Varieties of Groups

201 Pages·1967·5.878 MB·English

by  Prof. Dr. Hanna Neumann (auth.)

Ergebnisse der Mathematik
und ihrer Grenzgebiete
Band 37
Herausgegeben von
P. R. Halmos . P. J. Hilton· R. Remmert· B. Szokefalvi-Nagy
Vnter Mitwirkung von
L. v. Ah1fors . R. Baer . F. L. Bauer· R. Courant· A. Dold
J. L. Doob . E. B. Dynkin . S. Eilenberg· M. Kneser· M. M. Postnikow
H. Rademacher· B. Segre . E. Sperner
Redaktion: P. J. Hilton
Varieties of Groups
Hanna Neumann
Springer-Verlag New York Inc. 1967
Prof. Dr. Hanna Neumann
Th~ Australian National University
The School of General Studies
Department of Pure Mathematics
Canberra, A.C.T. / Australia
ISBN 978-3-642-8860\-0 ISBN 978-3-642-88599-0 (eBook)
001 \0.\007/978-3-642-88599-0
All rights reserved, especially that of translation into foreign languages. It is also
forbidden to reproduce this book, either whole or in part, by photomechanical
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permission from the Publishers
© by Springer-Verlag, Berlin. Heidelberg 1967
Softcover reprint of the hardcover 1st edition 1967
Library of Congress Catalog Card Number 67 -IS 607
Title No. 4581
To Bernhard
Preface
Varieties of algebras are equationally defined classes of algebras,
or "primitive classes" in MAL'CEV'S terminology. They made their first
explicit appearance in the 1930's, in Garrett BIRKHOFF'S paper on "The
structure of abstract algebras" and B. H. NEUMANN'S paper "Identical
relations in groups I". For quite some time after this, there is little
published evidence that the subject remained alive. In fact, however,
as part of "universal algebra", it aroused great interest amongst those
who had access, directly or indirectly, to PHILIP HALL'S lectures given at
Cambridge late in the 1940's. More recently, category theory has provided
a general setting since varieties, suitably interpreted, are very special
examples of categories. Whether their relevance to category theory goes
beyond this, I do not know. And I doubt that the category theoretical
approach to varieties will be more than a fringe benefit to group theory.
Whether or not my doubts have substance, the present volume owes its
existence not to the fact that varieties fit into a vastly more general
pattern, but to the benefit group theory has derived from the classification
of groups by varietal properties. It is this aspect of the study of varieties
that seems to have caused its reappearance in the literature in the 1950's.
Since then varietal methods in group theory have rapidly gained impetus
and, as is wont to happen, they have thrown up their own problems and
developed into a branch of group theory which is as much of intrinsic
interest as it is powerful as a tool.
A course of lectures given in 1963 at the Manchester College of Science
and Technology started the process of gathering and sorting the results
on varieties of groups. Perhaps this very process has had some share in
accelerating progress. In any event, when, after too long a time, the
manuscript was ready for the publishers, it was also ready to be re
written. Experience shows that a report of this kind, delayed too long,
becomes useless (if, indeed, it appears at all I). I therefore decided against
re-writing which would have meant a long delay. It may then be in order
to indicate briefly in this preface two of the points that I would have
wanted to attend to.
The first is mainly, though not entirely, a matter of organization of
material. Knowledge of metabelian varieties was scant until quite recently
when it expanded so rapidly that, to do it justice, a separate chapter
should now have been given to this topic. Instead I have merely added
to the information provided in various contexts - a plan that was natural
VIII Preface
and adequate a short time ago. I have tried to ensure that these scattered
references to the metabelian case are easily located by means of the index.
Secondly, I want to draw attention to an alternative development
which has great advantages. SMEL'KIN'S embedding theorem (22.48)
should, I believe, be made the starting point of the treatment of product
varieties. Its proof is direct, needing no more than basic facts and the
construction of the verbal wreath product; but its use would shorten and
simplify much of Chapter 2 and several other topics besides (for example
in Chapter 4).
