Table Of Contentde Gruyter Expositions in Mathematics 8
Editors
O. H. Kegel, Albert-Ludwigs-Universität, Freiburg
V. P. Maslov, Academy of Sciences, Moscow
W. D. Neumann, Ohio State University, Columbus
R.O. Wells, Jr., Rice University, Houston
de Gruyter Expositions in Mathematics
1 The Analytical and Topological Theory of Semigroups, K. H. Hofmann,
J. D. Lawson, J. S. Pym (Eds.)
2 Combinatorial Homotopy and 4-Dimensional Complexes, H. J. Baues
3 The Stefan Problem, A. M. Meirmanov
4 Finite Soluble Groups, K. Doerk, T. O. Hawkes
5 The Riemann Zeta-Function, A.A.Karatsuba, S.M. Voronin
6 Contact Geometry and Linear Differential Equations, V. R. Nazaikinskii,
V. E. Shatalov, B. Yu. Sternin
1 Infinite Dimensional Lie Superalgebras, Yu.A. Bahturin, A.A.Mikhalev,
V. M. Petrogradsky, M. V. Zaicev
Nilpotent Groups
and their Automorphisms
by
Evgenii I. Khukhro
W
DE
G
Walter de Gruyter · Berlin · New York 1993
Author
Evgenii I. Khukhro
Institute of Mathematics
Siberian Branch of the Russian Academy of Sciences
630090 Novosibirsk - 90, Russia
7997 Mathematics Subject Classification: Primary: 20-01; 20-02; 17-02.
Secondary: 20D15, 20D45, 20E36,20F18, 20F40,17B40
Keywords: Nilpotent group, p-group, operator group, Lie ring, commutator, (regular)
automorphism, Hughes subgroup
® Printed on acid-free paper which falls within the guidelines of the ANSI
to ensure permanence and durability.
Library of Congress Cataloging-in-Publication Data
Khukhro, Evgenii I., 1956-
Nilpotent groups and their automorphisms / by Evgenii I.
Khukhro.
p. cm. — (De Gruyter expositions in mathematics; 8)
Includes bibliographical references and index.
ISBN 3-11-013672-4
1. Nilpotent groups. 2. Automorphisms. I. Title. II. Series.
QA177.K48 1993
512'.2-dc20 93-16401
CIP
Die Deutsche Bibliothek — Cataloging-in-Publication Data
Chuchro, Evgenij I.:
Nilpotent groups and their automorphisms / by Evgenii I.
Khukhro. - Berlin ; New York : de Gruyter, 1993
(De Gruyter expositions in mathematics; 8)
ISBN 3-11-013672-4
NE:GT
© Copyright 1993 by Walter de Gruyter & Co., D-1000 Berlin 30.
All rights reserved, including those of translation into foreign languages. No part of this book
may be reproduced in any form or by any means, electronic or mechanical, including photocopy,
recording, or any information storage and retrieval system, without permission in writing from the
publisher.
Printed in Germany.
Disk Conversion: D. L. Lewis, Berlin. Printing: Gerike GmbH, Berlin.
Binding: Lüderitz & Bauer GmbH, Berlin. Cover design: Thomas Bonnie, Hamburg.
