Groups - St Andrews 1981 (London Mathematical Society Lecture Note Series (No. 71))

LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES Managing Editor: Professor I.M. James, Mathematical Institute, 24-29 St Giles,Oxford I. 4. 5. 8. 9. 10. II. 12. 13. 15. 16. 17. 18. 20. 21. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48.

General cohomology theory and K-theory, P.HILTON Algebraic topology, J.F.ADAMS Commutative algebra, J.T.KNIGHT Integration and harmonic analysis on compact groups, R.E.EDWARDS Elliptic functions and elliptic curves, P.DU VAL Numerical ranges II, F.F.BONSALL & J.DUNCAN New developments in topology, G.SEGAL (ed.) Symposium on complex analysis, Canterbury, 1973, J.CLUNIE & W.K.HAYMAN (eds.) Combinatorics: Proceedings of the British Combinatorial Conference 1973, T.P.McDONOUGH & V.C.MAVRON (eds.) An introduction to topological groups, P.J.HIGGINS Topics in finite groups, T.M.GAGEN Differential germs and catastrophes, Th.BROCKER & L.LANDER A geometric approach to homology theory, S.BUONCRISTIANO, C.P. BOURKE & B.J.SANDERSON Sheaf theory, B.R.TENNISON Automatic continuity of linear operators, A.M.SINCLAIR Parallelisms of complete designs, P.J.CAMERON The topology of Stiefel manifolds, I.M.JAMES Lie groups and compact groups, J.F.PRICE Transformation groups: Proceedings of the conference in the University of Newcastle-upon-Tyne, August 1976, C.KOSNIOWSKI Skew field constructions, P.M.COHN Brownian motion, Hardy spaces and bounded mean oscillations, K.E.PETERSEN Pontryagin duality and the structure of locally compact Abelian groups, S.A.MORRIS Interaction models, N.L.BIGGS Continuous crossed products and type III von Neumann algebras, A.VAN DAELE Uniform algebras and Jensen measures, T.W.GAMELIN Permutation groups and combinatorial structures, N.L.BIGGS & A.T.WHITE Representation theory of Lie groups, M.F* ATIYAH et al. Trace ideals and their applications, B.SIMON Homological group theory, C.T.C.WALL (ed.) Partially ordered rings and semi-algebraic geometry, G.W.BRUMFIEL Surveys in combinatorics, B.BOLLOBAS (ed.) Affine sets and affine groups, D.G.NORTHCOTT Introduction to Hp spaces, P.J.KOOSIS Theory and applications of Hopf bifurcation, B.D.HASSARD, N.D.KAZARINOFF & Y-H.WAN Topics in the theory of group presentations, D.L.JOHNSON Graphs, codes and designs, P.J.CAMERON & J.H.VAN LINT Z/2-homotopy theory, M.C.CRABB Recursion theory: its generalisations and applications, F.R.DRAKE & S.S.WAINER (eds.) p-adic analysis: a short course on recent work, N.KOBLITZ Coding the Universe, A.BELLER, R.JENSEN & P.WELCH Low-dimensional topology, R.BROWN & T.L.THICKSTUN (eds.)

49. Finite geometries and designs, P.CAMERON, J.W.P.HIRSCHFELD & D.R.HUGHES (eds.) 50. Commutator calculus and groups of homotopy classes, H.J.BAUES 51. Synthetic differential geometry, A. KOCK 52. Combinatorics, H.N.V.TEMPERLEY (ed.) 53. Singularity theory, V.I.ARNOLD 54. Markov processes and related problems of analysis, E.B.DYNKIN 55. Ordered permutation groups, A.M.W.GLASS 56. Journees arithmetiques 1980, J.V.ARMITAGE (ed.) 57. Techniques of geometric topology, R.A.FENN 58. Singularities of smooth functions and maps, J.MARTINET 59. Applicable differential geometry, F.A.E.PIRANI & M.CRAMPIN 60. Integrable systems, S.P.NOVIKOV et al. 61. The core model, A.DODD 62. Economics for mathematicians, J.W.S.CASSELS 63. Continuous semigroups in Banach algebras, A.M.SINCLAIR 64. Basic concepts of enriched category theory, G.M.KELLY 65. Several complex variables and complex manifolds I, M.J.FIELD 66. Several complex variables and complex manifolds II, M.J.FIELD 67. Classification problems in ergodic theory, W.PARRY & S.TUNCEL 68. Complex algebraic surfaces, A.BEAUVILLE 69. Representation theory, I.M.GELFAND et. al. 70. Stochastic differential equations on manifolds, K.D.ELWORTHY 71. Groups - St Andrews 1981, C.M.CAMPBELL & E.F.ROBERTSON

London Mathematical Society Lecture Note Series : 71

Groups - St Andrews 1981 Revised Edition

Edited by C.M.CAMPBELL and E.F.ROBERTSON Lecturers in Pure Mathematics University of St Andrews

CAMBRIDGE UNIVERSITY PRESS Cambridge London

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CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www. Cambridge. org Information on this title: www.cambridge.org/9780521289740 © Cambridge University Press 1982 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1982 Revised Edition Re-issued in this digitally printed version 2007 A catalogue record for this publication is available from the British Library Library of Congress Catalogue Card Number 82-4427 ISBN 978-0-521-28974-0 paperback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party Internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

CONTENTS Preface Twenty-five years of Groups St Andrews Conferences CM. Campbell & E.F. Robertson Original Introduction 1. An elementary introduction to coset table methods in computational group theory J. Neubuser

viii

ix xiii

1

2. Applications of cohomology to the theory of groups D.J.S. Robinson

46

3. Groups with exponent four S. J. Tobin

81

4. The Schur multiplier: an elementary approach J. Wiegold

137

5. A procedure for obtaining simplified defining relations for a subgroup D.G. Arrell, S. Manrai & M.F. Worboys

155

6. GLn and the automorphism groups of free metabelian groups and polynomial rings S. Bachmuth & H. Y. Mochizuki 160 7. Isoclinisms of group extensions and the Schur multiplicator F.R. Beyl

169

8. The maximal subgroups of the Chevalley group C?2(4) C Butler

186

9. Generators and relations for the cohomology ring of Janko's first group in the first twenty one dimensions G.R. Chapman 201 10. The Burnside group of exponent 5 with two generators M. Hall Jr. & CC Sims 11. The orient ability of subgroups of plane groups A.H.M. Hoare & D. Singerman

207

221

VI

12. On groups with unbounded non-archimedean elements A.KM. Hoare & D.L. Wilkens

228

13. An algorithm for the second derived factor group J.R. Howse & D.L. Johnson

237

14. Finiteness conditions and the word problem V. Huber-Dyson

244

15. Growth sequences relative to subgroups W. Kimmerle

252

16. On the centres of mapping class groups of surfaces C. Maclachlan

261

17. A glance at the early history of group rings C. Polcino Milies

270

18. Units of group rings: a short survey C. Polcino Milies

281

19. Subgroups of small cancellation groups: a survey S.J. Pride

298

20. On the hopficity and related properties of some two-generator groups S.J. Pride & A.D. Vella

303

21. The isomorphism problem and units in group rings of finite groups K. W. Roggenkamp

313

22. On one-relator groups that are free products of two free groups with cyclic amalgamation G. Rosenberger 328 23. The algebraic structure of No-categorical groups J.S. Wilson

345

24. Abstracts

359

25. Addendum to: "An elementary introduction to coset table methods in computational group theory" CM. Campbell, G. Havas & E.F. Robertson

361

Vll

26. Addendum to: "Applications of cohomology to the theory of groups" D.J.S. Robinson

365

27. Addendum to: "Groups with exponent four" S.J. Tobin

368

28. Addendum to: "The Schur multiplier: an elementary approach" J. Wiegold

373

PREFACE

We would like to thank Cambridge University Press for encouraging us to produce this new edition of the Proceedings of Groups St Andrews 1981. At the suggestion of Roger Astley of Cambridge University Press we have asked the four main speakers at the 1981 conference to provide brief addenda to their articles. We are delighted that they have all responded positively to this task. Three of the authors have provided their own new pages. The fourth article on 'An elementary introduction to coset table methods in computational group theory' has been prepared by us with our friend and collaborator George Havas after some helpful suggestions from Joachim Neubiiser. We have also added a short article looking back at twenty-five years of Groups St Andrews conferences. Although for the 1981 Proceedings we put all the references into a standard form, we have, twenty-five years later, adopted a more relaxed approach and have kept the refereeing style of the addenda as provided by the authors. Thanks are also due to our colleague Martyn Quick for his help with the preparation of the additional material. Colin M. Campbell Edmund F. Robertson St Andrews, August, 2006

TWENTY-FIVE YEARS OF GROUPS ST ANDREWS CONFERENCES COLIN M. CAMPBELL and EDMUND F. ROBERTSON

In 1979 we held a small meeting in St Andrews at which Joachim Neubiiser from RWTH Aachen spoke on Counterexamples to the class-breadth conjecture. At this time we discussed the possibility of organising a much larger group theory meeting in St Andrews in 1981. Preliminary dates were suggested to fit the German school holidays. Indeed choosing dates for all the meetings has proved an interesting task: fitting in with the end of the English academic year, the start of the American academic year, the Galway races (Galway 1993), the Open University Summer School (Bath 1997), the Open Golf Championship (St Andrews 2005). For Groups 1981 we invited main speakers whose mathematical interests were close to our own. By chance, three of the four — Joachim Neubiiser (RWTH Aachen), Sean Tobin (Galway), and Jim Wiegold (Cardiff) — had been friends from postgraduate days in Manchester. The fourth, Derek Robinson (Urbana), was originally from Montrose (visible on a good day across the Tay estuary from the Mathematical Institute in St Andrews). Despite our planning of the 1981 dates, the wedding of Prince Charles and Princess Diana was announced to take place during the period of the conference. Residences provided only packed lunches on the wedding day. However Jim Wiegold, the 'Mathematical Prince of Wales', provided our own star attraction! We had intended the conference to last a week but some participants wanted to stay in St Andrews for a further week. Thus began our style of a twoweek conference, the main speakers giving lecture courses in the first week with a more informal seminar programme taking place in the second week. During the 1981 conference, carried away by the excitement of the moment, we discussed, while we were driving participants round the Highlands (actually on a stop at Killin), holding another meeting in four years time. The 1985 meeting proved the largest of all the Groups St Andrews meetings with 366 participants from 43 countries. For this meeting we tried to make the topics broader to cover as much of group theory as possible. With this aim in mind we invited Seymour Bachmuth (Santa Barbara), Gilbert Baumslag (CUNY), Peter Neumann (Oxford), Jim Roseblade (Cambridge) and Jacques Tits (Paris) to be the main speakers. It is always a challenge to people to ask them what is wrong with the photograph of the main speakers, and ourselves, as it appears in the Conference Proceedings. (Clue: does it look better in a mirror?) Again, carried away by the vitality of the Conference, we announced another meeting to be held in 1989. Following the 1985 conference we asked a number of the participants which group theorists they would like to see as main speakers for 1989. Taking their advice we invited Sandy Green (Warwick), Narain Gupta (Manitoba), Otto Kegel (Freiburg), Sasha Ol'shanskii (Moscow), and John Thompson (Cambridge). Typical of problems organisers have to face, there was a "heightening of tension" between the UK and the Soviet Union in the spring of 1989. Consulates were closed, as was

TWENTY-FIVE YEARS OF GROUPS ST ANDREWS CONFERENCES

X

the British Airways Office in Moscow. The Principal of St Andrews University, Professor Struther Arnott, wrote to his colleague, the Rector of Moscow State University, in an attempt to make Sasha Ol'shanskii's visit possible. It was a great surprise and pleasure when Sasha achieved what we thought was impossible at that time, and actually arrived in St Andrews. So many of the participants were, by this time, friends we were delighted to meet again. Indeed it was (and is) great to see so many people from so many different countries renewing friendships and mathematical contacts at our conferences. After three conferences in St Andrews, we thought that a change of scene might be appropriate. We had enjoyed a very successful mathematical collaboration between RWTH Aachen, Galway and St Andrews (with an EU twinning grant for three twins!) and our Galway colleagues had (and still have) such a successful annual Group Theory meeting that we were happy to accept their invitation to hold Groups St Andrews 1993 in Galway. We now became part of an organising committee of five (us together with Ted Hurley, Sean Tobin and James Ward from Galway). The five organisers met at a Warwick conference in March 1991, and decided to invite as principal speakers Jon Alperin (Chicago), Michel Broue (Paris), Peter Kropholler (Queen Mary College, London), Alex Lubotzky (Hebrew University of Jerusalem) and Efim Zelmanov (Wisconsin-Madison). (We were especially delighted when Efim was awarded a Fields Medal exactly one year later at the 1994 ICM in Zurich.) We also invited our Aachen twin Joachim Neubiiser to organise a GAP workshop at the 1993 conference. The workshop became effectively a fully-fledged parallel meeting throughout the second week, with over thirty hours of lectures by experts together with practical sessions. Another unusual feature of this conference was the setting aside of one day for a special programme of lectures to honour the 65th birthday of Karl Gruenberg (London), in recognition of his many contributions to group theory. In 1989, at the conference, Geoff Smith (Bath) said that he would be interested in hosting a future Groups St Andrews meeting. At the 1993 conference Peter Neumann (Oxford) asked if Oxford might be considered as a venue for 1997. By this time, however, we had already accepted the invitation to hold the 1997 Conference in Bath, but it was agreed that Oxford would be the venue for 2001. For the Bath Conference Geoff Smith joined us on the organising committee. The main speakers for the 1997 conference were Laszlo Babai (Chicago), Martin Bridson (Oxford), Chris Brookes (Cambridge), Cheryl Praeger (Western Australia) and Aner Shalev (Hebrew University of Jerusalem). This time the second week featured two special days, a Burnside Day and a Lyndon Day, organised by Efim Zelmanov (now at Yale) and Chuck Miller III (Melbourne), respectively. An interesting innovation at this conference was twelve editions of %ty Baity ^rowp tEfcocift providing details of lectures, seminars, the social programme, together with other items of interest. The organizing committee for Groups St Andrews 2001 in Oxford consisted of the two of us together with Danny Groves (Merton, Oxford), Patrick Martineau (Wadham, Oxford), Peter Neumann (Queen's, Oxford), Geoff Smith (Bath), Brian Stewart (Exeter, Oxford) and Gabrielle Stoy (Lady Margaret Hall, Oxford). For the Oxford conference the main speakers were Marston Conder (Auckland), Persi

TWENTY-FIVE YEARS OF GROUPS ST ANDREWS CONFERENCES

xi

Diaconis (Stanford), Peter Palfy (Eotvos Lorand, Budapest), Marcus du Sautoy (Cambridge), and Mike Vaughan-Lee (Christ Church, Oxford). In the second week there was a day of special lectures in celebration of Daniel Gorenstein coordinated by Richard Lyons (Rutgers) and a "Groups and Sets" day coordinated by Simon Thomas (Rutgers). Again the %ty Jiaffp ^roup tEfcorift provided entertaining and informative reporting items such as "Punting tragedy", "News from the Netherlands", "Dancing in a group", and "Editor in Bath... pictures follow". We realised that Groups 2005 would be the last of the series that we would organise before retiring so we made the decision to bring the conference 'home'. Unlike the earlier St Andrews conferences we had an extended organising committee, being joined by Nick Gilbert (Heriot-Watt), Steve Linton (St Andrews), John O'Connor (St Andrews), Nik Ruskuc (St Andrews), and Geoff Smith (Bath). We also reverted to the format of the 1981 conference and put the whole programme into a week. The main speakers were Peter Cameron (Queen Mary, London), Slava Grigorchuk (Texas A&M), John Meakin (Nebraska-Lincoln) and Akos Seress (Ohio State). Additionally there were seven one-hour invited speakers together with an extensive programme of over a hundred seminars; a lot to fit into a week! The seven conferences have contained a wide selection of social events. "Groups St Andrews tourism" has taken us to a variety of interesting and scenic venues in Scotland, Ireland, England and Wales. Bus trips have included Kellie Castle, Loch Earn and Loch Tay, Falkland Palace and Hill of Tarvit, Crathes Castle and Deeside, Loch Katrine and the Trossachs, House of Dun, Connemara and Kylemore Abbey, the Burren and the Cliffs of Moher, Tintern Abbey and Welsh Valleys, the Roman Baths in Bath, Salisbury Cathedral, Rufus Stone and the New Forest, Stonehenge, Wells Cathedral and the Cheddar Gorge, Blenheim Palace, Glamis Castle. We have been on boats on Loch Katrine, the Thames, and Galway Bay to the Aran Islands. There have also been: musical events with participants as the musicians, Scottish Country Dance evenings, barn dances, piano recitals, organ recitals, theatre trips, whisky tasting, putting, chess, walks along the Fife Coast, walks round Bath, walks round Oxford, and there would have been a cricket match in Bath but it was rained off. All these have provided opportunities for relaxation, but also opportunities to continue mathematical discussions. For example after the trip to Falkland Palace we collected from the buses several copies of the guidebook, each filled with fascinating group theory theorems. Conference dinners have taken place in David Russell Hall (old and new) in St Andrews, the Corrib Great Southern Galway, the Ardilaun House Hotel Galway, the Assembly Rooms in Bath ("a grand Georgian affair with chandeliers etc."), Cumberwell Park Golf Club (in Bradfordon-Avon and not St Andrews!) and Lady Margaret Hall Oxford. The twenty-five years since 1981 have been an important period in the development of group theory following the classification of finite simple groups. Although attempting wide coverage of group theory topics, we made a conscious decision for the early conferences not to have a lecture series devoted to the classification. Despite the intervening twenty-five years, the papers from the first conference are still proving influential and hence the appearance of this second edition of the Groups St Andrews 1981 Proceedings. The main speakers at this conference spoke on: An

