Z(G)
[X, X]
1/, I/p
OK
o
0*I IT or
71"
F (k
w(P) wa(P) w(P) wa(P) o(P) oz(P) J-Li
p-group [H'k G ] (x, y) W(x, y)
+L [ , ]L L(H)
£(P) A(H)
d(N) Uz(N) ~(N)
the greatest integer S a the least integer ~ a limes superior wreath product semi-direct product of centre of the group G commutator group if X is a group Lie commutator of X is a Lie algebra = (gPjg E G) for a group G the closure of the i-th term of the lower central series of G, i.e. 1'1 := G,1'i+1 := bi' G] if G is a finite group or a finitely generated pro-p-group then the terms of the lower central series are closed automatically the closure of the i-th term of the lower p-series of G, i.e. Al := G, Ai+l := [AI' G]Az(G)P if G is a finite group or a finitely generated pro-p-group then the terms of the lower central series are closed automatically valuation, valuation where p is of value 1 ring of integers of field K ring of integers of some field = {x II/(x - 1) ~ i} prime elements residue class field primitive k th root of unity width of P (1.1) average width of P (1.1) ultimate width of P (1.1) upper average width of P (1.1) obliquity of P (1.5) i-th obliquity of P (1.5)
(1.5) p-adic analytic just infinite pro-p group (1.5) k-fold commutator cf. (11.1) Baker-Campbell-Hausdorff formula, cf. Chapter III a) commutator Baker-Campbell-Hausdorff formula, cf. Chapter III a) addition on a uniform subgroup, cf. Chapter III b) Lie bracket on a uniform subgroup, cf. Chapter III b) Zp-Lie algebra assigned to H, cf. Chapter III b) Lie algebra assigned to P, cf. Chapter III b) log(H), cf. Chapter III c) number of generators of N, cf. Chapter IV U I (N) = HP, Ui +l = U I (Uz(N)), cf. Chapter IV = (n E N I n P = 1)
XVI Notation
14
1>
Q G(1), K) (A,O)
AC PL , CPL, CLP, eLP
rad(A)
o p
di(a}, ... , an) J~~ = Ji-(A) Ui(A)
Br F Kab,p
root system, cf. Chapter V weight lattice, cf. Chapter V Chevalley group, cf. Chapter V algebra A with involution 0, cf. VI a) = {x E A I a O = -a} for some order or algebra A with involution 0, cf. VI a) Cayley maps, cf. Chapter VI a) radical of some order A maximal order of a division algebra D, cf. Chapter VI a) radO, cf. Chaper VI a) matrix defined in (VI.10) = (radA)~ n A - (VIA) (1 + (radA)~) n U (VIA) r th dimension matrix, cf. (VI.15) Gram matrix cf. after (VI.16) maximal abelian extension of exponent p of a local field K, cf. Chapter IX division algebra of dimension i 2 over the center K cf. XI a), e) cf. Chapter XII Nottingham group, cf. XIII a)
XVII
Index
;5-group, 4 p-adically simple group, 4
non-split groups, 73 Nottingham group, 106
algebra with involution, 30 automorphism group of a Lie algebra, 79 automorphism of Lie algebra, 16
obliquity, 3, 19,74 obliquity, ultimate, 19
Baker-Campbell-Hausdorff formula, 12 Cartan numbers, 26 Cayley map, 30, 31 centroid, 78 Chevalley basis, 26 Chevalley group, 26 Chevalley lattice, 26 commutator Baker-Campbell-Hausdorff formula, 14 diagram automorphism, 79 dimension matrix, 37 dimension of an analytic pro-p-group,
20 exponent matrix, 36 field extension, maximal abelian of exponent p, 62 Grigorchuk group, 2 hereditary just infinite, 5 involution of first kind, 30 involution of second kind, 30 just infinite, 2 Lie algebra for p-group, 14 Lie lattice, 14 lower p-series, 10 lower central series, 1 lower central series of a Chevalley group, 28
lower central series of a classical group, 37 maximal p..group, 19
periodicity, defect of, 18 periodicity, ultimate, 18 powerful, strongly hereditarily, 21 rational automorphism, 79 saturated, 35 semi-saturated, 35 settled, 21 similitude, 89 split groups, 70 uniform, 14 uniformly powerful, 14 weight, 68 width, 1 width, average, 1 width, finite, 1 width, ultimate, 1 width, upper average, 1
