Table Of ContentLecture Notes ni
Mathematics
Edited by .A Dold dna .B Eckmann
958
E Rudolf lyeB
negriLJ Tappe
Group ,snoisnetxE
rese ep R oitatn ,sn
dna the Schur Multiplicator
£alreV-regnirpS
Berlin Heidelberg New kroY 1982
Authors
E Rudolf Beyl
Mathematisches Institut der Universit~t
Im Neuenheimer Feld 288, 6900 Heidelberg, Germany
Jirgen Tappe
Lehrstuhl f~r Mathematik
Rhein.-Westf. Technische Hochschule Aachen
Templergraben 55, 5100 Aachen, Germany
AMS Subject Classifications (1980): 20C25, 20E22, 20J05, 20C20,
20E10, 20J06
ISBN 3-540-11954-X Springer-Verlag Berlin Heidelberg New York
ISBN 0-387-11954-X Springer-Verlag New York Heidelberg Berlin
This work si tcejbus to .thgirypoc All rights era ,devreser whether eht whole or part of the lairetam
si ,denrecnoc specifically those of ,noitalsnart ,gnitnirper esu-er of ,snoitartsulli ,gnitsacdaorb
noitcudorper yb photocopying enihcam or similar ,snaem dna storage ni data .sknab rednU
§ 54 of eht namreG Copyright waL where copies era made for other than private ,esu a fee si
elbayap to tfahcsllesegsgnutrewreV" ,"troW .hcinuM
© yb galreV-regnirpS Berlin Heidelberg 2891
detnirP ni ynamreG
gnitnirP dna binding: Beltz ,kcurdtesffO .rtsgreB/hcabsmeH
012345-0413/6412
TABLE OF CONTENTS
Introduction
Chapter .I Group Extensions with Abelian Kernel
.I The Calculus of Induced Extensions 5
.2 The Exact Sequence for Opext 91
.3 The Schur Multlplicator and the Universal
Coefficient Theorem 28
4. The Ganea Map of Central Extensions 4O
.5 Compatibility with Other Approaches 47
.6 Corestrictlon (Transfer) 58
Chapter II. Schur's Theory of Projective Representations
.I Projective Representations 67
.2 The Problem of Lifting Homomorphisms 77
.3 Representation Groups 91
.4 Representation Groups of Free and Direct Products 101
.5 The Covering Theory of Perfect Groups 113
Chapter III. Isoclinism
.I Isoclinic Groups and Central Extensions 123
.2 Isoclinism and the Schur Multlplicator 137
.3 The Isomorphism Classes of Isoclinic Central
Extensions and the Hall Formulae 144
.4 On Presentations of Isoclinlc Groups 155
.5 Representations of Isoclinic Groups 169
VI
Chapter IV. Other Group-Theoretic Applications of the
Schur Multipllcator
.I Deficiency of Finitely Presented Groups 179
.2 Metacycllc Groups 193
.3 The Precise Center of an Extension Group and
Capable Groups 204
4. Examples of the Computation of Z*(G) 213
.5 Preliminaries on Group Varieties 227
.6 Central Extensions and Varieties 233
.7 Schur-Baer Multipllcators and Isologlsm 244
Bibliography 261
Index of Special Symbols 271
Subject Index 274
INTRODUCTION
The aim of these notes is a unified treatment of various group-
theoretic topics for which, as it turns out, the Schur multiplicator
is the key. At the beginning of this century, classical projective
geometry was at its peak, while representation theory was growing
in the hands of Frobenius and Burnside. In this climate our subject
started with the two important papers of Schur ~I~,~2~ on the pro-
Jective representations of finite groups.
But it was only in the light of the much more recent (co)homology
theory of groups that the true nature of Schur's "Multlplicator" and
its impact on group theory was fully realized; the papers by GREEN
EI~, YAMAZAKI EI~, STALLINGS ~I~, and STAMMBACH ~I~ have been most
influential in this regard.