TWENTY-FIVE YEARS OF GROUPS ST ANDREWS CONFERENCES

elementary introduction to coset table methods in computational group theory; Applications of cohomology to the theory of groups; Groups with exponent four; and The Schur multiplier: an elementary approach. The topics of the lectures given at the conferences show the development of group theory during the quarter century. This is illustrated by the themes of the main speakers at the last two conferences. At Groups St Andrews 2001 the lecture series were: Group actions on graphs, maps and surfaces with maximum symmetry; An introduction to random walks on finite groups — character theory and geometry; Groups and Lattices; Zeta functions of groups and counting p-groups; and Lie methods in group theory, and at Groups St Andrews 2005: Aspects of infinite permutation groups; On self-similarity and branching in group theory; Interactions between group theory and semigroup theory; and Graphs, automorphisms, and product action. The influence of the series of conferences is, we believe, illustrated by the fact that 'Groups St Andrews' is mentioned in 285 reviews in MathSciNet and over 1500 papers in a beta version of the Google Scholar search. None of this would have been possible without the support of many people and organisations. Research students and colleagues have provided invaluable help in running each of the conferences. The British Council, the Edinburgh Mathematical Society, the London Mathematical Society, the Royal Society of London have provided grants, while the universities of St Andrews, Galway, Bath and Oxford have all contributed in financial and other ways to the success of the conferences. Of course the bulk of the funding has come from universities and other organisations worldwide that have supported the participation of the delegates. We are grateful to Cambridge University Press, in particular to David Tranah and Roger Astley, for the care and expertise with which they have published the Proceedings. The Proceedings of each of the first six conferences have been published by CUP as numbers 71, 121, 159 & 160, 211 & 212, 260 & 261, 304 & 305 in the London Mathematical Society Lecture Note Series. The Proceedings of Groups St Andrews 2005 will again be published in two volumes in the same series. Additionally there was a special part (Volume 30 Part I) of the Proceedings of the Edinburgh Mathematical Society consisting of sixteen of the papers presented at Groups St Andrews 1985, together with an introduction. We have enjoyed the twenty-five years of Groups St Andrews conferences and look forward to the continuation of the series. We hope to meet many of our friends at Groups St Andrews in Bath in August 2009.

ORIGINAL INTRODUCTION

An international conference 'Groups - St. Andrews 1981' was held in the Mathematical Institute, University of St. Andrews during the period 25th July to 8th August 1981. The main topics of the conference: combinatorial group theory; infinite groups; general groups, finite or infinite; computational group theory are all well-represented in the survey and research articles that form these Proceedings. Four courses each providing a five-lecture survey, given by Joachim Neubiiser, Derek Robinson, Sean Tobin and Jim Wiegold have been expanded, subsequently, into articles forming the first four chapters of the volume. Many of the themes in these chapters recur in the survey and research articles which form the second part of the volume. Methods and techniques such as homology, geometrical methods and computer implementation of algorithms are used to obtain group theoretical results. Computational methods are surveyed in several articles in particular the major survey by Joachim Neubiiser and find application in papers on Burnside groups and finite simple groups. In fact Burnside groups are discussed in two rather different papers, a survey of groups of exponent four by Sean Tobin and a major contribution to the exponent five case by Marshall Hall and Charles Sims. Derek Robinson exploits the way in which cohomology groups arise in group theory to establish some splitting and near-splitting theorems. Rudolf Beyl also uses homological techniques to discuss group extensions. The Schur multiplicator which arises naturally in this context is given a 'nonhomological' treatment (and is called the Schur multiplier!) in a survey by Jim Wiegold. Splitting results are also studied by Klaus Roggenkamp when he considers the splitting of the natural injection from a group to the group of units in its group ring. The structure and group-theoretical properties of the group of units of a group ring feature in another survey article. Presentations of groups are studied in many of the articles already described. Several other authors discuss groups with presentations of a specific type, for example small cancellation groups and one-relator groups. The reader will find extensive bibliographies with many of the papers. Many open problems are also cited in the papers often with possible methods of attack. We hope that this not only makes the volume a useful record of the current state of the art but also points the way to future developments. During the two weeks of the conference, group theory programmes on the Aberdeen Honeywell and the St. Andrews VAX were widely used by conference participants. The CAYLEY group theory package was demonstrated from the Queen Mary College computer via a link to the St. Andrews VAX. We would like to thank those in the University of St. Andrews Computing Laboratory who helped make these facilities available. We would like to express our thanks for the assistance we received from our col-

leagues in the Mathematical Institute and in particular John Howie, John O'Connor and Peter Williams. We thank the British Council for grants for two conference participants and the London Mathematical Society for their early pledge of financial support which enabled the conference to proceed. Our thanks are also given to the London Mathematical Society and Cambridge University Press for their help and encouragement in the preparation of this volume, and to Shiela Wilson for so willingly undertaking the daunting task of typing a volume of this length and for the high quality of the final typescript. Our final thanks go to those who have contributed articles to this volume. We have edited these articles to produce some uniformity without, we hope, destroying individual styles. We will have introduced inevitably errors into the text. For these errors we take full responsibility. Colin M. Campbell Edmund F. Robertson St Andrews, June 1982

AN ELEMENTARY INTRODUCTION TO COSET TABLE METHODS IN COMPUTATIONAL GROUP THEORY J. Neubiiser RWTH Aachen, 5100 Aachen, West Germany

0.

PROLOGUE

"...; in fact the method can be reduced to a purely mechanical process, which becomes a useful tool with a wide range of application.

..., we venture to predict that our

method will prove quite practicable for most groups (at any rate such as occur naturally in geometry or analysis) of order less than a thousand, and for many groups of much higher order." J.A. Todd> H.S.M. Coxeter> 1936, [57].

The paper 'A practical method for enumerating cosets of a finite abstract group1 from which the quotation is taken, may very well be thought of as starting the subject of a series of 5 survey lectures which were given at "Groups -St. Andrews 1981" under the title "Computational methods in group theory".

The quotation itself was the

guiding principle for them; I neither dealt with the question of algorithmic solubility of problems - this will in fact often be obvious nor with the use of computers for solving specific group-theoretic problems in an ad hoc fashion but restricted attention to methods which are designed (and have been implemented) for practical use in a variety of cases. Of course in 1936 Todd and Coxeter proposed and used their method for hand calculations.

As far as I know the first proposal to use

a computer on a group-theoretic problem appeared in print in the 'Manchester University Computer Inaugural Conference' in 1951 [49] where M.H.A. Newman discussed how a computer could be used to investigate 2groups along the lines of P. Hall's approach.

Although this proposal

apparently has never been followed in detail, computers have produced quite spectacular results about finite p-groups during the last 7 years.

Neubiiser:

Coset table methods

2

The first actual implementation of a group theoretical program seems to have taken place about two years later (1953), when C.B. Haselgrove implemented a Todd-Coxeter method on the Cambridge EDSAC 1 computer.

No documentation of that program seems to be left, but J. Leech

gives some description of it and subsequent implementations including his own, in a later survey article [35]. In 1960 a first paper on an implementation of a group theoretical method, this time for finding the lattice of subgroups of a finite group, was published [45]. Since then the number of publications on the subject has grown steadily reporting on the invention of a wide range of "practicable methods" and applications of increasing relevance. While in surveys published about 12 years ago [15], [46] a rather complete description of all activities in the field could still be given, in those 5 lectures I could only introduce some main lines of development which may be indicated by the titles of the lectures:

'Coset Tables1, 'Permutation

Groups1, 'Collection1, 'Subgroup Structure', and 'Characters'.

Also the

bibliography distributed at the conference contained only a selection of titles.

A much more complete bibliography which is kept current in Aachen

by V. Felsch (of course on a computer), can be obtained on request [25]. The lectures were given with the intention of introducing "theoretical" group theorists to computational methods.

So facts from group theory and

representation theory were assumed to be known.

On the other hand the

description of computational methods started from scratch, assuming only very vague understanding of the way a computer works. I have to apologize that this paper does not, as originally intended, cover (with some more details and care) all the topics touched in the lectures; when trying to write them up I soon found the manuscript growing beyond and the progress behind schedule. confined the paper mainly to the first topic.

So in the end I have

As an excuse I can offer

that for all other four topics comprehensive treatments have been published recently or are being prepared for publication. The state of the art with Sims' powerful techniques for the construction of large permutation groups from a set of generators is very clearly described in Leon's papers [39] and [40] of which the first is an easier introduction to the subject, the second a more detailed report. Of the literature quoted there [51] and [52] deal with methods for further calculation in the permutation groups thus constructed.

A variation of

these methods for matrix groups is given in Butler's thesis [6] and forthcoming joint papers of Butler and Cannon [7], [8], [9].

Neubuser:

Coset table methods

3

The study of collection methods and in particular the nilpotent quotient algorithm can best be started with the papers [29] of Havas and Nicholson and [48] of Newman, where also good references to the origin of these methods are given. For the present state of methods for the closer investigation of the subgroup structure of a given group I have to refer to two papers that are planned for the proceedings of the August 1982 Durham Conference on computational group theory.

One, by V. Felsch, will describe details

of the newest implementation of a lattice-of-subgroups program, the other, a joint paper of several authors, largely interactive "top-down" methods for soluble and nilpotent groups which are based on collection techniques. Finally for the same proceedings a report, again by several authors, is being prepared on a character table system and its use. While preparing the lectures for Groups -St. Andrews 1981 as well as while writing this report, I have freely used the papers quoted as well as possibly other ones and private communications and notes of many of their authors.

Acknowledging this I want to thank them for their

help and cooperation over many years and ask their indulgence if I have missed out details, in particular historical ones.

1.

TEE TODD-COXETER METHOD Let us start by discussing the method Todd and Coxeter

proposed for working from a finite presentation: G = < gi ,...,gjri (gi,...,gn) = l,...,rm(gi >...>gn) = 1 > or shorter G = < E|R > with E = {gi ,...,gn> and R = {r^gi ,.. .,g n )| i=l,..,m} where each "relator" r.(gi,...,g ) is a word in gi,...,g .

(For simplicity

we shall use the same notation, e.g., r.(gi,...,g ) , for a word in elements gi>- — »gn

G

G

n

» i-e. formally a finite sequence g. ,.g. ,...,g. with 1 x i 12 t

g. e ( g i , . . . » g _ , g"1 ,.-.jg"1 > and its value, i.e., the element i. n * n g. g. ...g. 1

i

X

2

1

^ G; if we have to emphasize the distinction we shall do so

t

by using the terms "word" or "element" in the text.)

Such a presentation

not only defines a group uniquely up to isomorphism, as the factor group of a free group F on free generators fi ,...,f

by the normal closure under

F of the set {n (fi ,... , f R ) , . . . ,rm(f, ,...,f n )}, see e.g. [41] , but it may arise naturally also in other branches of mathematics as the following example illustrates.

Neubiiser:

Example 1.

Coset table methods

From Fig. 1 we can read off that the fundamental group of

the octahedron space has the presentation G = < a,b,c,d|abc = bdc = bad = acd = 1 >, the relations coming from II, III, I and IV respectively. d = a^b"

1

and putting A := a, a, B := b"

1

Eliminating

we have

G = < A,B | A3 = B3 = (AB)2 >. The relations A6 = B6 =1 are consequences and elements of G have normal form A v , 0 < v < 5; A V BA y , 0 < v < 2, 0 < y < 5.

Hence |G| < 24.

So it is a question of relevance even outside group theory to ask, for any given finite presentation, if the group presented is finite, and if so what is its order. We may generalize the situation slightly:

let an additional

finite set S = {si (gi ,...,gn),...,s (gi,...,gn)} of words in the generators be given and let U be the subgroup of G generated by the elements Si (gi ,...,gn),...,s (gi ,...,gn) e G.

We may ask if U is of

finite index in G and if so what is this index G:U.

The Todd-Coxeter

method attempts to find the index, if it is finite, by enumerating cosets of U in G in a systematic trial and error procedure. based on two simple facts:

This procedure is

if s(gi,...,g ) € S then clearly

Us(gj,...,g ) = U, and if r(gi ,...,g ) £ R then for any coset Uh, h £ G

Fig. 1

III

Neubuser:

Coset table methods

one has Uh r(gi , . . . ,g ) = Uh. ii

gi

e E := ( g i , . . . , g n ,

Hence i f r(gi , . . . ,g ) = g n

gj"1 , . . . jg" 1 } , and i f a sequence o f

U0 := Uh, Ui

:= U o g.

,...,U

:= U

g

1

is defined, then U

ij

... g

l^

,

cosets

,... j

= Uo and the analogous statement holds for Uo = U

and s(gi,...,gn) e S. This is used in the following way: g.

... g. , g.

for each s(gi,...,g ) =

£ E, a "subgroup table"

providing a single line for entering t +1 numbers representing cosets is set up, the entry

H. meaning that the coset of U which has got number k in the Todd-Coxeter enumeration procedure multiplied by g. with number 1.

from the right yields the coset

Let the subgroup U itself be given the number 1, then the

above remark makes clear that the first and the last entry in that line of coset numbers are 1. Further for each relator r(gi ,...,*) = g. n

... g. , g.

ij

it

eg,

i..

a "relation table" is set up organized analogously but providing one line for each coset given a number in the enumeration process. given above, the k-th line then starts and closes with k:

1

1

2

2

k

k

For the reason

Neubiiser:

Coset table methods

Finally to facilitate the bookkeeping of how cosets are numbered a "coset table" is set up, listing for each coset C that has been given a number k in the process, and each generator g. and its inverse which number is given to Cg. and to CgT 1 , respectively:

*n

<

1 2

k

A Todd-Coxeter procedure then consists of defining in some sequence numbers for cosets of U, starting with 1 := U, then proceeding e.g. with 2 := l g ^ U g J , 3 := lg 2 (=Ug 2 ), 4 := 2g2"1 (»Ug1 g," 1 ),..., with the only rule that a coset number I must be defined by an equation £=kg, k <£, g € E, where the place of kg in the coset table is still vacant.

As soon

as the coset number Jl has been defined the Jl-th rows of the coset table and the relation tables are initialized and then this definition and its trivial consequence Jig"1 =k are filled into all possible vacant places of the various tables.

The aim of the process is to obtain information about

equality or inequality of cosets that have been given different numbers, i.e. whose coset representatives are given by different words in the generators, from the fact that lines in the subgroup tables or relation tables close up.

Whenever that happens, an equality of the kind kg =I for

some g £ E is obtained and such an equality is called a "deduction". When such a deduction is reached three possibilities can occur: (i)

either

the places of both kg and Jig"1 in the coset table are still empty.

In this case we fill the number I into the place of kg and k into the place of Jig"1 in the coset table and also insert this information into all other relevant places in the other tables; or (ii)

the place of kg in the coset table is already filled by the number

I (and hence then the place of Jig"1 by the number k ) .

In this case our

deduction brings no new information; or (iii)

at least one of the places in the coset table is filled with a

number different from that given by the deduction.

In this case we conclude

that we have given different numbers, a and b, say, to the same coset.

Neubliser:

Coset table methods

This phenomenon is called a "coincidence" a = b .

When a coincidence is

found, we have to replace the bigger one of the two numbers a and b by the smaller one in all our tables.

We postpone the discussion how to do

this. The above description does not determine in which order a definition or a deduction is inserted into the various places in the tables into which it fits.

In fact this order is irrelevant for conclusions

about the Todd-Coxeter method that we shall draw, although it may influence the efficiency of the method.

In the following examples we shall adopt

the rule to take first the subgroup tables and then the relation tables in turn and fill into them line by line, first from the front and then from the end whatever information is available in the coset table.