This report would not have been written but for the interest and active
participation of my audiences in the lectures at Manchester, mentioned
earlier, and in similar lectures given at the Australian National Uni
versity, Canberra. JOHN P. COSSEY and IAN D. MACDONALD checked
parts of the manuscript; so did L.G. KovAcs, and M.F. NEWMAN read
most of it at some stage or other. KovAcs and NEWMAN contributed
much more in the way of elucidation, simplification and correction than
can be apparent from the text. B. H. NEUMANN and PETER M. NEUMANN
read the proofs and, aven at that late stage, prevented a number of minor
and major accidents. I wish to' acknowledge in particular the efforts of
the latter who seemed to interpret "proof reading" to mean "checking
all proofs". I record here my gratitude to all of them, and affirm, as is
usual but still important, that any errors that remain are mine. Last,
but not least, I acknowledge SPRI'NGER'S cooperation that was indeed all
that one has come to expect of this name.
Canberra, November 1966· HANNA NEUMANN
Contents
Note ......... . XI
Chapter 1. The Basic Facts.
1. Preliminaries . . .
2. Words, Laws, Verbal Subgroups. 3
3. Relatively Free Groups. . . . . 9
4. Varieties .......... 12
5. Varieties as Closed Classes of Groups 14
6. The n-Generator Groups and the n-Variable Laws of a Variety. 20
7. Discrimination and Residual Properties 27
8. Verbal Products . . . . 32
Chapter 2. Product Varieties. . . . . . . . 38
1. The Algebra of Varieties ..... 38
2. Wreath Products and Discrimination . 45
3. The Uniqueness of Factorization 55
4. Some Classes of Indecomposable Varieties 58
5. Product Varieties Generated by a Finitely Generated Group 64
6. Residual Properties of the Free Groups of Product Varieties 73
Chapter 3. Nilpotent Varieties . . . . . . . . . . 77
1. Summary of Properties of Nilpotent Groups 77
2. Residual Properties. . . . . . . . . . . . 80
3. A Lemma on Words with an Application to Free Products 83
4. The Laws of a Nilpotent Variety and Related Topics. . . 89
5. Generating Groups of Finite Rank .......... 99
6. The Variety of All Metabelian Nilpotent Groups of Class c 104
Chapter 4. Miscellaneous Properties of Relatively Free Groups. 110
1. Remarks on Automorphisms and the Hopf Property. . . 110
2. Free Subgroups of Free Groups . . . . . . . . . . . . 114
3. Theorems Like Auslander and Lyndon'S: the Schreier Property 126
4. The Splitting Property; Direct Decomposability 136
Chapter 5. The Laws of Finite Groups . . 145
1. Critical Groups and Cross Varieties 145
2. The Theorem of Oates and Powell . 151
3. Critical Groups and Subvarieties 161
4. Critical p-Groups and Locally Finite Varieties; a Summary of Develop-
ments 174
References 181
Author Index 187
Subject Index 189
Note
The numbering is decimal; thus for example the statement numbered
34.12 refers to Chapter 3, Section 4, Subsection 1, Item 2. Subsections
are not formally marked in any other way, and a subsection may contain
no more than a single numbered item - such as 32.1.
The presentation is intended to be self-contained. Facts have been
used without proof only if they are "known" in the sense that they have
found their way into at least one textbook. For less familiar facts of this
kind - here the choice is of necessity subjective - a textbook reference
is given explicitly, and referring to one textbook does not mean that
others may not also serve. Apart from this statements without proof
occur, I hope, only on the fringe of the main development, in examples
and in summaries of further results.
The textbooks referred to are the following; any reference containing
merely the author's name and a page or section number is to these books.
CoHN, P.M.: Universal algebra. Harper's Series in Modern Mathematics. New York:
Evanston and London: Harper & Row 1965.
CURTIS, C.W., and I.R. REINER: Representation theory of finite groups and associative
algebras. New York and London: Interscience Publ. 1962.
HALL Jr., MARSHALL: The theory of groups. New York: Macmillan 1959.
KUROSH, A.G.: The theory of groups, vol. I and II. Translat. from the Russian and
edit. by K.A. HIRSCH, 1st or 2nd ed. New York: Chelsea 1956,1960.
