Infinite Linear Groups: An Account of the Group-theoretic Properties of Infinite Groups of Matrices

242 Pages·1973·4.52 MB·English

by  Bertram A. F. Wehrfritz (auth.)

Ergebnisse der Mathematik
und ihrer Grenzgebiete
Band 76
Herausgegeben von P. R. HaImos . P. J. Hilton
R. Remmert· B. Szl)kefalvi-Nagy
Unter Mitwirkung von L. V. Ahlfors· R. Baer
F. L. Bauer· A. Dold . J. L. Doob . S. Eilenberg
K. W. Gruenberg· M. Kneser· G. H. Muller
M. M. Postnikov . B. Segre . E. Spemer
Geschaftsfuhrender Herausgeber: P. J. Hilton
B. A. F. Wehrfritz
Infinite Linear Groups
An Account oft he Group-theoretic Properties
of Infinite Groups of Matrices
Springer -Verlag Berlin Heidelberg New York
1973
Bertram A. F. Wehrfritz
Queen Mary College, London University, London/England
AMS Subject Classifications (1970):
Primary 20E99, 20H20 • Secondary 15A30, 20F99, 20G15, 20H25
ISBN-13: 978-3-642-87083-5 e-ISBN-13: 978-3-642-87081-1
DOl: 10.1007/978-3-642-87081-1
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the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Ber
lin Heidelberg 1973. Library of Congress Catalog Card Number 72-96729.
Softcover reprint of the hardcover 1s t edition 1973
Preface
By a linear group we mean essentially a group of invertible matrices with
entries in some commutative field. A phenomenon of the last twenty
years or so has been the increasing use of properties of infinite linear
groups in the theory of (abstract) groups, although the story of infinite
linear groups as such goes back to the early years of this century with the
work of Burnside and Schur particularly.
Infinite linear groups arise in group theory in a number of contexts.
One of the most common is via the automorphism groups of certain types
of abelian groups, such as free abelian groups of finite rank, torsion-free
abelian groups of finite rank and divisible abelian p-groups of finite
rank. Following pioneering work of Mal'cev many authors have studied
soluble groups satisfying various rank restrictions and their automor
phism groups in this way, and properties of infinite linear groups now
play the central role in the theory of these groups. It has recently been
realized that the automorphism groups of certain finitely generated
soluble (in particular finitely generated metabelian) groups contain
significant factors isomorphic to groups of automorphisms of finitely
generated modules over certain commutative Noetherian rings. The
results of our Chapter 13, which studies such groups of automorphisms,
can be used to give much information here.
A third way that linear groups arise is via an earlier theorem of
Mal'cev which states that a group G is isomorphic to some linear group
of degree n if each of its finitely generated subgroups is isomorphic to a
linear group of degree n (see 2.7 for a proof). It is principally via this
result and its generalizations that infinite linear groups have influenced
the theory of locally finite groups. With more information about which
linear groups the finitely generated subgroups of G are isomorphic to,
one can sometimes specify precisely a linear group isomorphic to G.
This led to very important characterizations of certain groups such as
PSL(2, F) over locally finite fields F, which now playa crucial role in the
theory oflocally finite groups, see reference [31 a]. I suspect that to date
we have only scratched the surface of the applications of infinite linear
groups to locally finite groups.
v
The class of linear groups is one of the few examples of a highly
non-trivial, highly non-soluble, but still relatively accessible class of
groups on which to test conjectures. It is quite common for counter
examples to be constructed as or out of linear groups, and for a problem
to be solved first for the linear case where the situation should be some
what simpler. The obvious examples of this are the Burnside Problem
and the minimal and maximal condition conjectures (see 1.23, 9.8 and
10.18). Finally it sometimes happens for purely ad-hoc reasons that one
knows that a particular group is isomorphic (or at least related) to a
certain linear group. When this happens one can often use this informa
tion to obtain painlessly properties of the original group. Examples of
this phenomenon that come to mind are free groups, certain relatively
free groups and certain generalized free products (see Chapter 2).
Much of the work on linear groups has been done by people with a
sufficiently non-group-theoretic background to make their articles
difficult for a group theorist to read. Also many other results on linear
groups are tucked away in papers with purely group-theoretic titles.
Thus there seemed to be a need to gather all this material together.