As

soon as new information is gained (cases (i) and (iii)) this is entered into the coset table and the process is started all over again. In the coset table, a number will be underlined, if it has been defined in this place; in the subgroup table and the relation tables deductions are underlined with a full line if they yield new information that can directly be put into the coset table (case (i) above), by a dotted line, if not (case (ii)) and by a wavy line, if they lead to a coincidence (case (iii)).

When we want to keep a record of the sequence

in which deductions occurred, we shall number them and put the number beneath the underlining. Let us follow these rules in our example (in which case (iii) above does not occur). d = a^b"

1

Putting A := a, B := b"1 one has c = b"

1

= A' B and the shorter presentation: G = < A,B | ABA"2B = 1,

1 >

= 1, ABAB"1 B'1

G = < A,B | BABA"^"

or

1 >.

With S = {A 2 } one has one subgroup table, two relation tables and the coset table, as shown below with the first definition 2 := 1A and the deduction 2A = 1, which follows from the closing of the subgroup table, being inserted. Subgroup table

Relation tables

B" 1 B"

1

1r 1 A-1 1 1 2

1

2

1

2

2

2

1

2

II

A

B

1

2

BA"

Neubiiser:

Coset table methods

The coset table A

A"1

B

1

B"1

2 1

2

1

Proceeding further by the sequence of definitions 3 := IB, 4 := IB"1 5 := 2B, 6 := 2B"1 , 7 := 3A"1 , 8 := 4A we get the following picture: Subgroup table

Relation tables 'A' 1 1

3

2

5

2 4

4

1

2

1

1

2

5

6

3

1

2

1

2

2

1

3

4_

5

5

2

3

6

6 1

2

5

4

8

7

S

8

7

8

4

3

3

4

1

2

5

8

4

4

8

7

3

1

6

5

5

6

142

1

4

5

7

8

5

2

6 7

7

3

2

B'1 B"

B

6

4

6

A

1 12

9

7 J1

10

3

7

5

4

3

3

6

5

8

6

6

7

7

7

3

6

is7

8

8

8

5

4

8

7

3 6

4

The coset table B

A"1

B"1

CM|

1

A

^

2

4_

2

1

15

1

6»

3

4

6

7_

1

4

00|

1

3

5

5

6

4

8

2

6

7

2

5

3

7

3

8

6

8

8

5

7

4

7

All lines of all our tables have closed simultaneously.

We shall show

later that when this happens all coset numbers represent different cosets and all cosets have been numbered. Before we do this let us look at another example which shows that coincidences (case (iii) above) cannot generally be avoided. Our first presentation is obviously equivalent to < A,B | BAB = A2 , ABA = B2 >. We change this, seemingly only slightly, to

Neubuser:

Coset table methods

Gi = < A,B | BAB'1 = A2 , ABA'1 = B2 > and enumerate again cosets of Ui = < A 2 > .

From the definition 2 := 1A

we obtain the deduction 2A =1 and from the further definition 3 := IB in turn the deductions 3A = 3, IB"1 =2 and 2B"1 =2.

At the time, when this

last one is found, the state of the tables is the following:

B 1

3

2

1

2

1

3

3

3

A

B

A"1

B"1

2

3

2

2

2

1

1

3

3

1

B"1 B"

B

B" 1

1

2 1

1

2

2

3

3

3

1

3 3

3

2

1 3

3

1 1

3

Trying to insert 2B = 2 into the coset table we discover the coincidence 1=2.

Multiplying this by B, we get immediately 3 = IB = 2B = 1, hence

the numbers 1, 2 and 3 all denote the same coset, we have encountered what is called a "total collapse". It is clear that in this example we could not avoid defining some redundant coset numbers, in fact it is not difficult to construct examples for which the number of redundant cosets exceeds any given bound see, e.g. [33, p.93].

Therefore we have to describe a systematic procedure

for the elimination of redundant coset numbers when coincidences have been found.

When two coset numbers a and b represent the same coset, of course

also the cosets ag and bg are equal for each element g € G.

Taking for g

the generators g. and their inverses, we see that from a comparison of the entries in the places of ag and bg in the coset table we may find deductions and in particular further coincidences, which we have to keep in mind while still dealing with the first one. We do this (in principle) by establishing, as soon as we encounter a coincidence, an equivalence relation between coset numbers, calling two coset numbers equivalent, when they have been shown to represent the same coset.

So this equivalence relation will change with

the progress of the procedure which we now describe. For this description we no longer assume that we have just found a first coincidence but rather that the coset numbers are already sorted into equivalence classes and that we want to eliminate one of a

Neubuser:

Coset table methods

pair of equivalent coset numbers.

10

We choose some pair of coset numbers a

and b, with b >a from some equivalence class and do the following: Cl.

We replace each entry b in all the tables by a.

C2.

For each g e E we compare the entries in the g-column of the coset

table in lines a and b (i.e. we compare the entries in the places of ag and bg) . a)

If the place for bg is still empty, we do nothing.

b)

If the places for ag and bg are both filled and if the entries are

equal or are unequal but in the same equivalence class, we do nothing. c)

If the place for bg contains some entry c, while the place for ag is

empty, we copy c into that place (i.e. we put ag =c as a new deduction). d)

If the place of ag contains the entry c, the place of bg the entry c'

with c and cf not in the same equivalence class then we join the two classes to which c and c' belong. C3.

When we have executed C2 for all g e E we delete the b-th lines in

all tables. Let us discuss the effect of this procedure on the entries of the coset table.

Before a first coincidence is found, the entries in the

coset table satisfy the following property P: (P)

Let a and b be coset numbers.

The coset table lists that a g = b for

some g e E iff it also lists bg"1 = a. At that time of course our equivalence classes consist of single coset numbers and whenever this is so, property ? may evidently be reformulated as (P')

Let A and B be equivalence classes of coset numbers and g G E.

There exist a € A and b ^ B for which the coset table lists ag =b iff there exist a f e A and b ' e B for which the coset table lists b'g"1 = a f . We claim that if we enter our coincidence procedure with an equivalence relation satisfying V1, the new equivalence relation produced by the procedure will also satisfy P f .

This is clear for steps Cl and

C2(c) because a and b are supposed to be in the same equivalence class; it is trivial for steps C2(a) and C2(b) and it is also true for step C2(d), because joining two equivalence classes of a relation satisfying P' in any case produces a relation satisfying V.

Finally step C3 preserves P'

because after step C2 for each non-empty place in line b the corresponding entry of line a is filled with an equivalent entry. Our procedure diminishes the number of coset numbers by one and is applicable whenever there exists an equivalence class consisting of more than one coset number, hence after applying the procedure

Neubuser:

Coset table methods

11

finitely many times we must have arrived again at all equivalence classes consisting of one coset number, i.e. the coset table satisfies property P again and no further coincidence is known. We further observe that another essential property of the coset table is preserved through the coincidence procedure. each coset number £ is defined as £ := kg with k < £, g £ E. time also k was inserted into the place Jig"1 of line £.

Originally At that

If row £ survives

the coincidence procedure, this entry k, if altered, can only have been diminished to some k r < k < £.

Since after the coincidence procedure

property P holds again, we then have £ = k 'g with k' < £, i.e. for each surviving coset number £ there still exists a surviving coset number k' < £ with the entry £ = k'g for some g e E in the coset table, a property which we shall call t. We may possibly have to go through several phases of defining new coset numbers, filling these into the tables and eliminating redundant coset numbers until we reach simultaneous closure of all tables, if we reach this at all.

Rather than illustrating this on further examples, of

which the reader can find several instructive ones e.g. in [22], [33], [38], [43], [57] of which he should do at least a few for himself, we want to settle now the question what we can infer from the simultaneous closure of all our tables. The numbers defined in the process represent cosets of U. Each of these cosets has been multiplied by each generator and its inverse and the cosets obtained by these multiplications have got a number, hence each element of G lies in one of the cosets of U which have been numbered in the process.

Therefore, if we are left with a set K of k coset numbers

at the time of simultaneous closure of all tables, we know that G:U < k. On the other hand since the coset table fulfills property P, i.e. ag. =b iff bg?1 =a, we see that for each of the'k coset numbers an image under each of gi,...,g ,g\l ,...,g~1 has been defined, i.e. to each generator g. a permutation g. of the set K has been assigned. it is clear because of property % that gi ,...,g group on K.

Further

generate a transitive

Thirdly from the relation tables we see that these g. satisfy

the relations of G, hence G acts transitively on K via the homomorphism g. -*• g7, hence G : StabG(l) = k.

The subgroup tables show that U < StabgCl),

whence G:U > k and finally G:U = k.

Theorem 1.

We have thus proved:

When a Todd-Coxeter method terminates^ it determines the index

G:U, moreover it provides its with the permutation representation of G on

Neub'user: Coset table methods

12

the right oosets of U. Let us next deal with a second natural question: in view of the possibility that phases of definition and elimination of coset numbers can alternate, under what conditions can we guarantee that the Todd-Coxeter method will eventually terminate with a simultaneous closure of all tables? In the description of the Todd-Coxeter method above we left undecided in which order we define coset numbers. For instance, in our first example, we might as well have started by defining 2 := IB, 3 := IB !, 4 := 1A, and would have reached simultaneous closure after defining 8 coset numbers as well. In fact we shall see in a moment that the success of the method only depends on a certain natural condition to be fulfilled by the sequence of definitions. Following Mendelsohn [43] we first prove: Lemma 1. If by the sequence of definitions in a Todd-Coxeter procedure it is guaranteed that for each coset number a defined at some stage and not eliminated by coincidences also all cosets ag, g € E, will have got numbers after a finite number of steps then for each k e IN there is a finite number of steps after which the first k rows of all tables (remaining after the preceding coincidences) "become stable"> i.e. are no longer affected by the procedure. To prove this, we first consider the first rows of the various tables. Since all these start with 1 which is defined in the beginning and never changed, the condition implies that after finitely many steps these will all be filled. There is then a highest coset number occurring in them. Entries can be changed only by steps Cl of handling coincidences and by these they can only be diminished. Hence the entries in the first rows of all tables can only be affected a finite number of times by coincidences, i.e. the first lines become stable after a finite number of steps. Assume thus, by induction, that at some stage at which no coincidence is pending, the (remaining) first k-1 rows of all tables have become stable and let i, be the number of the next remaining line. Because of property t there exists an I < i, and g G E with i, = £g, i.e. i^ occurs in one of the stable lines and is hence no longer affected by any further coincidence. Therefore we may now argue for the i, -th lines K

as we did before for the first. From this lemma we have:

Neubliser:

Theorem 2.

Coset table methods

13

If the index G:U ts finite, any Todd-Coxeter procedure for the

enumeration of the cosets of k that satisfies the condition of Lemma 1 wiVL terminate with simultaneous closure of aVl tables after a finite number of steps* For the proof assume that the procedure would not stop.

Then

by the lemma, each remaining row will eventually become stable and hence the number of such rows would grow beyond any given bound.

The same

argument as given in the proof of Theorem 1 now shows that we obtain a transitive permutation representation of G on a countably infinite set and that U is contained in the stabilizer of 1 in this permutation representation, which contradicts the assumed finiteness of G:U.

A few remarks about Theorem 2 are appropriate.

The result is

satisfactory from a theoretical point of view, it is of little help for practically performing Todd-Coxeter procedures because it does not give any bound for the number of steps necessary to reach closure in terms of the input data (say the number and lengths of relators and subgroup generators) and a hypothetical index.

In fact such an a priori bound

cannot be given at all because it would turn the Todd-Coxeter method into an algorithm for deciding e.g. in a finite number of steps if a finite presentation represents the unit group, but the non-existence of such an algorithm has been proved [50]. The condition of Lemma 1 is the one used (in slightly disguised form) by Mendelsohn [43], examples given there and by J.N. Ward [59] demonstrate that some condition of this kind is needed. The Todd-Coxeter procedure, that condition and the proofs have been formulated here with the intention of showing that a great deal of variation of the procedure is possible. which the reader should be aware.

There is, however, one point of

In our description of the coincidence

routine, we made things easy for the proof by introducing condition P'. In actual implementations a more sophisticated routine is followed that allows condition P to be maintained even when coincidences are pending, the practical advantage of this formal complication being that the validity of condition P at any given time allows one to read off from the j-th row all occurrences of j elsewhere in the coset table.

Formal

descriptions of such routines can be found in e.g. [18], [24], [37], [38].

Neubiiser:

2.

Coset table methods

14

SOME ASPECTS FOR THE IMPLEMENTATION Discussing at least some features of implementations, the

first observation is that really all information is kept in the coset table (and a listing of equivalence classes while coincidences are pending).

Therefore, and since experience has shown that in all

implementations of the Todd-Coxeter method storage space rather than computing time tends to be the bottleneck, in all implementations known to me subgroup and relation tables are never physically stored but their rows are reconstructed using the entries in the coset table in order to find new deductions or coincidences.

This process is called scanning.

The most important difference between various implementations is in the sequence in which definitions are made. In a first method the vacant places in the coset table are filled line by line.

In order not to miss any deduction, after each

definition a scan of all relations is made.

Since new information can

only be hoped for from this new definition, the scan is started at all significantly different places of all relations, into which this definition fits, regarding relators as being cyclically permuted accordingly.

Of course deductions discovered in such a scan have to be

kept and treated analogously.

Leech reports in [35] that this method was

first programmed by Bandler in 1956 [2], still without a coincidence routine.

It has been called the Felsch method e.g. in [18] because the

first description of a full program following this method was published by H. Felsch [24].

In many cases this method gets away with comparatively

few redundant definitions, but the many scans tend to be time-consuming. A second method, according to J. Leech [35] first programmed by C.B. Haselgrove as early as 1953, follows the strategy to define coset numbers with the primary aim to close at least one line of some relation table in order to get at least one deduction as soon as possible. Cannon [18] calls it the HLT method, because after Haselgrove it has been developed by in particular Leech [35] and Trotter [58]. The HLT method tends to define more redundant cosets than the Felsch method, but wastes much less time in fruitless scans. A method that to some extent combines the merits of the two previous ones is the so-called "lookahead" method in which periods of definition a la HLT alternate with periods of intensive scan a la Felsch. In modern implementations, e.g. the one by G. Havas and W.A. Alford in Canberra this alternation can be steered by input parameters and/or interactive handling.

With their new program Havas and Alford have also

Neubiiser:

Coset table methods

15

experimented with the strategy of defining cosets in such a way as to systematically close "minimal gaps" in the relation tables.

The

interesting results are at the time of writing only available through private communication. Just a few words must suffice here about the data structures in implementations.

The storing of the entries of the coset table is

straightforward, problems arise only through coincidences.

While in the

early implementations "deleted" rows were recovered for new use by physically pushing the later rows forward and actually renumbering cosets so that whenever no coincidences are pending the remaining cosets had consecutive numbers l,...,k, it is more common now to use linked lists. The usually two extra columns for the links can also be used to store the equivalence relation by link-lists.

A description of this technique

as well as a formal algorithm description is given in [18]; on the other hand, developing a suggestion of I.D. Macdonald, M.J. Beetham has designed a method to save the space for these extra columns (private communication). His idea has entered, for example, the above mentioned program of Havas and Alford. In [18] an extensive experimental study of the performance of "Felsch", "HLT" and "Lookahead" method is described, further comparisons by Cannon and Havas including "minimal gap" strategies are still unpublished.

Although these experiments support certain rules of thumb

that also look plausible when one envisages the different strategies, they also produce exceptions to practically all rules.

They certainly

show that the efficiency of all implementations may be substantially influenced by such subtle changes as cyclically permuting relators, changing the order in which the different relators are scanned or adding redundant relators or subgroup generators.

Observations of this kind of

course weaken one's hope for the development of a theory that would allow practically useful predictions with regard to efficiency or suggestions for an optimal choice of strategy and preparation of the input. There is, however, an interesting description of a (partially filled) coset table as a mathematical structure in its own right, which bears some analogy to automata theory.

Steps of a Todd-Coxeter procedure

are described as transformations from one table to another.

This idea has

been outlined by Sims in several talks e.g. [55], [56], a formal description, contained in [40], is used there to formalize both the description of the procedure and the proofs given above in a less formal way.

Neubiiser:

3.

Coset table methods

16

INFORMATION OBTAINABLE FROM A COSET TABLE In Theorem 1 we have stated that the columns of a complete

coset table form the images of the generators gi,...,g permutation representation of G on the cosets of U.

of G in the

The permutation

group generated by them, which is isomorphic to G/ n u g , can be g6G investigated using Sims' method which will be sketched in §7.

However,

there are several things that can be read even more directly from a complete coset table.

The basic operation for this is to determine the

product of a coset with coset number k by a word g. ...g.

with g.

€ E

by successively looking up the coset numbers of kg. , (kg. )g. ,... . 1

i

x

i

1

2

This is called "tracing" the word from coset k through the coset table. We just mention four possibilities of using it [16], [19]: A coset Ug is contained in the normalizer N G (U) iff g" 1 Ug=U, i.e.

