Lecture 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.