Table of Contents
Notation viii
Preface ix
Part I Linear Methods 1
Chapter 1
Preliminaries 3
§1.1 Groups 3
§ 1.2 Rings and modules 6
§ 1.3 Lie rings 9
§1.4 Mappings, homomorphisms, automorphisms 15
§ 1.5 Group actions on a set 15
§ 1.6 Fixed points of automorphisms 17
§ 1.7 The Jordan normal form of a linear transformation of finite order 20
§1.8 Varieties and free groups 22
§ 1.9 Groups with operators 24
§ 1.10 Higman's Lemma 25
Chapter 2
Nilpotent groups 30
§ 2.1 Commutators and commutator subgroups 30
§ 2.2 Definitions and basic properties of nilpotent groups 34
§ 2.3 Some sufficient conditions for soluble groups to be nilpotent 37
§ 2.4 The Schur-Baer Theorem and its converses 43
§ 2.5 Lower central series. Isolators 47
§ 2.6 Nilpotent groups without torsion 51
§ 2.7 Basic commutators and the collecting process 53
§ 2.8 Finite /7-groups 59
Chapter 3
Associated Lie rings 70
§3.1 Results on Lie rings analogous to theorems about groups 71
§ 3.2 Constructing a Lie ring from a group 73
vi Table of Contents
§ 3.3 The Lie ring of a group of prime exponent 78
§ 3.4 The nilpotency of soluble Lie rings satisfying the Engel condition 81
Part II Automorphisms 85
Chapter 4
Lie rings admitting automorphisms with few fixed points 87
§ 4.1 Extending the ground ring 87
§ 4.2 Regular automorphisms of soluble Lie rings 90
§ 4.3 Regular automorphisms of Lie rings 94
§4.4 Almost regular automorphisms of prime order 102
§4.5 Comments 117
Chapter 5
Nilpotent groups admitting automorphisms
of prime order with few fixed points 121
§ 5.1 Regular automorphisms of prime order 121
§5.2 Nilpotent /^-groups with automorphisms of order p 123
§ 5.3 Nilpotent groups with an almost regular automorphism of prime order 128
§5.4 Comments 148
Chapter 6
Nilpotency in varieties of groups with operators 155
§6.1 Preliminary lemmas 157
§6.2 A nilpotency theorem 161
§6.3 A local nilpotency theorem 164
§6.4 Corollaries 174
§6.5 Comments 177
Chapter 7
Splitting automorphisms of prime order
and finite p-groups admitting a partition 180
§ 7.1 The connection between splitting automorphisms of prime order and
finite /7-groups admitting a partition 181
§7.2 The Restricted Burnside Problem for groups with a splitting auto-
morphism of prime order 185
§ 7.3 The structure of finite p-groups admitting a partition and a positive
solution of the Hughes problem 202
§ 7.4 Bounding the index of the Hughes subgroup 208
Table of Contents vii
§7.5 Comments 216
Chapter 8
Nilpotent /;-groups admitting automorphisms
of order pk with few fixed points 226
§8.1 An application of the Mai'cev correspondence 227
§ 8.2 Powerful /^-groups 232
§ 8.3 A weak bound for the derived length 234
§ 8.4 A strong bound for the derived length of a subgroup of bounded
index 236
References 240
Index of names 248
Subject Index 250
Notation
AutG, 15
(M), 3 B(m,n), 23
a*, 3 2l*, 23
M · N, 3 S3„, 23
MN,3 01,, 23
C, 3 <m , 25
p
N, 3 /„(A/), 50
Q, 3 /* (W), 50
M, 3 /.(D, 51
Z, 3 [a, „61, 55
a = b (mod W), 4 Φ(Ρ), 60
Λ < G, 4 L(G), 73
B X A, 4 B(m,n), 76
N (M), 4 'L, 87
G
C (g), 4 ' , 89
G
[M, JV], 4 /z(/7), 102
(Λ/6'), 4 tfi, 110
Ά: (s), in
CG (M), 4
[a, b], 4 'jc(.y), 111
G', 4 i?(Jc, ), 132
[ ],a , ...,a ], 5 A"(jc), 132
2 k
G^' 5 x(s), 134
ft (G), 5 /T (i), 134
Ω,(Ρ), 5 (q(H),q(H),t(H)), 135
Z(G), 5 P(G), 135
n (G), 5 OT, 155
G", 5 , 166
§„, 6 //„(G), 181
GF(q), 6 ««o,« ,flp-i», 188
'π, 6 <ίίΜο, HI, . . . , iip^i^, 191
A®* , 8
<ίίΜ|, Μ2, ..·, M .v, (^ — ί)??ϊϊ>, 191
α ® ί>, 8 tf*, 192
w W, H V* (§1,^2,..., It), 209
+ (M), 11 /*, 209
α* = φ(α) — αφ, 15 degjc,, 212
CG/ N (ψ), 15 Iνι /\ μ,/\ , . . . , μ,, ϊ)\, . . . , T]ic-r)j *·**\ O 1 T
< Γ
C(<p), 15 ///(G), 220
c
[G, H 15 f N*TJl 973
A-/ J τ «V l n f £*^-J
Preface
In group theory it is natural to distinguish classes of groups according to the extent
to which the group operation is commutative. At one extreme we have the class of
commutative (abelian) groups and at the other - free groups or groups close to them,
as well as nonabelian simple groups, that is, groups without any nontrivial normal
subgroups. Commutativity of elements a and b is equivalent to the triviality of
their commutator la, b] = a~lb~*ab. So abelian groups are defined by the identity
[jc, y] — I and more complicated commutator identities define classes of groups
which are close to abelian, but less commutative. The identity
defines the variety of nilpotent groups of class c, and the identity 8^ — 1 of 2*
variables, where 8^ is defined recursively by
5] = [X],x \, &k+] — \&k(x], ···- *2*Mt(*2*+i, ... , Jr t.i)],
2 2
defines the variety of soluble groups of derived length k. Another way to define
these classes of groups is in terms of the existence of a series of normal subgroups
with central or commutative factors, respectively.