(i) -1

iff g Ug < U and gUg"1 < U. 1

Ug" s^ = Ug"

1

These two conditions are satisfied iff

and Ugs. = Ug, respectively, for all elements s. in a

generating set S = {si ,...,s } of U. the two conditions suffices.)

(Of course, if G is finite, one of

However, a generating set S of U is given

as words in the generators of G and so this test can be performed by tracing these words from the coset numbers of Ug"1 and Ug, respectively, through the coset table. (ii)

In particular it is possible to test, if U < G by applying the

previous test with g running through the given generating system {gi,...,gn> of G. (iii)

If an element h £ G is given by a word in the generators {gi,...,g }

of G, the number k of the coset Uh is found by tracing the word from coset number 1 through the coset table.

Restarting the Todd-Coxeter procedure

with the additional coincidence k =1 will shrink the coset table of G over U to the coset table of G over < U , h > . (iv)

In particular one obtains the coset table of G over the normal

closure of U by applying the previous procedure in turn to all elements h = gjSi^j1 ^ t h g. e E, s± s S. Some of these ideas are useful, even if a Todd-Coxeter procedure has not successfully been finished and one is left with an incomplete coset table.

For instance one can still try to find normalizing

cosets using the first test.

If it is still successful for all subgroup

generators s., a normalizing coset has been found.

Of course with an

incomplete table the tracing of a word may run into a dead end, because

Neubiiser:

some product kg. 1

Coset table methods

17

has not got a coset number yet and then no conclusion

5

can be drawn.

Note, however, that no conclusion can be drawn either, if

for some s. £ S the coset numbers of Ug and Ugs. are found to be different, because a hitherto undiscovered coincidence might still equate them.

Such

an attempt to find normalizing cosets can in particular be useful if one is running out of storage space and hopes to find at least the index of a bigger subgroup using the method indicated under (iii) .

4.

FIRST VARIATION ON THE THEME OF TOW PRESENTATIONS FOR A SUBGROUP

AND COXETER:

For this section we recall some basic facts about presentations of a subgroup U of finite index in a group G with the finite presentation G = <E|R>.

Let T be a transversal of U in G and let the coset

representative function x -* x of G onto T be defined by Ux =Ux, then Schreier's theorem [41, p.88], [33, p.13] states that U is generated by the set of all elements s e

G

i

k

product u = g. ...g. 1

t

€ U with g.

= g

Moreover a

G E can be written as

x

\

u = s^

= tgtg"1 with t <= T and g € E.

e k ...s.

...g

J

, where

if e. =1,

t

= g

...g

J

if e. =-1.

This process, known as Reidemeister rewriting [41, p.90], can also be understood as a transformation T of the word u = g. ...g. e

G

i

TU = s

k

...s

into the word

'

and we use this interpretation in the formulation z

t, ,g.

\r>*i

1

of Reidemeister s theorem.

Theorem (Reidemeister [41, p.94], [33, p.107]).

With assumptions and

notations as defined above the subgroup U has the presentation

U = < ^gkttr.t- 1 ) = 1, s ^ g = T(tgti"1)> where t runs through T, g runs through E and r. through R and where tgtg"

Neubiiser:

Coset table methods

18

has first to be written as a word in g. € E in order to he transformed 1 3

by T. Ifj in addition^ the transversal T is a Schreier transversal3 the presentation for U simplifies to U=< s

| x ( t r f 1 ) = 1, s

= 1 i f tg = tg >.

The special form of Reidemeister's theorem is used in a program of Havas [27] to construct a presentation of U in terms of Schreier generators from a completed coset table of G modulo U.

Because

property % holds for the coset numbers, we may recursively define a Schreier transversal T of U starting with the unit element for U itself and using, for example, the first occurrence (in a linewise scan of the coset table) of a coset number k as k = ig with an i for which t. has already been defined for the definition of its representative t, . Any entry ig = k, with i,k coset numbers and g € E will correspond to a Schreier generator, which can hence be characterized by the pair (i,g). Those which are equated to 1 in the Reidemeister presentation correspond to pairs which have been used in the definition of t. or t . r l k

Since the

coset table allows us to determine recursively the number of the coset to which any given product of the generators of G belongs, we may now apply Reidemeister rewriting to all trt"1 with t € T, r e R and obtain a presentation of U in terms of Schreier generators.

Of course the sets of

both generators and relators obtained in this way increase with the index of U in G and will in general be highly redundant.

Havas1 Reidemeister-

Schreier program is therefore equipped with means of sorting out dependent relators and of eliminating generators by Tietze transformations.

Also it

is possible (and has proved useful in applications [30]) to abelianize the presentation obtained for U and get the structure of U/U' using an abelian decomposition program as described e.g. in [31]. The large number of Schreier generators entering Reidemeister1s theorem in most cases compares unfavourably with the usually few given s

-(gi>««»>g ) e

s

that suffice to generate U.

It is therefore desirable

to get a presentation of U in terms of generators s. corresponding to these.

Methods for reaching this goal seem to have developed independently

in several places [4], [5], [10], [33], [34], [36], [38], [42], [43], [44].

While not following any of these exactly, I shall try to bring out

their common background. In Havas1 Reidemeister-Schreier program, coset representatives were defined a posteriori^ i.e. after the Todd-Coxeter procedure had

Neubiiser:

finished.

Coset table methods

19

We may as well modify the Todd-Coxeter method by considering

the numbers used in the procedure as representing coset representatives from the beginning.

The fact that each new coset representative k is

defined by an equation k = £g with I < k and g G E then means that we are defining coset representatives which satisfy the Schreier property [41, p.93], [33, p.80].

In fact if we succeed in finishing the Todd-Coxeter

procedure without any coincidence, we have in the end obtained a Schreier system of coset representatives of U. Let us consider next what happens to deductions and coincidences when our procedure is performed with coset representatives instead of cosets.

There is one essential change:

if s = s(g, ... ,g ) is

an element of U we can no longer conclude that Is = 1 but only that Is = si.

Hence at the end of the one row of a subgroup table we have to

insert si instead of 1.

Assuming that this row is the first row of a

subgroup table to close, it will therefore yield a deduction of the form kg.

= s£ which may then get inserted into the coset table, other subgroup

tables and the relation tables which therefore when their rows close, will produce deductions that may involve products of the given subgroup generators as left hand factors.

Let us demonstrate this effect in Example

1, i.e. for the group with the presentation < A,B | BABA"*2 =1, ABAB"2 =1 > and the subgroup U = < A2 >.

Let us denote A2 by h, then our subgroup

table is started as

hi From the first definition 1A = 2 we now get the deduction 2A = hi and hence LA"1 = h - 1 2 .

Inserting this, e.g. into our first relation table, we see

that the factor h is dragged into further entries of the table.

Following

exactly the same rules and the same sequence of definitions as before, we get as our 16-th deduction h28B"1 = 7. follows:

hi

At that time our tables look as

Neubiiser:

1

3

2

5

2

Coset table methods

h4

hi

2

1

h6

h2

hi

2

2

\ 7

3

h6

4

1

2

5

h24

h28

9

1

20

2

2

hi

7

IT'S

h^S

8

7

12

3

3

1

5

2

5

2

hi

h3

h24 3

h4

3

3

h4

hi

h2

h6

3

5

8

4

4

8

7

3

1

^4

h6

5

5

h6

h2

h2l

h24

5

5

2

6

hZ7

2

7

10

6

h6

h8

h27 1J

5

14

h3

h7

h4

3

h6

5

6 8

6

h7

7

8 15

7

7

8

3

h6

h8

8

I6g

13

A

B

A"1

B-1

1

CM|

3

4

2

hi

S

IT1 2 1

3

h4

h6

7

1 h" 2 5

6

00|

1

5

h6

h4

IT1 3 8

6

h7

2

IT1 5

h- J 3

7

3

8

5

7

h" ! 6 4

h7

4

2 8

Since the coset table holds the entry 8B"1 = h7 and since now the numbers are coset representatives, i.e. group elements, we conclude h3 = 1 . The same result also follows if we now fill in the last row of the second relation table. Although coincidences do not occur in our example it is clear that they also will in general take the form b =sa with b and a coset representative numbers and s some word in the generators of U.

Therefore,

after coincidences have been processed the remaining coset representatives need not form a Schreier system any longer but only satisfy equations of the form k = slg where I < k, g € E and s is a word in the given subgroup generators again. We have seen that in the working of the modified Todd-Coxeter procedure we have already picked up relations between the given subgroup generators s.. We shall now show that in fact the "augmented" coset table allows us to determine a set of defining relations for U in terms of the s.. For each coset representative I and each g € E, the augmented table provides an equation

Neubiiser:

tg = v

Coset table methods

21

(s.)k X* »&

(l)

J

where k is also a coset representative and v generators s. of U.

(s.) is a word in the * >& 3 (Note that we are nowhere using relations between the

s. in the construction of the augmented table, so the v (s.) as well as 3 ^>S 3 the words derived from them are determined up to free equivalence by the modified Todd-Coxeter procedure.)

By repeated use of such equations we

obtain for each coset representative I and each word w(g.) an equation = v £ w(s.)k.

(2)

We see that the augmented table provides us with a "rewriting rule" for word w(g.), depending on the chosen coset representative.

Moreover, since

by the construction of the augmented table equation (1) is equivalent to

we see that likewise (2) is equivalent to

i '

- Vk,w-' ^

With V

k,w-> ^

= \,^SP"

•

f

Applying this rewrite rule to a relator r (g.) of G, we obtain for each coset representative I

and since r (g.) = 1 in G, we have for each I a relation

(6) for the subgroup generators s..

Further, applying the rewrite rule to a

word s, (g.) which expresses the generator s. of U in terms of the g., we obtain for the coset representative 1

Since, on the other hand, Is, = s, 1, we have the further relation

Neubiiser: Coset table methods

22

for the generators s. of U. We now claim: Theorem 3. The relations v

*>r x

(s ) = 1; £ = 1,...,G:U; x = l,...,m; s = v (s ) ; h = l,...,p 3 n i,s h 3

(which have been seen to hold in U see (6), {!)) are a set of defining relations of U w££/z respect to the generating set S = {st ,... ,s } o / U . For the proof, which develops the basic idea of McLainfs paper [42], let V be the group defined by the presentation V = < al ,... ,a I v (a.) = 1; I = 1,...,G:U; x = l,...,m; P *,r x 3 a

h

= V

l,s h ( a h ) ; h = 1,...,P>.

Since the defining relations of V in terms of the a. are satisfied by the s., the mapping a. -> s. defines an epimorphism <{> of V onto U. We are now going to define a mapping ty from U into V, which will turn out to be an epimorphism that maps s. to a., thus showing that U and V are isomorphic. To define ij/, let u be some element of U. Then u can be expressed as some word u(g.) in the generators g. of G. Using the augmented coset table we can rewrite u(g.), that is, find u = v (s.). We now define ty : u + v

(a.) e V.

Since the expression of u as word u(g.) is by no means unique, we have to show that i[> is well-defined by this definition. Now, if u(g.) = u'(g^) in G, we have u(g.)u f (g.)" 1 = 1, i . e . the word ufg.Ju'fg.)" 1 is a relator of G with respect to the generators g. , hence a finite product of conjugates of the defining relators r (g.)> x = l , . . . , m . So, modulo an induction over the number of conjugates of defining relators involved, we may assume that

where r (g.) is one of the defining relators and w(g.) is some word in the generators. We now apply the rewriting rule to both sides, starting from coset representative 1. Let lw(g.) = v. (s.) k, hence because of (4) we i

also have kwfg.)" 1 = v i

•*• > w

3

Cs.)"!l, hence the right hand side yields

-i >w

3

Neubliser:

Coset table methods

23

lw(g.) rx(g.) w(gi)-'u'(g.) = v 1 ) W C s . ) v k > r ^ s . ) v 1 ) W ( s . ) - ' v 1 > u f ( s . ) ,

so we see that the results of the rewriting differ by a conjugate of the relator v,

(s.) and since this is a relator also in V, the words in the

k>r

3

x

a obtained by the definition of ij; represent the same element of V, i.e. ip is well-defined.

(Note that so far we have used only the relations (6)!)

Now we apply i|> to a generator s, , using its given expression s, (g.) in terms of the g. . Through the rewriting rule we obtain v. s (s.)> n 1 l * h ^ hence iKs.) = v.

(a.)* hence because of the relation a, = v.

V we have ip(s, ) = a, .

(a.) in

In the same way, if u is any element in U we obtain

ip(u s.) = i|j(u) a, and hence finally ty(u u') = i|/(u)ij>(uf), which proves Theorem 3.

We first relate Theorem 3 to Reidemeister's theorem.

Let

g.) be an expression of the coset representative I in terms of the generators g±> then l£(gi) = v x £(s.)£ and £ £(g i )" 1 = Vj £ ( s . ) " ! , hence for a defining relator r (g.) we have

i.e. v. 1* of U.

x

_i(s.) and v J

(s.) are conjugate, hence equivalent relators x ^

Restricting now the rewriting rule used above to the coset

representative 1 and to w(g.) e U and denoting v (s.) by T^CW) we 1 1, W J o therefore see that Theorem 3 is equivalent to:

Theorem 3'.

With the above definitions and notations the relations

T

S(Ugi)rx(giU(girl)

= l

(6f)

with x = l,...,m and £(g.) running through some system of coset representatives and

h

S

h

i

with h = 1,...,p are a set of defining relations of U with respect to the generating system {si,...,s }.

Neubiiser:

Coset table methods

24

This formulation brings to light a formal analogy with Reidemeister's theorem.

In fact Reidemeister1s theorem is a special case

f

of Theorem 3 , which is seen as follows:

if g €= E, quite generally

(s.) is the Schreier generator s expressed in terms of the s.. £>g 3 x* >% 3 Moreover, for a coset representative k and a generator g we have

v

kg~ ! g = k

so

(kg -1 g k g ^ g - 1 ) " 1 = kg"1 kg"1"1 .

Therefore again generally v

-i(s.) is the inverse of the * *& 3 Schreier generator s_ expressed in terms of the s.. From these r Tcg^S 3 remarks we see that the Reidemeister rewriting is nothing but the special case of our general rewriting rule with the Schreier generators taken as generators s.. One may also think of using the connection just outlined to prove Theorem 3 f (and hence Theorem 3) from Reidemeister's theorem using Tietze transformations. Quite generally if a group has a presentation < x. | w.(x.) =1 >, a presentation in terms of another generating system (y^.) is given by < y k |w^.(xi(yk)) = 1, y k = y k C x i ( y k ) ) > *

where x

i^

are words

expressing

the x. in terms of the y, and y, (x.) are words expressing the y, in terms 1

K

K

X

K

of the x. [33, p.39].

Let < x. | w.(x.) = 1 > be the Reidemeister

presentation of U: U =< s

I T(tr t"1) = 1, s

= x(tgti- 1 )>

then the above considerations show that the relations (6') in Theorem 3' are obtained from the relations x(tr t"1) = 1 by inserting s x

t,g

=v

(s.)

t,g j

while the relations (7') are nothing but the "extra" relations brought in by the Tietze transformation, i.e. obtained by inserting s = v (s.) into s. = x(s.(g.)). It is interesting to note that because of Theorem 3' the relations obtained by inserting s =v (s.) into the Reidemeister *>g t,g 3 1 relations s = xftgtg" ) must be consequences of the others, but I did t >g not see an easy direct proof of this fact. After this digression into theory let us return to the proposals made for computing a presentation in terms of "given" subgroup generators.

Benson and Mendelsohn [5], Campbell [10], W. Moser, described

the construction of the augmented tables.

In [44] Mendelsohn proposed to

Neubiiser:

Coset table methods

25

do two independent modified coset enumerations in order to find a presentation of U in "given" generators s..

McLain [42] proves a somewhat

weaker version of Theorem 3 by changing from the presentation < g- I ro(&-)

=

1 > of G with subgroup U = < s.(g.) > to the more complicated

presentation < g . , s . |r(g.) = 1, s. = s.(g.)> for G then showing that 1 J 36 1 J J 1 the relations of type (6) obtained from this suffice to define U.

His

relations always contain the relations (6) and (7) of Theorem 3, but in general, i.e. if the s. are not originally a subset of the g., are much more numerous.

Beetham and Campbell [ 4 ] show that if in the course of

the generation of the augmented table all opportunities for picking up relations both in deductions, as in the example above, and in the treatment of coincidences are used systematically, one obtains a presentation of U. Even without going through their formal proof we see that this is highly convincing in view of Theorem 3 since all information contained in the augmented coset table comes from closures of rows of subgroup and relation tables and they are collecting the reasons why this happens.