This book is an attempt to give an account of those properties of
linear groups that seem relevant to infinite group theory, assuming by
and large no more than a postgraduate student of group theory might
reasonably be expected to know. This has not always been possible of
course. In the main we require the reader to know no more linear algebra
than usually appears in undergraduate courses, and little more commu
tative algebra. We assume no knowledge of algebraic geometry (with
the exception of the final chapter which is an appendix on algebraic
groups) developing what little we need as we go along. In contrast we
assume a considerable quantity of (infinite) group theory. For the most
part we need no more than might reasonably be taught in M. Sc. courses,
but occasionally we need considerably more. I have tried to give full
(and often alternative) references whenever this occurs and it is very rare
for me not to be able to refer the reader to a textbook.
A word about the arrangement of this book; the basic chapters are
1,5,6 and to some extent 2, which provides the reader with some ex
amples oflinear groups to bear in mind. These should in a way have been
labelled Chapters 1 to 4. However, assuming the reader to be interested
in the group theoretic properties of linear groups there seemed to be a
considerable risk of him abandoning the book through boredom before
he reached the meat. I have therefore split the introduction into very
roughly the ring theoretic part in Chapter 1 and the geometric part in
Chapters 5 and 6. It is possible to go a good way with the theories of
soluble linear groups and finitely generated linear groups without the
VI
latter part and this we do in Chapters 3 and 4. The determined reader is
welcome to read Chapters 5 and 6 immediately after Chapter 1.
The remainder of the book is composed as follows. Chapter 7 studies
Jordan decomposition in linear groups and contains some structure
theorems for locally nilpotent linear groups, results that are the basis of
much of Chapter 8 on upper central series and a little of Chapter 11 on
locally supersoluble linear groups. The bulk of the material on periodic
linear groups is in Chapter 9, some further material being in the second
part of Chapter 12. Chapter 10 is devoted to properties of linear groups
with a (sometimes very vague) varietal flavour. In Chapter 13 we show
how properties of groups of automorphisms of finitely generated modules
over commutative rings may be derived from properties of linear groups
and Chapter 14 is little more than a description of the main properties
of algebraic groups over algebraically closed fields with very few of the
proofs.
This book has been written with the requirements of the reader
wishing to make a serious study of the subject much to the forefront of
my mind. However the needs of those wishing to use it as a reference
work have not entirely been ignored. A reader who desires to look up
a particular type of result or property should look first in the index. If
this is unhelpful, as of course it often will be if for example he does not
know precisely what he is looking for, he should turn to the earliest
chapter with the most promising looking title. In the first paragraph or
two he will find a brief summary of the properties of linear groups
discussed in that chapter (apart from Chapters 1, 5 and 7 where this is
not really practicable). If this is still unhelpful he should turn to the end
of the chapter where, when relevant, he will find a brief summary of where
the main types of groups studied in the chapter appear later in the book.
Approximately half of the material in this book has been available
for some time as lecture notes issued in the Queen Mary College Mathe
matics Notes series. Chapters 10, 11, 12 and 14 have no counterpart in
the Q.M.C. notes. Chapter 13 is a largely rewritten and considerably
extended version of the Q.M.C. Chapter 10. Chapters 2, 4 and 9 are
greatly extended versions of the corresponding Q.M.C. chapters. The
remaining six chapters are only minor variants of their Q.M.C. counter
parts, the main changes being correction of mistakes, the insertion of
more examples, exercises and references to the literature, and the in
clusion of the proofs of certain results that had only been stated before.
It had been my intention to give everybody his due credit. This has in
practice proved quite impossible, particularly where the idea occurs as
part of a proof, and the book is sprinkled with uncredited ideas and
tricks that I have lifted from various people and places. Much of the
material I have given in courses and seminars in London at one time or
VII
another and my audiences are responsible for many improvements in
content, presentation and notation. I am especially indebted to Karl
Gruenberg and Otto Kegel in this respect. My thanks are also due to
Kurt Hirsch, who, among other contributions, originally stimulated my
interest in linear groups with a course he gave on the subject during the
academic year 1963-64. Finally Norman Massey helped me with the
proof reading; I claim full credit however for all the errors left behind.