Study of nilpotent groups often aims to prove that they possess some degree
or other of commutativity. For example, the positive solution of the Restricted
Bumside Problem for groups of exponent pk (Kostrikin, 1959, for k = 1, and
Zel'manov, 1990, for all k) means that there is a function f(p,k,m) such that
the nilpotency class of any w-generated finite p-gmup of exponent pk does not
exceed f(p, k,m).
The fact that nilpotent groups are close to being commutative means that it is
possible to apply linear methods to their study. Using the group operations one
can define the structure of a Lie ring on the direct sum of the factors of the lower
central series. The action of a group on an invariant commutative section looks like
the action of a matrix group on a vector space. However, although for such more
linear objects as Lie rings or matrix rings powerful results and highly developed
techniques are available, there may be difficulties in using them when it comes to
going from groups to rings and back again.
In this book linear methods in the theory of nilpotent groups are applied to
the study of automorphisms of nilpotent groups. We prove the analogue of the
Preface
positive solution of the Restricted Burnside Problem for groups with a splitting au-
tomorphism of prime order. This gives rise to a structural theory of finite ^-groups
admitting a partition which includes the positive solution of the Hughes problem for
almost all (in some precise sense) finite /^-groups. The Higman-Kreknin-Kostrikin
Theorem on the boundedness of the nilpotency class of Lie rings (or nilpotent
groups) admitting a regular automorphism of prime order, is generalized to the
case where the number of fixed points is finite: almost regularity of the auto-
morphism of prime order implies almost nilpotency - that is, the existence of a
nilpotent subring (or subgroup) of bounded index and of bounded nilpotency class.
Kreknin's Theorem on the solubility of Lie rings with regular automorphisms of
finite order is used to prove the "almost solubility", in an analogous sense, of a
nilpotent /?-group with an almost regular automorphism of order pk. Linear and
combinatorial methods are used to prove a theorem of a rather general nature
which gives a positive solution to the Restricted Burnside Problem for a variety of
operator groups under the hypothesis that this problem has a positive solution for
the ordinary variety obtained from this variety by replacing all operators by 1 in
its identities.
The first part "Linear methods" is, in fact, a textbook. The existence of many
books or chapters of books devoted to nilpotent groups - for instance, those of
Baumslag [6], Gorenstein [25], M. Hall [27], P. Hall [28], Huppert and Blackburn
[49], Kargapolov and Merzlyakov [51], Kurosh [85] and Warfield [153] - makes it
a difficult task to write something new. We have tried to select only that material
which is necessary for the exposition of the aforementioned results from the second
part of the book, "Automorphisms". Practically every result proved in the first part
is in some way used in the second: either there is a reference to it or to its proof,
or its statement and proof prepares the reader for more complicated arguments of a
similar nature in the second part. But, of course, we have not followed this rule too
strictly in the hope that the reader may get to know at least some of the classical
methods in the theory of nilpotent groups. However, many important results are
only briefly mentioned; for example, Kostrikin's Theorem on Engel Lie algebras
is stated here without proof. On the other hand, in the interests of completeness,
we reproduce proofs of the theorems of Higman, Kreknin and Kostrikin on regular
automorphisms.
The author hopes that the second part is something more than a collection of
several research papers under one cover. The material is presented here with more
detail and perhaps more intelligibly than originally. Moreover, some proofs are
longer in an attempt to make the book more self-contained. Repetition of typical
arguments should help with their mastery.
This book is based on a special course given at Novosibirsk University in 1988-
90. The author thanks Andrei Vasil'ev and Natasha Makarenko for several valuable
remarks.
The book consists of seven chapters.