In [38]

Leech describes essentially the same method as Beetham and Campbell for finding a complete presentation of U, in his previous papers [34], [36] he describes a method for proving single relations between the generators s. of the subgroup that amounts in essence to a backward trace and aimed reconstruction of those parts of the augmented coset table that are involved in the proof starting from a normal coset table and a record of its generation.

Because the construction of the full augmented table

needs much more effort both in storage space and time, his earlier proposal is in particular interesting in case e.g. only the verification of a few given relations between the s. is requested.

Interesting applications are

given by Leech himself in the quoted papers and by Havas [28], [32]. Finally, Johnson [33] describes what may be called a mixture of the Beetham-Campbe11 method and that following from Theorem 3; however, the proof given by him requires that the chosen coset representatives form a Schreier system, which is not guaranteed when coincidences are involved. To insure the validity of his proof it is necessary, when a coincidence occurs, to redefine the coset representatives involved in a suitable way. A main practical problem in working on the computer with the modified table is the storing of the words v

(s.) whose length is i J In these proceedings Arrell, Manrai and Worboys [l] >g

unpredictable.

discuss the use of tree structures for a recursive storing of these words.

Neubuser:

5.

Coset table methods

26

SECOND VARIATION ON THE THEME OF TOW PRESENTATIONS FOR A CONCRETE GROUP

AND COXETER:

In all previous sections we assumed that we want to investigate a group given by a presentation.

However, Todd and Coxeter really

started from a counterpoint to this theme:

in a "concrete" group, say a

permutation or matrix group, it is easy to write down some relations for the elements of a generating set, e.g. those giving the orders of the generators, products of generators etc., but in general it is not at all easy to prove that a set of relations is a defining set.

Now any set R

of relations holding for a generating set {gi,...,g } of G defines a group G* of which G is a homomorphic image.

If by the Todd-Coxeter method one

can show that G* has the same order as G, then G* and G are isomorphic and R is a defining set of relations for G.

This technique has been frequently

used, in particular by Leech [35]. The task of finding a set of relations with a good chance of being defining relations is left to intuition with this approach.

A

systematic method for constructing a defining, though highly redundant, set of relations and for reducing the amount of Todd-Coxeter computations necessary to test if subsets are defining was described and applied by Grover, Rowe and Wilson [26]. Finally Cannon [17] developed an algorithm that uses already Todd-Coxeter-like methods for the construction of a defining set of relations which, moreover, is close to being irredundant. We describe essentially his method, however using a different terminology. Throughout let a group G be given by a set of "concrete" generators, the only prerequisite being that inverses and products of elements can be formed and elements can be compared. Let us first describe what Cannon calls the "one-stage" algorithm.

For this we assume that the order of G is small enough for

keeping the coset table and usually also a list of all elements in store, say |G| not much bigger than 10 4 .

With reasonable effort this list is

computed from the generators and from it the coset table of G with respect to the unit subgroup by just multiplying each element by each generator and looking up the result (using hash coding if necessary) in the list of all elements.

Next we define inductively from the coset table for each element

h of G a representation by a word w, (g.) in the generators g.. Havas

1

As in

Schreier-Reidemeister program we begin by assigning the empty word

to the identity, find for an element h its first occurrence (in a linewise scan of the coset table) as a product h = kg of some element k to which a word wj,(g«) has already been assigned and some g £ E, and define

Neubuser:

w, (g.) := w, (g.)g.

Coset table methods

27

At the same time the entries h = kg in the coset

table that have been used for the definition of these words and the corresponding entries k = hg"1 are marked (e.g. by replacing them by their negative value). Having defined the words w, (g.) for all h e G, we see that each entry in the coset table giving the information Jig = j yields a relation w (g.)gwT1 (g.) = 1 in terms of the generators g. which may or may not be trivial.

We start the relation finding algorithm by looking up the

first unmarked entry in the coset table.

We mark it and if it yields a

nontrivial relation, say rj(g.) = 1, we set up a relation table for ri and insert into it all possible marked entries from the coset table.

If

a row of the relation table closes, yielding a "deduction" ag = b, then the entry in the a-th line, g-th column of the coset table must be b since we deal with the coset table of a concretely known group; if this entry is not already marked, we now mark it and the corresponding one bg"1 = a. When we have thus passed through all rows of the relation table of xx we search for further unmarked entries in the coset table; if there are any we treat the first we encounter as before, getting an additional relation and now at each step filling all marked entries into all relation tables that have been set up so far. The process comes to an end, when all entries in the coset table have been marked and at that time also all relation tables will have closed.

We claim that the relations, say rj (g.) = l,...,r (g.) = 1 found

to hold for the g. in the process form a defining set of relations for G. To see this, let G* be the group defined by the presentation G* = .

Clearly g* •*- g. defines an epimorphism of G* onto G.

Consider now that a

Todd-Coxeter procedure is started for G* modulo the unit subgroup {1}. As we have discussed in the first section, the success of a Todd-Coxeter procedure does not depend on some finite initial sequence of definitions and preliminary postponement of taking account of deductions.

Hence we

may start the Todd-Coxeter procedure for G* with a sequence of definitions that correspond exactly to the definitions we have chosen for the elements of G as words in the g..

Since all further entries in the coset table of

G got marked because of deductions from relations also present in the presentation of G*, we see that the coset table and the relation tables of G* will have closed at the latest after the definition of |G| "cosets"

Neubiiser:

Coset table methods

28

of G* over {1}, hence |G*| < \G\ and hence G* and G are isomorphic and the relations gathered in the process define G in terms of the g. . The application of this one-stage algorithm is limited by the order of G.

However if generators are given for a group G that is too big

for this algorithm, we may try to find certain elements hj,...,h

in G,

expressed as words hj(g.),...,h (g.) in the generators g., that generate a subgroup H, small enough for the one-stage algorithm. supply a presentation < ht ,. .. ,h

This will then

| r^fh.) = 1,...,r'(h.) = 1 > of H in

terms of the h. and for each h G H a word w, (h.). h 3 3

Since each h. in turn 3

was supposed to be expressed in terms of the g., we also have available for each h € H a word vj-(g-) expressing h in terms of the g.. Let us further suppose that we can still construct the coset table of G modulo H.

Then by the following "two-stage" algorithm we can

again find a presentation of G.

Following the same rule as in the one-

stage algorithm, we start by defining coset representatives for all cosets I as words t (g.) and marking the entries in the coset table that are 36

1

used in these definitions.

Since the numbers occuring in the coset table

now represent cosets of H, an equation Ig = k read from an entry k in the Jl-th row and g-column of the coset table now only yields G may evaluate t1 (g.)gt,(g.) obtain element elements, h of H which t (g.)gt, (g.)" H. However,and since the ag.concrete are "concrete" we in X>

1

K

Jo

1

1

K

1

1

^ -1 v, (g.) by the information we had obtained turn is expressed as a word

earlier about H.

Hence from Ig = k we get the relation

r(gi) = Vg^gt^r'v^g.)- = 1 which again may or may not be trivial. With the modification that we use these more complicated relations instead of the easier ones in the one-stage algorithm we now proceed as in the latter one until all entries in the coset table have been marked.

Let r (g.) = l,...,r (g.) = 1 be the relations gathered in

the process.

We then claim that the relations ^'(h.) = l,...,r'(h.) = 1,

*i ( g ^ = 1 > - - ' » r m ^ i ) = 1> \

= h1(gi),...,h

= h (gi) define G in terms

of the set {gj ,... ,g ,h1 ,... ,h } of generators of G. To see this we again compare our concrete group G with a group G* with the presentation G* = < g*,...,g*,h*,...,h* |r/(h*) = l,...,rj(h*) = 1, rf (g*) = 1,..., rn(gj) - 1, h f . h 1 ( g j ) , . . . . h ; - h p ( g * ) > .

Neubiiser:

Coset table methods

29

As the defining relations of G* hold for the corresponding elements of G it is again clear that g* -* g., h* -»• h. defines an epimorphism of G* onto G.

Moreover G* has a subgroup H* = < h*,...,h* > whose generators satisfy

in particular the relations r/(h*) = l,...,r'(h*) = 1, which are defining relations of H in terms of h,,...,h . Hence h. •* h* defines an epimorphism of H onto H* so that |H*| < |H|. We now again consider a Todd-Coxeter process being applied to G* modulo H*,, using the relation tables for all relations given and the subgroup tables for hj(g*),...,h (g*). Using again the freedom we have for designing a special Todd-Coxeter procedure, we start with a sequence of definitions that correspond exactly to the definition of the coset representatives t^(g*).

By the same argument as

with the one-stage algorithm we see that the relations ^ (g*) =l,...,r (g£) = 1 will suffice to produce enough deductions to fill all g*-columns (g* G E*) of the coset table (note that our supposed coset table for G* modulo H* has also columns for h*,...,h* and their inverses), the relation tables of the r , the subgroup tables of the h.(g?) and, starting from the x j l right and, using the g*-columns of the coset table only, also all entries of the relation tables of the relations hth.Cg*)"1 = 1.

Each row of these

will then yield a deduction of the form kh* = £, so that we get all entries of the h-columns of the coset table and these allow us to fill all entries of the relation tables of the relations ^'(h*) = l,...,r'(h*) = 1.

Since

at worst we could encounter coincidences in this process we conclude that G*:H* < G:H and so finally |G*| < |G|.

Therefore G* and G are isomorphic

and our claim is proved. A few comments are appropriate. (i)

The presentation obtained for G can be simplified by eliminating

the redundant generators h,,...,h using h. = h.(g.).

by a sequence of Tietze transformations

Doing this we arrive at the presentation

G =< gl ,...,g n I r . C g . W , . . . , ^ . ) ^ , r/(h.(g.)) = l,...,r(j(hj(gi))=l>. (ii)

In the description of the two-stage algorithm we have made no

explicit use of the fact that the presentation of H had been obtained by a one-stage algorithm.

Really for the two-stage algorithm only the

following had to be known (from whatever source): (a)

a concrete generating set {gi,...,gn> of G;

(b)

generators hj,...,h

of a subgroup H of G expressed as words

h. = h.(g.) in terms of the generators g. of G;

Neubiiser:

Coset table methods

30

(c)

a presentation of H in terms of the h.;

(d)

a possibility of expressing a concretely given element h e H in terms

of the generators g. £ G; (e)

a coset table of G modulo H (for the generating set {gj ,...,g }). This consideration opens up the possibility of a "multistep"

algorithm, working up a chain of subgroups 1 < Ht < ... < H

= G.

As we

have seen the output of the i-th step provides most of the data needed for the i+lst step, the main problem remaining is the calculation of the coset table of a fairly big subgroup in a still bigger group, which already occurred with the two-stage algorithm. (iii)

In the calculation of the coset table the critical point is to

identify the coset to which a concrete element belongs which has been obtained e.g. as a product.

If the elements of H and coset representatives

of H can be stored, then this can be done by choosing a "canonical" coset representative which can be computed with not too much effort from any member of its coset.

For arbitrary groups of order not much bigger than

106 Cannon [17] suggests defining a "canonical" coset representative via the internal representation of the group elements as bit strings in the machine, while for permutation groups (also bigger ones) he points out how Sims1 techniques can be used for this purpose.

We shall come back to

that point in the last section where we shall discuss briefly the interplay of these with coset table methods. (iv)

In his paper [17] Cannon describes his algorithm (as well as the

original Todd-Coxeter method in [18]) using the Cayley graph and Schreier coset graphs of G.

In this paper he states that the two-stage algorithm

need not always return correct presentations of G and hence its results have to be checked by an extra Todd-Coxeter run.

He also lists examples

where an implementation of the algorithm actually failed.

I think,

however, that the description given here is a straight translation of his graph theoretical one and hence his algorithm does not need any disclaimer. Moreover the implementations of it that are contained in the CAYLEY system at least since 1976 do not reproduce the alleged failures.

6.

THIRD VARIATION ON THE THEME OF TOW ALL SUBGROUPS OF LOW INDEX

AND COXETER:

While in the previous sections, whenever a subgroup was involved, it was specified by the user, now we ask to find all subgroups of index up to a given bound in a finitely presented group G = < gi ,.. .,g n |r t (g i ) = l,...,rm(gi) = 1 >.

We have discussed in §1 how

Neubiiser:

Coset table methods

31

each subgroup U of index z in G determines a complete coset table of G with z rows, in which U is given the number 1.

Such a complete coset

table, i.e. a table with z rows, and with columns gj,...,g , gj" 1 ,...^" 1 corresponding to gj,...,g , g^^.^jg" 1 * has three characteristic properties: 1.

Its columns are permutations of the set K of the z remaining coset

numbers, and g~l = g7 2.

.

The columns gt,...,g

generate a transitive subgroup of the symmetric

group Sj. on K. 3.

The columns gt»...,g

satisfy the defining relations of G.

In other words, g. ->- g7 determines an epimorphism of G onto a transitive subgroup (f of the symmetric group S«, and U is the stabilizer of 1 in that permutation representation:

U = {g|g € G, l g = 1}.

On the other hand, we can assign to a complete coset table of G, i.e. a table with z rows and with columns corresponding to gi,...,gn>gTl,...,g-1

and satisfying properties 1,2, and 3 (wherever this

table is obtained from), the subgroup U = {g|g € G, lg = 1}.

As we have

seen at the beginning of §4, we can, for example, read off from the coset table the Schreier generators of U as words in the g.. While these remarks make clear that it is possible to find all subgroups of index z in G by constructing all coset tables of G with z rows which is clearly a finite task, cf. [55], for example, one has to remember that the correspondence between coset tables and subgroups is far from being bijective, in fact each non-trivial permutation of i2,...,i applied to the rows i2,...,i will not change U.

of a coset table as well as to its entries

So the aim must be to design a method which will

produce each subgroup U of index z just once.

On the other hand such a

method will often have to produce several subgroups of the same index, hence the process will have to branch off to different alternatives at certain points.

We shall first describe the principle of such a method

due to C.C. Sims and come to its variations and history later. Under §3 (iii) we have seen how the user can utilize the techniques for handling a coincidence in order to join cosets of a smaller subgroup U to build up the coset table of a bigger subgroup < U,h >.

In

Sims' method coset tables for the different subgroups of a given index in G are constructed by going through a whole tree of such "forced coincidences".

Neubliser:

Coset table methods

32

Let us then assume that we want to construct just one coset table for each subgroup of index < b in G.

For the start we provide an

empty coset table with columns corresponding to gt ,- .. ,g ,gr* ,... ,g~* . The process then is started using some Todd-Coxeter strategy enumerating the cosets of 1, but is allowed to work with at most a prescribed number f(b) of coset numbers at a time, where f(b) > b (and usually between b+1 and 2b). Now either this Todd-Coxeter run will find that G is of some order < f(b) or the process will come to a halt being left with f(b) coset numbers without all remaining f(b) lines of the coset table having closed.

That this must eventually happen follows from Lemma 1 in §1,

which implies that a Todd-rCoxeter procedure cannot go on indefinitely with a limited number of remaining lines.

In the first case, if G is of order

< b, we print its coset table modulo the unit subgroup as the first of the tables for which we are searching. If G is found to be of order 1, of course we finish.

Other-

wise we have reached a branch point B° to which we assign the level 0.

If

the table has not closed (with f(b) > b remaining coset numbers), clearly for any fixed subgroup U of index < b at least two of the coset numbers must denote the same coset of the subgroup U, hence if we try through all possible forced coincidences in turn, at least one of them must be correct for the cosets of U.

If G has been found to be of some order < f(b), the

same argument holds for any fixed subgroup U < G with {1} ^ U. Now the rationale behind the method for finding each subgroup U of index < b only once is as follows.

An entry in the coset table really

means the statement that a certain word in the generators g!,...,g

(which

is defined as described in §4 by induction using property Z) belongs to a certain coset represented by a certain number.

In particular we can

assign to each coset k the word t, (g.) that was originally used to define K

k.

1

To introduce the forced coincidence k :=: £ then means looking only for

the subgroup U for which t, (g.) and t (g.) lie in the same coset of U which K

1

is tantamount to t, (g.)t (g.)"

36 1

1

e U.

We now separate all subgroups U of

index < b in G into mutually disjoint classes by such conditions corresponding to certain forced coincidences. Let 1 = ij < i 2 < ... < i

be the coset numbers remaining at

B° . We order the (unordered) pairs (i^,i ) of different coset numbers, and hence the possible forced coincidences at B°, by assuming w.l.o.g.

i < i and definin

k

£

i,f < i,.

s (V'V- 1 <

(

W

iff

V

< i or

i

To each such pair we define a class C° .

V

= i and

£

. of subgroups U

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Coset table methods

33

(which may be empty) by = {U I U < G, G : U < b , t . t : 1 e U, t .

C°. . IT.'1!? K.

1

H

X/

K

tT 1

1

Q

1r

K>

f

1

K

0

f U '

X/

for a l l ( i . , , i , f ) < ( i v , i . ) ) .