SCOTT, W.R.: Group theory. Englewood Cliffs, New Jersey: Prentice Hall Inc. 1964.
Since this report went to press, the book by W. MAGNUS, A. KARRASS
and D. SOLITAR, Combinatorial Group Theory, Interscience has appeared.
Had it been available earlier, reference to it would have been frequent,
for example wherever commutator calculus is used.
Chapter 1
The Basic Facts
1. Preliminaries
The following notation will be adhered to throughout.
Groups: capital Roman letters.
Group elements: small Roman letters. But the identity of every group
is denoted by 1.
Varieties 0/ groups: capital German letters; these will also be used for
more general sets and classes of groups.
Sets 0/ group elements: small German letters.
Mappings and/unctions: small Greek letters.
However, small Roman and small Greek letters will also occur as
exponents, and as subscripts and superscripts. In this context the capital
Roman letter I will always denote the set of positive integers.
The order of a group, or the cardinal of a set, is indicated by vertical
bars: 1A I, 1x I. If A and B are groups, and there exists a monomorphism
of A into B, we often write A ~ B: A is isomorphic to a subgroup of B.
If it is relevant that A is actually a subset, we write A £ B. If A is a normal
subgroup of B, we write A<l B. The derived group of the group A is A'.
For any set S and any positive integer n, S" denotes the cartesian
product of n copies of the set S. If Tc;;;, S, then S\T is the set of elements
of S that do not belong to T.
Arrows denote mappings as usual. In the case of homomorphisms,
a two-headed arrow (-+) indicates an epimorphism, an arrow with a
tail (~) indicates a monomorphism.
A/actor of a group A is any factor group H/Kwhere {l}~K<lH~A;
the factor is proper unless K={l} and H=A.
We shall make frequent use of the following notation which is a
slight modification and extension of that used by P. HALL: If X is a
class of groups, we denote by
s X the class of subgroups of groups in X,
QX the class of homomorphic images of groups in X (Q for quotient
group!),
eX the class of cartesian products of groups in X (cf. Section 5 of this
chapter),
2 The Basic Facts
D X the class of finite direct products of groups in X (where 'finite
direct product' does, of course, not imply the finiteness of the group,
but only that the product is formed with a finite number of factors).
Thus, for example, QS X consists of all factors of groups in X.
We summarise some simple facts onfree groups. All that is used can
be found in textbooks, for example those by A. G. KUROSH and by
MARSHALL HALL.
The letter F will denote a free group of unspecified rank; we write Fn
if the free group is of finite rank n, Fro if it is of countably infinite rank.
A set of free generators is denoted by f; thus
Each element =l= 1 of F has a unique representation in the form
g;
11.11 h~'h~2 ... i;.=l=iHl for 2=1, ... ,[-1
and the j1;. are non-zero integers.
Let A =gp(a) be an arbitrary group and F=gp(f) a free group
whose free generating set f has the same cardinal as a. Then there is a
1-1 mapping cc f~a of f onto a.
11.12 The mapping r:t. can be extended to an epimorphism r:t.: F--'i>A.
This extension is, of course, obtained by mapping f~'·· .fir' on
(fi, r:t.)'ll ... (fi,r:t.)Il'. Note that the mapping of the free generating set and
the homomorphism extending it are denoted by the same letter. Moreover
we shall, unless otherwise stated, use the convention that if the generating
set f is indexed by the elements of some index set, then the set a of images
will be indexed by the same set in such a way that elements of f and a
bearing the same subscript correspond under r:t..
Going beyond 11.12 we have more generalIy:
11.13 Every mapping r:t.: f ~ f r:t. !:; A can be extended to a homomorphism
of F into A.
In particular
11.14 Every mapping 8: f ~ f 8!:; F can be extended to an endomor
phism of F.
We go back to 11.12; it shows that A ~ Fj R where R = ker r:t.. Put
differently, we have a presentation A =gp(a; R). Here every element
rE R is called a relator of the generating set a of A, and if r =ft fiz2 .. .ft.'
according to 11.11, then a~' ai,2 ... ar.' = 1 is a relation satisfied by the
generators of A.