The first chapter provides the setting for these notes. We start
out with the concepts of group extension (handled in terms of
diagrams) and Schur multiplicator (here defined by the Schur-Hopf
Formula) to obtain a group-theoretlc version of the Universal
Coefficient Theorem. All these concepts and the Ganea map have a
homological flavor, but are here developed in a rather elementary
group-theoretic fashion; the (co)homology theory of groups is not
a prerequisite for reading most of these notes. The first chapter
also includes a full translation from our approach to the usual
group (co)homology for the reader's convenience. (We feel that our
presentation is very suited for the applications to follow, but this
view is to some extent a matter of taste.)
In the second chapter we consider projective representations,
which can be regarded as homomorphisms into projective groups.
Schur showed that the projective representations of the finite group
Q over the complex field C can be described in terms of the (linear)
representations of certain central extensions by Q, and thus In-
vented the "Darstellungsgruppen yon Q" or, in English, the repre-
sentation groups of Q. In the course of this chapter, several
variations of Schur's theme are discussed. (The problem is that of
lifting homomorphisms where the projective representations are
replaced by more general homomorphisms.) In most of the literature
on Schur's theory, representation groups etc. are defined only for
finite groups. In these notes this attitude is seen to be unneces-
sarily restrictive, provided one carefully distinguishes between
M(Q), as the common kernel of all representation groups, and the
competing condidate H2(Q,C *) ~ Hom(M(Q),t*) We find that repre-
sentation groups of arbitrary groups are important in many parts of
group theory beyond the original aspects of representation theory.
For example, the final section of Chapter II gives a comprehensive
treatment of the covering theory of perfect groups. This theory is
related e.g. to finite simple groups, but has recently gained par-
ticular attention for its relevance to Milnor's K 2 functor - where
Q is an infinite matrix group.
In the third chapter we study the notion of isoclinism, which
P. Hall introduced in his GGttingen lectures on the classification
of groups. In spite of World War II having begun, summaries of
Hall's lectures were published in "Crelle's Journal", cf. P. HALL
I,2,3,4; it may be due to these circumstances that Hall did
not publish further details. The notion of isoclinism is extended
to central extensions; this step is more or less technical, but
provides for clarity and enables us to apply the machinery of
Chapter I. The first major result of this chapter is a description
of isoclinism classes in terms of the subgroups of the Schur multi-
pllcator° We then study a refinement of the isoclinism concept and
prove formulae of P. Hall ~3S. Our treatment in terms of central
extensions differs from Hall's, which employs free presentations.
In any case, Chapter III brings out some connections between both
views.
In the final section of Chapter III, we work out implications of
the isocllnism relation for the ordinary and the modular representa-
tions of finite groups.
The last chapter contains further group-theoretic applications
of the Schur multiplicator, in some aspects it supplements STAMMBACH
~3S. We first resume the question of group deficiency, a concept
grown out of the desire to present a group with as few relators as
possible. Our emphasis is on worked-out examples and their inter-
pretation. Among other results, we give rather elementary treat-
ments of (i) Swan's examples of finite groups with trivial multi-
plicator and large deficiency; (ii) an interesting representation
group of the non-abelian group of order p3 and exponent p, for p an
odd prime; (iii) metacyclic groups and their multiplicators. The
next topic is a group invariant Z*(G), the central subgroup that
measures how much G deviates from being a group of inner auto-
morphisms. (This concept is unrelated to Glauberman's Z*, any
serious confusion seems to be unlikely.) We then obtain rather
explicit results on the question whether a central group extension
lies in a given variety of exponent zero. These sections again
show the importance of Schur's "hinrelchend erg~nzte Gruppen", i.e.
central extensions having the lifting property for complex projec-
tive representations; they are called "generalized representation
groups" in these notes. The chapter ends with a development of
isologism, a related concept of P. Hall, in analogy with our treat-
ment of isoclinism in Chapter III. The reader will now be prepared
to study LEEDHAM-GREEN/McKAY ~I and other papers on varietal
cohomology.