In other words, each class C°. . is defined by a set P° . of positive 1 1

V*,

V*

conditions, here consisting of the single condition t. tT1 € U, and a set N° 1 . of negative conditions, namely t.

V*

tT1 ? U for all (i, k ,,i ,) <

V V

*

The definition of these classes already suggests how we are going to proceed. From B° edges corresponding to all pairs (i,,i ) K

originate.

Jo

In a backtrack search through a tree, rooted at B°, cf. Fig. 2, Fig. 2

Neubiiser:

Coset table methods

34

these will be entered in the order just defined; the edge corresponding to (i. ,i ) will lead into a subtree in which all subgroups in C°. . and k I i k ,i £ only these will be found and which must be completely finished before the next edge is entered. Entering the edge corresponding to (i,,i ) will mean starting a Todd-Coxeter procedure with the forced coincidence i, :=: i. being introduced into the table we had at B°. definitions and coincidences.

This may produce new deductions,

Now in order to keep in line with the

definition of C°.1 . , whenever the Todd-Coxeter produces a coincidence

V*

violating some negative condition in N*!

. , we completely break off

computation in this subtree and start working with the next edge.

In

Fig. 2 this event is symbolized by an edge ending in a double bar without reaching a point at the next level. If none of the negative conditions is encountered, the ToddCoxeter will eventually come to a halt again for the same reason as before. We have then reached some point B1.

of level 1, in which again we have

three possibilities. For the "leftmost" edge corresponding to (l,i2) the ToddCoxeter may return index 1; we have reached an endpoint of the tree. is symbolized in Fig. 2 by a double bar on top of the point.

This

In this case

we return to B° and resume computation there. If the Todd-Coxeter returns some index 1 / r f < f(b), we print the corresponding subgroup if r' < b.

In this case as well as in the last

case that the Todd-Coxeter comes to a halt again without having closed the coset table, B1.

is another branchpoint in which we argue much in the same

way as we did in B . Of course the set of subgroups that we want to subdivide further is now C°.1 . , and hence each of its mutually disjoint subsets

V*

inherits all the positive and negative conditions that defined C°. . .

V1*

x Again from B. J

there originate edges corresponding to all (unordered) pairs

i

of coset numbers remaining at B. , the first of these will be a subset of ^i

ii,...,i

(remaining after the coincidences that may have been found in

the Todd-Coxeter leading from B° to B1. )., but there will also be additional ones.

Let them all be called i/(= il = 1 ) , i2',...,iff, then we order their

pairs as before and for each edge get additional positive and negative conditions, which are used (together with the inherited ones) as before.

Neubiiser: Coset table methods

35

Without introducing still more indices and notations, it has now become clear what the conditions are for the groups to be found in a k at subtree starting with an edge that leads from some B.k-1 to some B. levels k-1 and k, say. Because we are working in a tree there is a unique path B°B1. ...B." B. leading from B° to B. . The positive conditions are just those of the edges of this path, while the set of negative conditions (to be used as criteria for breaking off in the ToddCoxeter leading from B." to B. ) is the union of sets N° , N1. ,...,N." where N*

consists of all those conditions corresponding to edges from B*

to points of level i+1 that are "left" of B.

in the order of edges

described above. In Fig. 2 the edges providing the negative condition of the edge B. B. are marked with a x. The process comes to an end when the subtree originating with the edge (i ,,i ) at B° has been finished. That this will happen after a finite number of steps follows since in each branch point there are at most £f(b) (f(b)-l) edges originating from it and since the length of any path from B° to an endpoint or a break-off is bounded which is seen much as in the proof of Lemma 1. It is clear from the description of the method that when it finishes each subgroup U of index < b has been found once and only once. For the practical implementations, however, a number of additional considerations are useful. We go through these in the following remarks. Remarks. 1. For one of the last (or rightmost) edges originating from B°, say, the set of negative conditions that have to be looked at whenever the Todd-Coxeter finds a coincidence may be fairly large. We shall see that this set can in fact be replaced by a single, very simple condition. For, let us consider an edge (i, ,i ) again assuming w.l.o.g. i, < i . The ToddK

K

Jo

Jo

Coxeter run corresponding to this edge is started with the forced coincidence i, :=: i . If the Todd-Coxeter finds that some coset number K

Jo

i,r with i,f < i denotes the same coset as i. then it will in due course also find that i,f and i, denote the same coset. Hence, instead of looking through the whole list of negative conditions, it suffices to use as breakoff criterion for the edge (i^'ip) t n e discovery of a coincidence of two coset numbers i, , and i , with both i, , < i and i , < i . The same K

Jo

K

Jo

Jo

Jo

remark applies in a similar fashion for all other branch points.

Neubiiser:

2.

Coset table methods

36

Another great technical improvement is the following.

are two coset numbers and g = g . g. ...g.

If i, and i

is some word in the generators

of G such that i,g and i g have both been defined in the coset table (i.e. K J6 the sequence of coset numbers for i,g. , (i,g. )g. ,... leading to i,g and the corresponding one leading to i^g can be read from the coset table) then clearly the two forced coincidences i^ :=: i the same deductions and coincidences.

and i.g :=: i g imply

Therefore, as soon as we enter the

edge (i,,i 0 ), we need not later on enter (i,g,i.g).

This is avoided by

crossing off from a (bit-) list of forced coincidences still to be dealt with, that is kept as long as needed at each branch point, those coincidences that need not be looked at because of the above remark. 3.

It should be noted that in our description we have used the natural

order of the coset numbers which also were left unchanged when some of them got eliminated due to coincidences.

In actual implementations both rules

may be abandoned; often coset numbers are changed to fill the gaps created by coincidences and an order on the remaining coset numbers is then artificially introduced.

We have not entered such technically quite

important questions in order to keep matters easy. 4.

If some words ut (g.),..., u (g.) in the generators of G are given the

method can be modified to determine only those subgroups U < G of index G : U < b that contain these words (and hence the subgroup Uo generated by them).

This is done by simply starting the first enumeration not modulo

the unit subgroup but modulo Uo.

It is plausible that this modification

not only reduces the output but that in general it will also reduce considerably the amount of computation necessary to obtain it. 5.

Since in one way or the other negative conditions are checked it is

also possible to feed such in, right from the beginning, i.e., one may require to obtain only those subgroups that do not contain a prescribed set of words (or at least one from a prescribed set of words).

However,

since these extra conditions have to be carried along with the computation, this kind of restriction, although useful for applications, is usually of less help in reducing the computing effort, 6.

So far we have discussed the determination of subgroups of index < b

one by one.

It is important to note that not only can conjugacy of these

subgroups under G be decided but in fact the method can be modified so that only one representative of each class of conjugate subgroups gets determined.

The key remark is this:

each subgroup is formed as the

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Coset table methods

37

stabilizer of 1 in a transitive permutation representation of G.

All

conjugates of U in G will therefore be stabilizers of some other coset number in that same permutation representation.

Hence we will obtain just

one representative from each conjugacy class of subgroups of index < b if we abandon the construction of the coset table of a subgroup U (which, as stressed before, is tantamount to the construction of the permutation representation <{>.. of G on the cosets of U) as soon as we can detect that the stabilizer of some coset number k has been formed earlier on, i.e., following a path to the "left" of the one we are presently following. This can be done for a given k by applying to it, level by level, all the words representing positive and negative conditions that are valid for the presently constructed graph.

If up to a certain level all positive

conditions are fulfilled and at that point a certain negative condition is violated then we know that at this point a path towards the stabilizer of k will branch off "to the left" of our path, which we may therefore abandon.

Details about two different realizations of this idea are given

in [53] as well as in the comments contained in the program texts of the relevant programs in CAYLEY [21]. 7.

It may just be mentioned that one can keep the forced coincidences

(i.e. the positive conditions) that led to each subgroup and use these to settle the question of containment between any two subgroups constructed. (Those contained in one path originating at B° anyhow form a chain in the lattice of subgroups.)

In fact the implementation of the method now

contained in CAYLEY [21] and its stand-alone predecessors by E. Lepique and R. Gallagher in Aachen allows one to print out the whole lattice of subgroups of a finite group or the upper semilattice of the subgroups of index < b, produced by the method. 8.

Finally a word about the history of the method.

It was, in slightly

different forms, discovered independently by C.C. Sims and M. Schaps. Early implementations by Sims (1965) and by Schaps and later A, Dietze (1968/1969) are no longer operational, newer ones have already been mentioned above.

The only printed full description [23] by Dietze and

Schaps reflects a point of view that was also used in their implementation. They uniquely assign to each subgroup a "normal numbering" of its cosets (defined with respect to the given ordered generating system of G).

This

"normal numbering" reflects the sequence of definitions that a Felsch ToddCoxeter would produce.

Subtrees are broken off as soon as it can be

recognized that the computation is leading to a violation of the normality condition.

Here I have followed much closer the viewpoint (although not

Neubiiser: Coset table methods

38

its description) in Sims' unpublished notes [53], in which no special requirement need be made on the Todd-Coxeter method to be used. 7. YET ANOTHER OCCURRENCE OF THE THEME OF TODD AND COXETER: A SIDE-GLANCE ON THE SCHREIER-TODD-COXETER-SIMS METHOD This powerful method, due to Sims, allows one to determine, inter alia, the order and a presentation of a permutation group G of fairly large degree (up to 10000 and higher) from a given set of generators of G. Since the method has recently got a comprehensive description in the two papers [39], [40] of J. Leon, some short remarks will suffice here, in which I shall follow [39] very closely. Let G be a permutation group, operating on a setft,and let Gai,...,a denote the pointwise stabilizer of iat ,...,a.} C Q. An ordered k k

set B = (6.1 ,. .. ,K & ) with 1 6. eftis called a base for G, if G o

Pi »•••»&]£

= 1,

a generating set S of G is called a strong generating set for G relative to B if G ^ := G o is generated by S ^ := S n G ^ for all o Pi'--" B i-l i = l,...,k. Since the index G ^ : G^ 1+ ' is equal to the length of the orbit of 3. under G , determination of the order of G is reduced to the calculation of orbits (which can easily be done for any permutation group from a generating set of it) once a base and strong generating set of G are known. Thus we have the following "fundamental problem". Given some generating set E = {st ,...,s } of G and an ordered set r = (Yj >.. • ,y^) °f points inft,extend Y to a base and E to a strong generating set of G relative to this base. In the solution of this problem a further benefit from having a base and a strong generating set will play an important role. For a base B = (3j,...,$k) and strong generating set S relative to B let A^1^ be the orbit of 6- under G ^ , and for 6 e A ^ let tr 1 ' (6) be a word in the elements of S which maps (3. to <S. Then for an element g in the symmetric group

ft :

iff

g GG guj"1

S

iff

GG l

and <

Ml*O G

where

and

where u2 = u ( 2 ) (82 (guj"1)),

Neubiiser: Coset table methods

39

and so on. Hence, if coset representatives u^(<S) are fixed for the cosets of G ' in G^1' and all i, both as permutations and as words in the strong generators, we can test containment in G for any given g € S , and moreover, if g 6 G, we obtain g = u,u, -.-••u1 and, using the expression of the u. as words in the strong generators, we can write g in terms of the strong generating set. (In practice the u. need not be stored explicitly but can be reconstructed from the so called "Schreier vectors" [40].) The crucial observation needed for a solution of the "fundamental problem" is the following lemma of Sims [39]: Lemma. Let S be a generating set of G, B = (p^ ,... ,8, ) an ordered set of "points in Si such that each s e S moves some point in B and let W ^ := < S n G Q >. Then B is a base and S a strong generating a Pj »• • • » p j _ i

set of G relative to B iff ^i

= H ^ + 1^ for all j or, equivalently, iff

for all j the index H ^ : H ^ + ^ is equal to the length of the orbit of 6. under the action of H ^ . The condition of this lemma is checked for i = k,k-l,...,l, i.e. from bottom to top, assuming, by induction, that B and S n Go Q Pi >..

are already a base and a strong generating set for G Pj ,...

definition H ( l + 1 ) C H ^ , hence the test if H ^

. ^

. B y >&i

= H ( l + 1 ) can be performed

by testing if the Schreier generators of H^ are contained in H . In 3 i the notation introduced above, these are u ^ ( 6 ) s ( u CSs))"1 where 6 € A J and s £ S n GQ ; using the fact that by induction we Q Pj ,..

•>P^_1

have a basis and strong generating set for H , containment can be tested working down the stabilizer chain, as explained above. If the containment test fails for some Schreier generator g then either B.guT* uT* .. .uT1 . £ A ^ for some j with i+1 < j < k or

In the first case the element guT* ...uT1- is added to S and we have to reset the inductive process to test H ^ = H ^ + * keeping p-i J J J H Vfi+11 ,...,H fkl unchanged. In the second case we have to add to B a point that is moved by guT* ...u"1 and to add the element guT* ...u"1 to S, and we have to start from the bottom again.

Neubiiser:

Coset table methods

40

Going with this test through all Schreier generators may be slow.

However Schreier generators as well as the u. are given as words in

the generators of G, hence the test, if gu7* .. .u'1 = 1, may be thought of as verifying that a relation holds for the strong generators of G.

Now,

if a certain small set of such relations has been verified, one may try to deduce from these alone by coset enumeration that H to the orbit length of 3. under H

:H

is equal

which by Sims lemma also suffices.

At this point, at which coset tables have appeared at least, I want to refer the reader to Leon's papers for details, which involve an interruptible Todd-Coxeter program in which the change between "look ahead" and "define" mode can be influenced by the user via several preset parameters.

There also most impressive applications are described. It remains to draw a connection to Cannon's method discussed

in §5.

Here the argument is that at level i the process is terminated

only when enough relations have been picked up to prove with the help of the Todd-Coxeter that the index H^1'' : H^ 1 + ' is what it is supposed to be in a stabilizer chain corresponding to a base of a permutation group and hence the union of all relations picked up at the different levels suffices to define G.

This is essentially the same argument as that used for the

verification of Cannon's multistep algorithm.

The use of the stabilizer

chain as the chain of subgroups simplifies the matter here greatly, because the otherwise very restrictive requirement stated at the end of §5, that a coset table modulo a big subgroup in a still bigger group must still be obtainable, becomes much easier for a stabilizer in a permutation group. Finally we may remark that the Schreier-Todd-Coxeter-Sims method is not restricted to groups given by generating permutations. G. Butler [6] has implemented a variation of it for matrix groups.

8.

EPILOGUE Let me start some concluding remarks by stating that I have

kept to coset table methods in computational group theory.

I have not at

all tried to look at their application to semigroups and related structures for which they also seem to get increasing importance. In the previous sections I have only occasionally hinted at technical problems in the implementation of the various methods, mainly the reader is referred to the papers quoted for these.

However, a few

words on availability and use of the implementations might be added. Except for the methods of §4 that determine a presentation for a subgroup in terms of given generators all methods are represented by

Neubiiser:

Coset table methods

41

routines contained in the CAYLEY system [21]. There they are accessible through a language and within a data organisation framework which has been designed with great care to give the user a maximum of comfort.

By

the same token, however, the user may sometimes not even realize which method CAYLEY uses to answer his question.

Somebody wanting to experiment

with these methods, to make direct use of built-in variations of the algorithm or even to implement variations of his own may therefore ask for stand-alone programs, which in most cases allow more direct access to the data and routines and in some cases pay for the less comfortable input and output by high and tunable efficiency.

We list, without any claim of

completeness, some persons/places which now (Spring 82) have such programs, although not all of these are fully portable.

The "ordinary" Todd-Coxeter

is the most widespread one, a particularly versatile and efficient program in standard Fortran is available from G. Havas (Department of Mathematics, Institute of Advanced Studies, Australian National University, P.O. Box 4, Canberra, A.C.T. 2600, Australia).

He also has a stand-alone version of

the Reidemeister-Schreier program discussed in §4 and a program for Tietze transformations.

For programs implementing the methods discussed in the

rest of §4, E.F. Robertson (Department of Pure Mathematics, University of St. Andrews, St. Andrews, KY16 9SS, Scotland) should be asked.

While I do

not know if a stand-alone version of Cannonfs method (§5) is available, there are stand-alone implementations for the low-index method of §6 available with the author at Aachen.

Finally for the Schreier-Todd-Coxeter-

Sims method J. Leon (Mathematics Department, University of Illinois, Box 4348, Chicago, Illinois 60680, USA) might be approached. Applications of the methods described to concrete group theoretical questions are too numerous to be listed completely here, moreover they are often hidden in a short remark or acknowledgement in an otherwise "pure" and unsuspicious paper.