London, January 1973
Bertram A.F. Wehrfritz
VIII
Table of Contents
1. Basic Concepts. . . . . . . . . 1
2. Some Examples of Linear Groups. 17
3. Soluble Linear Groups . . . . . 41
4. Finitely Generated Linear Groups. 50
5. CZ-Groups and the Zariski Topology 72
6. The Homomorphism Theorems. . . 82
7. The Jordan Decomposition and Splittable Linear Groups. 90
8. The Upper Central Series in Linear Groups. . . . . 101
9. Periodic Linear Groups . . . . . . . . . . . . . 112
10. Rank Restrictions, Varietal Properties and Wreath Products 134
11. Supersoluble and Locally Supersoluble Linear Groups.. 155
12. A Localizing Technique and Applications . . . . . .. 174
13. Module Automorphism Groups over Commutative Rings 186
14. Appendix on Algebraic Groups. 202
Suggestions for Further Reading. 219
Bibliography 221
Index . . . 227
IX
Notation
If G is a group, then
G' denotes the derived group of G,
G = yl G 2 y2 G 2··· the lower central series of G,
{1} = (o(G)s; (l(G)S;··· the upper central series of G,
((G ) the hypercentre of G,
Yf(G) the Hirsch-Plotkin radical of G,
Yfl(G) the Fitting subgroup of G,
A(G) the product of all the G-hypercyclic normal subgroups of G
(see p. 156),
cjJ(G) the Frattini subgroup of G,
t5(G) the intersection of the non-normal maximal subgroups of G,
or G itself if none such exist,
tjJ(G) the intersection of the centralizers of the chief factors of G,
tjJl (G) the intersection of the centralizers of the finite chief factors
of G, or G itself if none such exist,
n(G) the set of primes p such that G contains an element of order p,
and O"(G) the intersection of the normal subgroups N of G such that
n(GjN)s;n.
If S is any subset of G then
lSI denotes the cardinality of S,
NG(S) the normalizer of Sin G,
CG(S) the centralizer of Sin G,
<S) the subgroup of G generated by S,
and <SG) the normal subgroup of G generated by S.
If a is any element of G, then aG denotes the set of conjugates of a in G.
If Hand K are subgroups of G, then
(G:H) denotes the index of H in G,
[H, K]=<[h, k] =h-1 k-1 hk: heH, keK) =[H, lK],
[H, S+lK] = [[H, sK], K]= [H, K, K, ... , K] wheretheKisrepeated
s+ 1 times,
and H eK means that for each h in Hand k in K there exists an integer r=
r(h, k)such that [h, rk] = [h, k, k, ... , k] = 1 where kis repeated rtimes.
XI
If G acts as a group of operators on the group A, then either A] G or
G [A denotes the split extension of A by G.
If Hand K are groups and {Hi: i e J} is a family of groups then
X
H x K and Hi denote the direct products of the corresponding
*ieI
groups,
H *K and Hi the free products,
n ieI
Hi the cartesian product,
ieI
H'LK the (restricted) wreath product of H by K
and H"G K the complete wreath product of H by K.
3'", ffi, ~, 91, 6 denote respectively the classes of all finite groups, finitely
generated groups, abelian groups, nilpotent groups and soluble
groups. S, Q, L, R denote respectively the subgroup, quotient, local
and residual operators.
If R is a commutative ring and Van R-module, then
EndR(V) denotes the ring of R-endomorphisms of V,
AutR( V) the group of R-automorphisms of V,
Rn the ring of n x n matrices over R, and if xeR
n,
tr(x) denotes the trace of x,
and det(x) the determinant of x.
F always denotes a field.
F is an algebraic closure of F,
F* the multiplicative group of F,
and char F the characteristic of F.
Sn is the symmetric group on n-symbols,
C a cyclic group of order n,
n
Cpoo a Priifer pro-group,
GL(n, F) the general linear group of degree n over the field F,
SL(n, F) the special linear group, = {geGL(n, F): det g= 1},
Tr(n, F) the triangular group, = {(gi)eGL(n, F); gij=O for i <j},
Tr (n, F) the unitriangular group, = {(gij} eTr(n, F): gii= 1 for each i},
1
D(n, F) the diagonal group, = {(gi)eGL(n, F): gij=O for i=l= j}.
If S is a subset of an F-algebra A then F {S} denotes the subalgebra of A
with identity generated by S.
If S is a subset of GL(n, F) then dF(S) denotes the minimal closed sub
group of GL (n, F) containing S; see p.74.
If g is an element of GL(n, F) then g=gu gd is the (multiplicative) Jordan
F;
decomposition of g over see p.91.
If G is a subgroup of GL(n, F) then GO denotes the connected component
of G containing 1.
XII