These notes partly present results from our "Habilitationsleistun-
gen" at "Ruprecht-Karls-Universit~t Heidelberg" and "Rheinisch-West-
f~lische Technische Hochschule Aachen", respectively. We thankfully
acknowledge the support we received from our institutions, as well
as partial support from the "Deutsche Forschungsgemeinschaft (DFG)",
the "Forschungslnstitut f~r Mathematik (ETH ZGrich)", and the
"Gesellschaft yon Freunden der Aachener Hochschule (FAHO)".
We remember with pleasure that we greatly profited from the feed-
back various seminar audiences gave us, in particular from discus-
sions with the late R. Baer, with P. Hilton, C.R. Leedham-Green,
R. Laue, J. NeubUser, J. Ritter, U. Stammbach, R. Strebel, and
J. Wiegold.
CHAPTER I. GROUP EXTENSIONS WITH ABELIAN KERNEL
The core of this chapter consists of Sections 3 and 4.
1. The Calculus of Induced Extensions
This section is preparatory. We introduce forward and backward
induced group extensions and discuss the relationship between exten-
sions and factor systems. An extension of the group N by the group Q
is a short exact sequence
(1.1) e = (~,~) : N~ G ~ ~Q
or, equivalently, an exact sequence
e : O--~N ~ •G w ~Q ~0
of groups. The arrows ; > and ~ denote injective and sur-
jectlve homomorphisms (monomorphlsms and epimorphisms), respectively,
and 0 = ~11 stands for the group of one element. (This terminology
accords with category theory and is easy to work with.)
In most of the cases treated here, N will be abelian. Then the
extension e gives rise to a Q-module structure on N, which is well-
defined by qn = ~-1(g.~n.g-1) , where q ~ Q , n e N and g ~ G
is any element with wg = q . Now let (N, :~ Q - Aut(N)) be a
Q-module. We call e a Q-extension of (N,~) if e is an extension
of N by Q which induces the given Q-module structure on N by the
method above. Two extensions I e and 2 e of N by Q are called
congruent, if there exists an isomorphism :~ 1 2 ~ G G such that the
following diagram is commutative; we then write tfe 2 ~ e .
6
e I : IN I > ~ t G 1 ~'- 1 Q 1
(1.2)
e 2 : N ; ; G 2 m Q
Congruence is an equivalence relation. When N is abelian, congruent
extensions define the same Q-module structure on N. Let
0pext(Q,N,~) denote the set of congruence classes e of Q-exten-
sions of (N,~) . The existence of the extension
K O
(1.3) e o : N~ ; N~ Q o ;;Q
implies that Opext(Q,N,m) is not empty. - Here N~Q is the
semidirect product of Q by the Q-module N with the multiplication
formula (n,q).(nl,ql) = (n.qnl,q-ql) The maps Ko and ~o are
defined by ~o(n) = (n,1) and Vo(n,q) = q . - The extension e as
in (1.1) is called split if there is a homomorphism a: Q - G with
~a= 1Q . The Q-extension e of (N,m) is split precisely when it
is congruent to e o .
A morphism (e,e,~): e I - e 2 of extensions is a commutative
diagram
e I : N I ~' G1 ~ 1Q
(1.4)
e 2 : N 2 ; ; G 2 ~11 Q2
This is called an isomorphism of extensions if ~ and ~ (hence also ~)
are group isomorphisms. Now assume N i abelian and let
mi: Qi " Aut(Ni) be the structual maps defined by e i , for i:=1,2.
If a morphlsm (~,8,~): e I - e 2 exists, then
(1.5) ~(qn) = ~(q)~(n) for all n e N I , q e Q1
or, equivalently, 5: (NI,~ )I - (N2,~2~) is Q1-homomorphic.