A large number may be found in

the bibliography [25], some overview of the capability of the methods is given in [47]. Just a few papers on very typical applications have been included in the list of references; in addition to those already mentioned in the previous sections, see e.g. [11], [12], [13], [14], [28], [30], [34]. In spite of all that has been undertaken with the theme of Todd and Coxeter, heuristic exploration, many successful applications, formulation of a few provable statements and formalization of the concept of an (incomplete) coset table, I think it is fair to admit that we are still far from a good understanding of the matters that influence its performance.

On the other hand we may also hope that we have not yet seen

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42

the last variation of this "useful tool with a wide range of applications".

Acknowledgement.

I wish to thank V. Felsch and W. Plesken for many helpful

discussions and the editors for their patience. REFERENCES 1. D.G. Arrell, S. Manrai § M.F. Worboys, A procedure for obtaining simplified defining relations for a subgroup, these proceedings. 2. P. A. Bandler, A method for enumerating the oosets of an abstract group on a digital computer, M.A. thesis, University of Manchester (1956). 3. M.J. Beetham, Space saving in coset enumeration, unpublished notes, St. Andrews (1976). Abstract: Sigsam Bull. IO_ (4) (1976), 14. 4. M.J. Beetham § C M . Campbell, A note on the Todd-Coxeter coset enumeration algorithm, Proc. Edinburgh Math. Soc. 2£ U 9 7 6 ) > 73-79. 5. C.T. Benson § N.S. Mendelsohn, A calculus for a certain class of word problems in groups, J. Combinatorial Theory _1. (1966), 202-208. 6. G. Butler, Computational approaches to certain problems in the theory of finite groups, Ph.D. thesis, University of Sydney (1979). 7. G. Butler $ J.J. Cannon, Computing in permutation and matrix groups I: normal closure, commutator subgroups, series, to appear. 8. G. Butler, Computing in permutation and matrix groups II: backtrack algorithm, to appear. 9. G. Butler § J.J. Cannon, Computing in permutation and matrix groups III: Sylow subgroups, to appear. 10. C M . Campbell, Enumeration of cosets and solutions of some word problems in groups, dissertation, McGill University (1965). 11. C M . Campbell $ E.F. Robertson, On a class of finitely presented groups of Fibonacci type, J. London Math. Soc. H_ (1975), 249-255. 12. C M . Campbell § E.F. Robertson, Remarks on a class of 2-generator groups of deficiency zero, J. Austral. Math. Soc. JL£ (1975), 297-305. 13. C M . Campbell § E.F. Robertson, Deficiency zero groups involving Fibonacci and Lucas numbers, Proc. Roy. Soc. Edinburgh 81A (1978), 273-286. 14. C M . Campbell § E.F. Robertson, On 2-generator 2-relation soluble groups, Proc. Edinburgh Math. Soc. Th_ (1980), 269-273. 15. J.J. Cannon, Computers in group theory: a survey, Comm. ACM 12_ (1969), 3-12. 16. J.J. Cannon, Computing local structure of large finite groups, in Computers in algebra and number theory9 edited by G. Birkhoff and M. Hall, Jr., SIAM-AMS P r o c , Vol. 4, Amer. Math. S o c , Providence, R.I. (1971), 161-176. 17. J.J. Cannon, Construction of defining relators for finite groups, Discrete Math. 5_ (1973), 105-129. 18. J.J. Cannon, L.A. Dimino, G. Havas § J.M. Watson, Implementation and analysis of the Todd-Coxeter algorithm, Math. Comput. TJ_ (1973), 463-490. 19. J.J. Cannon, A general purpose group theory program, in Proc. Second Intemat. Conf. Theory of Groups^ edited by M.F. Newman, Lecture Notes in Mathematics, Vol. 372, Springer-Verlag, Berlin (1974), 204-217.

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Coset table methods

43

20. J.J. Cannon, Software tools for group theory, in The Santa Cruz Conference on Finite Groups, edited by B. Cooperstein and G. Mason, Proc. Symp. Pure Math., Vol. 37, Amer. Math. Soc., Providence, R.I. (1980), 495-502. 21. J.J. Cannon, CAYLEY, a system of group theoretical programs developed by and obtainable from J.J. Cannon, Department of Pure Mathematics, the University of Sydney, Sydney N.S.W. 2006, Australia. 22. H.S.M. Coxeter § W.O.J. Moser, Generators and relations for discrete groups, 4th ed., Springer-Verlag, Berlin (1980). 23. A. Dietze § M. Schaps, Determining subgroups of a given finite index in a finitely presented group, Canada J. Math. 26_ (1974), 769-782. 24. H. Felsch, Programmierung der Restklassenabzahlung einer Gruppe nach Untergruppen, Numer. Math. 3_ (1961), 250-256. 25. V. Felsch, A bibliography on the use of computers in group theory and related topics: algorithms* implementations* and applications, kept current and obtainable from Lehrstuhl D fur Mathematik, RWTH Aachen, D-5100 Aachen, Federal Republic of Germany. 26. J. Grover, L.A. Rowe $ D. Wilson, Applications of coset enumeration, in Proceedings of the Second Symposium on Symbolic and Algebraic Manipulation, edited by S.R. Petrick, Assoc. Comput. Mach., New York (1971), 183-187. 27. G. Havas, A Reidemeister-Schreier program, in Proc. Second Internat. Conf. Theory of Groups, edited by M.F. Newman, Lecture Notes in Mathematics, Vol. 372, Springer-Verlag, Berlin (1974), 347-356. 28. G. Havas, Computer aided determination of a Fibonacci group, Bull. Austral. Math. Soc. 15_ (1976), 297-305. 29. G. Havas $ T. Nicholson, Collection, in SYMSAC 76, Proceedings of the 1976 ACM Symposium on Symbolic and Algebraic Computation, edited by R.D. Jenks, Assoc. Comput. Mach., New York (1976), 9-14. 30. G. Havas, J.S. Richardson § L.S. Sterling, The last of the Fibonacci groups, Proc. Roy. Soc. Edinburgh 83A (1979), 199-203. 31. G. Havas § L.S. Sterling, Integer matrices and abelian groups, in Symbolic and Algebraic Computation, edited by E.W. Ng, Lecture Notes in Computer Science, Vol. 72, Springer-Verlag, Berlin (1979), 431-451. 32. G. Havas, Commutators in groups expressed as products of powers, Comm. Algebra 1 (1981), 115-129. 33. D.L. Johnson, Topics in the theory of group presentations, London Math. Soc. Lecture Note Series, Vol. 42, Cambridge University Press, Cambridge (1980). 34. J. Leech, Some definitions of Klein's simple group of order 168 and other groups, Proc. Glasgow Math. Assoc. S_ (1962), 166-175. 35. J. Leech, Coset enumeration on digital computers, Proc. Cambridge Philos. Soc. 5£ (1963), 257-267. Reprint with supplementary notes and references distributed at the Conference on Computational Problems in Abstract Algebra, Oxford (1967). 36. J. Leech, Generators for certain normal subgroups of (2,3,7), Proc. Cambridge Philos. Soc. 6j_ (1965), 321-332. 37. J. Leech, Coset enumeration, in Computational Problems in Abstract Algebra* edited by J. Leech, Pergamon, Oxford (1970), 21-35. 38. J. Leech, Computer proof of relations in groups, in Topics in Group Theory and Computation, edited by M.P.J. Curran, Academic Press, London (1977), 38-61.

Neubuser:

Coset table methods

44

39. J.S. Leon, Finding the order of a permutation group, in The Santa Cruz Conference on Finite Groups, edited by B. Cooperstein and G. Mason, Proc. Symp. Pure Math., Vol. 37, Amer. Math. S o c , Providence, R.I. (1980), 511-517. 40. J.S. Leon, On an algorithm for finding a base and a strong generating set for a group given by generating permutations, Math. Comp. 3S_ (1980), 941-974, 41. W. Magnus, A. Karrass § D. Solitar, Combinatorial group theory: presentations of groups in terms of generators and relations, Pure and Appl. Math., Vol. 13, Interscience, New York (1966). Second revised edition, Dover, New York (1976). 42. D.H. McLain, An algorithm for determining defining relations of a subgroup, Glasgow Math. J. _1£ (1977), 51-56. 43. N.S. Mendelsohn, An algorithmic solution for a word problem in group theory, Canad. J. Math. 16^ (1964), 509-516. Correction, Canad. J. Math. \J_ (1965), 505. 44. N.S. Mendelsohn, Defining relations for subgroups of finite index of groups with a finite presentation, in Computational Problems in Abstract Algebra, edited by J. Leech, Pergamon, Oxford (1970), 43-44. 45. J. Neubuser, Untersuchungen des Untergruppenverbandes Endlicher Gruppen auf einer Programmgesteuerten Elektronischen Dualmaschine, Numer. Math. 1_ (1960), 280-292. 46. J. Neubuser, Investigations of groups on computers, in Computational Problems in Abstract Algebra, edited by J. Leech, Pergamon, Oxford (1970), 1-19. 47. J. Neubuser, Computing with groups and their character tables, Computing> Suppl. £, to appear. 48. M.F. Newman, Calculating presentations for certain kinds of quotient groups, in SYMSAC 76, Proceedings of the 1976 ACM Symposium on Symbolic and Algebraic Computation, edited by R.D. Jenks, Assoc. Comput. Mach., New York (1976), 2-8. 49. M.H.A. Newman, The influence of automatic computers on mathematical methods, Manchester University Computer Inaugural Conference, Manchester (1951), 13. 50. M.O. Rabin, Recursive unsolvability of group theoretic problems, Ann. of Math. 61_ (1958), 172-194. 51. C.C. Sims, Determining the conjugacy classes of a permutation group, in Computers in algebra and number theory, edited by G. Birkhoff and M. Hall, Jr., SIAM-AMS P r o c , Vol. 4, Amer. Math. S o c , Providence, R.I. (1971), 191-195. 52. C.C. Sims, Computation with permutation groups, in Proceedings of the Second Symposium on Symbolic and Algebraic Manipulation, edited by S.R. Petrick, Assoc Comput. Mach., New York, (1971), 23-28. 53. C.C. Sims, Some algorithms based on coset enumeration, unpublished notes, Rutgers Univ., New Brunswick, N.J. (1974). 54. C.C. Sims, The role of algorithms in the teaching of algebra, in Topics in Algebra, edited by M.F. Newman, Lecture Notes in Mathematics, Vol. 697, Springer-Verlag, Berlin (1978), 95-107. 55. C.C. Sims, Some group-theoretic algorithms, in Topics in Algebra, edited by M.F. Newman, Lecture Notes in Mathematics, Vol. 697, Springer-Verlag, Berlin (1978), 108-124. 56. C.C. Sims, Group-theoretic algorithms, a survey, in Proceedings of the International Congress of Mathematicians, Helsinki 1978, Vol. 2, edited by Olli Lehto, Acad. Sci. Fennica, Helsinki (1980), 979-985.

Neubiiser:

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45

57. J.A. Todd § H.S.M. Coxeter, A practical method for enumerating cosets of a finite abstract group, Proa* Edinburgh Math. Soo. S_ (1936), 26-34. 58. H.F. Trotter, A machine program for coset enumeration, Canad. Math. Bull. 1 (1964), 357-368. 59. J.N. Ward, A note on the Todd-Coxeter algorithm, in Group Theory, edited by R.A. Bryce, J. Cossey and M.F. Newman, Lecture Notes in Mathematics, Vol. 573, Springer-Verlag, Berlin (1977), 126-129.

46 APPLICATIONS OF COHOMOLOGY TO THE THEORY OF GROUPS

D.J.S. Robinson University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, U.S.A.

INTRODUCTION Homological concepts have long been used implicitly in the theory of groups.

They occur, for example, in the work of Holder (1895)

and Schreier (1926) on group extensions and of Schur (1904, 1907) on projective representations.

The significance for group theory of the

cohomology groups in low dimensions appears to have first been recognized by Eilenberg and MacLane in the 1940's.

On the other hand, actual

applications of homology to establish theorems of a purely group theoretical nature are of more recent origin, dating largely from the last twenty five years.

Perhaps the most famous result to be discovered

by the use of such methods is Gaschiitz's theorem on the existence of outer automorphisms of finite p-groups (1965-6). Recently there has been an increasing awareness among group theorists of the utility of homology, and many have been led to equip themselves with homological tools.

In this respect the lecture notes of

K. W. Gruenberg [12] and U. Stammbach [37] have been influential. The present work is not intended to be a survey; rather its aim is to review the ways in which the cohomology groups arise in group theory and then to exploit this connection by proving some theorems about groups. We begin with the familiar relation between derivations and complements in semidirect products and prove a generalization of Mal'cev's theorem on the near conjugacy of maximal torsion subgroups in soluble groups of finite rank.

Then, after a review of the classical

extension theory of Schreier-Eilenberg-MacLane, we discuss the more novel concept of near splitting and proceed to state a series of splitting and near splitting theorems.

Most of these were first established in [28],

and they have since turned out to be useful in infinite soluble group theory.

Applications to the structure of infinite soluble groups follow.

Robinson:

Applications of cohomology

47

A detailed account of automorphism groups of group extensions is given, leading up to the fundamental exact sequence of Wells.

The

treatment here is based on the classification of extensions with nonabelian kernel, an approach which involves a minimum of calculation. various special forms of the sequence are discussed in detail.

The

The great

use of the Wells sequence and its derivatives is to construct outer automorphisms and to provide information about the structure of the outer automorphism group.

This is illustrated by a discussion of outer auto-

morphisms of free abelianized extensions, outer automorphisms of finite p-groups and the recent theory of complete groups.

CONTENTS 1.

Conjugacy of complements and the first cohomology group. Conjugacy.

2.

Group extensions and the second cohomology group. splitting theorems.

3.

Applications of splitting and near splitting to infinite soluble groups.

4.

Automorphisms of group extensions.

5.

Constructing outer automorphisms.

1.

Near

Splitting and near

The Wells exact seauence.

CONJUGACY OF COMPLEMENTS AND THE FIRST COHOMOLOGY GROUP Consider a semidirect product G = Q K A in which 0 operates

on the dbeiian group A in some prescribed manner, so that A is a Q-module. Recall that a complement of A in G is a subgroup K such that G = KA and K n A = 1. A.

Of course, Q and all its conjugates in G are complements of

It is an important problem in group theory to decide whether the

converse is true. Let K be any complement of A in G. a unique element a £ A such that xa £ K. 6K

Then, if x G Q, there is

Thus we may define a function

: Q •+ A by means of the rule

xx

G K.

Straightforward multiplication reveals that

(xy) so that 6

R

= (x V y

\

(x,y € Q ) ,

is a derivation from Q to A.

Observe that 6 n = 0.

Robinson:

Applications of cohomology

48

Conversely, if we start with a derivation <5 : Q -> A, we can find a corresponding complement of A in G, namely the set K 6 = {xx6|x e Q}. This is very simple to check.

K I

> 6

and 6 I

It is also easy to see that the functions

> K are mutually inverse.

Consequently there is a

bisection between the set of all complements of A in Q K A and the set Der(Q,A) of all derivations from Q to A. So far no use has been made of the commutativity of A; however, this property becomes relevant when one discusses conjugacy. Consider two complements Ki , K2 with associated derivations 6i , 62 .

If Ki and K2 are conjugate in G, there is an element a of A such

that K2 = Ki . x[x,a]x

61

Conjugating the relation xx

l

G Ki by a, we obtain

6

<= Ka , with the result that x * = [x,a]x61 , or x 6 '" 6l = [x,a] .

Thus 62 - 61 is an inner derivation if Ki and K2 are conjugate. Conversely two derivations that differ by an inner derivation determine conjugate complements, as may be seen by reversing the previous argument. Now the set of all inner derivations is a subgroup Inn(Q,A) of the abelian group Der(Q,A) (where, of course, the group operation is given b y x 1

2

= x

1

x

2

) .

So the foregoing considerations demonstrate

that the conjugacy classes of complements of A in Q K A stand in one-one correspondence with the elements of the group Der(Q,A)/Inn(Q,A). In the language of cohomology a derivation is just a 1coeycle and an inner derivation is a 1-coboundary.

The above quotient

group is therefore Z1(Q,A)/B'(Q,A) = H'CQ.A), the first cohomology group of Q with coefficients in A.

(Here, of

course, we write Zn(Q,A) and Bn(Q,A) for the additive groups of ncocycles and n-coboundaries respectively where A is a given Q-module.) These conclusions may be summed up as follows.

(1.1)

There is a bisection between the set of conjugacy classes of

complements of A in Q K A and the group H1 (Q,A).

A complement is

conjugate to Q if and only if it corresponds to the zero element of

Robinson:

Near conjugacy.

Applications of cohomology

49

We shall now introduce a weak form of conjugacy which is

in some respects more natural in infinite group theory, where conjugacy of complements is something of a rarity. Let K be a complement of A in G = Q K A which determines an element 6 = 6 + Inn(Q,A) of H1 (Q,A).

Assume that m6~ = 0 for some m > 0.

The question is, what special properties does such a complement possess? Consider the endomorphism a of G in which xa is mapped to xa , (x £ Q, a £ A). derivation m6. a in A.

Now xx

£ K, so that (xx )

Let T be a transversal to A

(t e T, b G A ) .

= xx

£ K

However, by hypothesis m6 is inner, and K in A and put a = tb

Then K a = b" a (Q t )b a and thus (Kb

be the preimage of Q

under a.

Then we have K

)

= K

a

= QZ.

and K

= Q

has

for some

= tb , Define K

to

(Ker a ) , while

obviously Ker a = {a G A|a m = 1} = A[m]. Thus K is conjugate to K

modulo A[m]. Notice that the set of subgroups

{K It £ T} depends only on m (and the choice of T ) . We shall now assume that if p is either 0 or a prime, the p-rank of A, r (A), is finite. finite.

(1.2)

Then both |A : A m | = |T| and A[m] are

The situation may be summed up in the following manner.

Let K be a complement ofkinQMk

which determines an element

~6 = 6 + Inn(Q,A) of H1 (Q,A) . Assume in addition that r (A) < °° if p = 0 or a prime. (i)

If m6 = 0 for some m > 0, then each K. ., i G TL, is conjugate modulo the finite subgroup A[m] to one of at most | A : A | subgroups depending only on m.

(ii)

Conversely, assume that the K ^ , i G 2, fall into finitely many conjugacy classes modulo some finite G-invariant subgroup of A. Then T has finite order.

Proof.

To prove (ii) note that for some i < j we have K.. and K..

conjugate modulo a certain finite G-invariant subgroup F of A. |F| = s.

Let

Then s(j6 - i<5) = s(j - i)6 is inner and 6 has finite order.

Of particular interest is the case where H1(Q,A) is a bounded abelian group.

For if, say, mH1 (Q,A) = 0 where m > 0, the

complements of A in Q IX A fall into at most | A : A | conjugacy classes modulo A[m]. With this fact in mind, we say that the complements of A

Robinson:

Applications of cohomology

50

in Q P< A are nearly conjugate if they fall into finitely many conjugacy classes modulo a finite normal subgroup of G contained in A. We record for future reference the following special case of (1.2).

(1.3)

Let r (A) < «> for p = 0 or a "prime. Then the complements of A in

Q K A are nearly conjugate if and only if H1 (Q,A) is a bounded abelian group*

Conjugacy of maximal torsion subgroups.

In our application of near

conjugacy we shall be concerned with groups G which have a series 1 = Go < Gi < ... < G = G m

(1)

such that each factor is either infinite cyclic or locally finite.

The

number of infinite cyclic factors in the series is an invariant called the torsion-free (or 0-) rank. finite.

The torsion subgroups of G are locally

We aim to show that the maximal torsion subgroups fall into

finitely many conjugacy classes.

This was proved by Mal'cev [23] in the

case of soluble groups (see also Bowers and Stonehewer [3]). In order to establish this result we shall need certain information about the structure of the group G.

(1.4)

(cf. Mal'cev [23], Theorem 3).

Let G be a group with a series of

finite length having infinite cyclic or locally finite factors.

If T is

the unique largest normal torsion subgroup of G, then G/T has a characteristic series of finite length whose infinite factors are torsion-free abelian groups of finite rank. Proof.

Clearly we may assume that T = 1.

(1) in G. on m.

If m < 1, all is clear.

Then N = G

Assume that there is a series

So let m > 1 and proceed by induction

has a series of the required type.

If G/N is

infinite cyclic, then we have at least a normal series in G of the type sought.

Suppose, however, that G/N is locally finite.

The idea is to

show that this factor can be moved down the series leaving behind factors that are finite or torsion-free abelian of finite rank. To achieve this consider H < K where H is torsion-free abelian of finite rank and K/H is locally finite.

Put C = C K (H).

Then

K/C is finite since by a well-known theorem of Schur a torsion subgroup of GL(n,Q) is finite.

Also H < Z(C), the centre of C, and C/H is locally

Robinson:

finite.

Applications of cohomology

51

Another theorem of Schur tells us that Cf is locally finite.

Therefore, if S is the (unique) maximal torsion subgroup of C, then C/S is torsion-free abelian of finite rank.

Of course, S is locally finite.

Replace H <1 K by S <3 C <1 K, noting that if H and K are G-invariant, so will be S and C. So far we have found a normal series 1 = G <1 G .. < .. . <3 Go _ x, J6-1 Replace d by Gi = n G/; then G/Gi is Y e Aut G _ either finite or torsion-free abelian of finite rank according as G/Gi is. = G of the type needed.

In the same way the series may be continued inside Gi .

We shall also require the well-known (1.5)

If Q is a finite group of order m and A is any ^-module, then

mHn(Q,A) = 0 for all n > 0. This is proved by the standard averaging process for cocycles: see [22] for example. We can now give our application of near conjugacy.

(1.6)

Let G be a group with a series of finite length whose factors are

either locally finite or torsion-free abelian of finite rank.

Then the

maximal torsion subgroups of G fall into finitely many conjugacy classes. Proof.

Since each maximal torsion subgroup contains the unique maximal

normal torsion subgroup, the latter may be factored out and assumed to be trivial.

It now follows from (1.4) that there is a normal series

l = G o < l G i < l . . . < l G = G with each G.

1+X

)C

free abelian group of finite rank. let I > 1 and use induction on I.

/G. either finite or a torsion1

If I < 1, the result is obvious, so Set A = Gi .

If A is finite, it is contained in every maximal torsion subgroup and the result follows by induction.

Assume, therefore, that A

is torsion-free. Suppose that Ti , T 2 , ... are infinitely many inconjugate maximal torsion subgroups of G.

Then T. n A = 1.

By induction the T.A

fall into finitely many conjugacy classes - observe that all torsion subgroups of G/A are finite and hence finitely inconjugate. relabelling the T. we may suppose that H = Ti A = T 2 A = ... . finite, so H1(H/A,A) is bounded, by (1.5).

By But T. is

It follows from (1.3) that

the complements of A in H fall into finitely many conjugacy classes, a contradiction.

Robinson:

2.

Applications of cohomology

52

GROUP EXTENSIONS AND THE SECOND COHOMOLOGY GROW We begin with a review of the theory of group extensions as

propounded by Schreier, Eilenberg and MacLane.

By an extension e_ of a

group N by a group Q is meant a short exact sequence of groups and homomorphisms £ : N >-^-> G — ^ - » Q. Here, of course, > respectively.

> and

>> denote a monomorphism and an epimorphism

The kernel N will be written additively, the groups G and

Q multiplicatively. By choosing a transversal to Im y = Ker z in G we obtain a function T : Q •> G with the property xe = 1.

Such a T is called a

transversal function for e_. If x 6 Q, conjugation by x

in G leads to an automorphism x

of N given by the equation (ax )y = (xT) where a £ N.

1

(ay)xT

Also, if x and y are elements of Q, then x y

differ by an element of Im y. x V where (x,y)<{> £ N.

and (xy)

Thus we can write

= (xy)T((x,y)cf0y

(2)

The function $ is known traditionally as a factor set.

In consequence, associated with e_ and a fixed transversal function T there is a pair of functions £ :' Q •* Aut N

and

(f> : Q * Q -> N

which will be referred to as an associated pair for £. The associative law x T (y T z T ) = (x T y T )z T and the definition of (J> lead directly to the familiar equation (x,yz) + (y,z)<|> = (xy,z)cj> + (x,y)<j).zC. Moreover, conjugating by x y

(3)

and using (2), we obtain in addition the

relation

x V = (xyr(x,y)cf>

(4)

where b denotes the inner automorphism of N induced by b € N. The functions E, and <j> depend on the choice of transversal function T.

Suppose that T 1 is another transversal function and write

Robinson: T'

x

Applications of cohomology

53

T

= x ((x)i|0v

where (X)I|J £ N.

Then xf gives rise to an associated pair of functions

(£f ,<}>') where x

= x (x)ip

(5)

and

as simple calculations show. Observe the consequence of (5) that x X = x (Inn N) is independent of the transversal function T . X

The function

: Q •*• O u t N

is known as the coupling of the extension; by (4) it is a homomorphism.

Constructing extensions.

Now suppose that £ : Q •> Aut N and

: Q x Q -+ N are given functions which satisfy (3) and (4) . Then there is an extension e of N by Q which has (£,<j>) as its associated functions with respect to an appropriate transversal function.

Define G(£,<j>) to be

the set of all pairs (x,a), x £ Q, a £ N, equipped with the binary operation (x,a)(y,b) = (xy,(x,y) + ay

+ b) . Then G(£,cf>) is a group.

Moreover there is an extension

where ay = (l,-(l,l)c{> + a) and (x,a) ? = x, (a e N, x e Q) . The transversal function x »

> (x,0) gives rise to the functions £ and $ as an

associated pair for ei_(£,<|>). All of this can be established by routine computations (cf. [18], §48). y. Eqivalent extensions.

Two extensions e_. : N >

e. > G.

»

Q, i = 1,2,

are said to be equivalent if there is an isomorphism 0 : Gi -* G2 making the diagram

I commute.

4 I

Here the left and right down maps are identity functions.

Robinson:

(2.1)

Applications of cohomology

54

Let ei and e_2 be two extensions of N by Q with associated

functions (£i,4>i) and (£2,4*2) with respect to certain transversal functions.

Then ei and ei are equivalent if and only if there is a

function \\) : Q •* N suc/z

(x,y)2 = -(xy)ij; + (x,y)c()i for all x,y in Q. This again is proved by straightforward calculations.

Notice

that every extension with associated functions (£,<(>) is equivalent to the constructed extension e_(£,cf)).

Also note that equivalent extensions have

the same coupling.

Extensions with abelian 'kernel. Consider now an extension A >——> G »

Q with abelian kernel A.

with the coupling x A.

anc

In this case the function £ is identical

* amounts to prescribing a Q-module structure for

The function (f> satisfies (3), so it is a 2-cocycle in Z2 (Q,A) where

the module structure is given by x-

Tne

element (j> + B2 (Q,A) is called

the cohomology class of the extension. The conditions for equivalence of two extensions e_i and e^ become X

= x > and

(x,y)(i) = -(xy)i|> + (x)ij>.yxi + (y)^, (see (2.1)). coboundary.

The expression on the right is the value at (x,y) of a 2Hence the condition is simply §7 = <\>i mod B2 (Q,A) . Thus we have the well-known result.

(2.2)

Let A be a ^-module via the coupling x ' Q * Aut A.

Then there is

a bisection between the set of equivalence classes of extensions of N by Q with coupling x and the group H2 (Q,A).

An extension splits if and only

if it corresponds to the zero element of H2 (Q,A).

Two splitting theorems.

We shall state two generalized splitting

theorems which will find application in §3 and §5.

These were first

established in Robinson [28] (see also [4] and [6] for special cases).

Robinson:

(2.3)

Applications of cohomology

55

Let Q be a nilpotent group and let R be a ring with identity. If

A is a noetherian ^-module

such that Ho (Q,A) = 0, then Hn(Q,A) = 0 =

H (Q,A) for all n > 0. (2.4)

Let Q be a nilpotent

A is an artinian

group and let R be a ring with identity.

If

n

RQ-module such that H°(Q,A) = 0, then H (Q,A) = 0 =

H (Q,A) for all n > 0. Recall the interpretations of the zero dimensional homology and cohomology groups, H0(Q,A) = A Q = A/[A,Q], and H°(Q,A) = A^ = {a e A|ax = a, V x e Q}. Here, of course, [A,Q] is the additive subgroup generated by all [a,x] = a(x - 1 ) , a £ A, x e Q. Of course, (2.3) and (2.4) are stated in much greater generality than we require.

It is only the vanishing of H1 (Q,A) and

H2 (Q,A) which interest us:

this means that every extension of A by Q

will split and all complements of A will be conjugate.

Remark. group.

In fact (2.3) and (2.4) are still true if Q is a hypercentral This same comment applies to all other splitting and near

splitting theorems in the sequel except (3.4).

Near splitting.

Let A >

> G

>> Q be an extension with abelian

kernel and denote the cohomology class of the extension by A. We plan to investigate the effect on the extension of assuming that A has finite order.

This will lead to the concept of a nearly split extension, which

has proved to be extremely useful in infinite group theory. Let the given extension have coupling x a n d write A = <j> + B2 (Q,A). m;

We shall assume that mA = 0 for some positive integer

there is no loss in identifying our extension with the constructed

extension e_(x,<|>).

Now x and m<j) satisfy the conditions (3) and (4) for an

associated pair of functions.

So_we may form the extension e^x,ni) >

which will be written A >——-> G

»

Q.

The mapping (x,a)

> (x,ma)

from G to G is a homomorphism 9 and it leads to the commutative diagram A >-

i

A >- y

-> G -

4

-> G - e

> Q

1

> Q

Robinson:

Applications of cohomology

Here the left down map is a i

56

> ma.

Since mA = 0, the lower extension splits and there is a subgroup X such that G" = XAy

and

X n A y = 1.

Writing Xi for 3f n G 6 , we have (XA y9 ) n G6 = Xi A y 0 and |G9 : X>A y9 | < |XAy : XA y6 | = |Ay : A m y | = |A : mA| . Define X to be the preimage of Xi under 6.

Then

| G : XA y | < |A : mA| .

(7)

Also (X n A y ) 9 < X n A y = 1, so that X n A y < Ker 0.

But clearly

y

Ker 6 = A[m] , and therefore X n A y < A[m] y .

(8)

Now if r (A) < oo whenever p is 0 or a prime, then |A : mA| and A[m] are finite.

Therefore,

|G : XA y | < -

and

|x n A y | < ».

When such a subgroup X exists we shall say that an extension A >——> G

>> Q nearly splits.

(2.5) Let A >

> G

» Q be an extension with abelian kernel. Let A

be the cohomology class of the extension and assume that r (A) < °° if p = 0 or a prime.

Then the extension nearly splits if and only if A has 2

finite order in H (Q,A). Proof.

Only the necessity of the condition requires proof (and for this

no assumption about the ranks of A need be made). A >—^-—> G

properties |x n A | < °° and |G : XA| < °°. notation.)

Assume that

>> Q is nearly split, so that there is a subgroup X with the (Here we omit the y to simplify

Since N (X n A) contains XA, it has finite index in G and the

normal closure M of X n A in G is finite. Write G" = G/M, X* = XM/M and A = A/M. Then |(f : XA| < « and X n A = 1. Also X is isomorphic via e with a subgroup Qo of finite index in Q.

Let A be the cohomology class of the extension A >

> G

»

Q.

Now the inclusion Qo C> Q induces a map of cohomology groups, namely the restriction

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Applications of cohomology

57

Res : H2 (Q,A) •> H2 (Qo ,A) , and (A)Res = 0 because A >

> XA

»

Qo is split.

Hence (A)Res * Cor

= 0 where Cor : H2 (00 ,A) -> H2 (Q,A) is the corestriction.

However Res ° Cor is well-known to be multi-

plication by m = |Q : Qo | (see [16], VI. 16) . Hence mA~ = 0.

Finally,

because M is finite, A has finite order.

Two neccr splitting theorems.

We shall state two criteria for the near

splitting of abelian by nilpotent groups. splitting theorems(2.3) and (2.4).

These are analogous to the

In what follows, if M is an R-module

and R is some ring, we say that M is bounded if there exists an r ^ 0 in R such that Mr = 0.

(2.6)

Let Q be a nilpotent group and R a principal ideal domain.

Suppose that A is an RQ-module which is R-torsion-free with finite rank as an R-module.

If H° (Q,A) = 0, then Hn(Q,A) and H (Q,A) are bounded

R-modules for all n > 0.

(2.7)

Let Q be a nilpotent group and R a principal ideal domain.

Suppose that A is an RQ-module which is R-divisible with finite total rank as an R-module. of R is to be finite.)

(The sum of the p-ranks of A where p = 0 or a prime If Ho(Q,A) = 0, then Hn(Q,A) and Hn(Q,A) are

bounded R-modules for all n > 0. These results were first proved in [28] with R = 7L. However, the proofs work for any p.i.d. R.

The case R = 2 < t > where < t > is an

infinite cyclic group has also turned out to be useful (see (3.5)). To obtain actual near splitting theorems from (2.6) and (2.7) it is necessary to make an additional assumption about the principal ideal domain R, namely that R/Rr

is finite if

r t 0.

(9)

For this will guarantee that a bounded R-module of finite rank is finite. The argument preceding (2.5) will then go through, as may be seen from (7) and (8).

(2.8)

Let Q,R and A be as in (2.6) and assume in addition that R

satisfies (9). Then every extension of A by Q is nearly split.

Robinson:

(2.9)

Applications of cohomology

58

Let Q,R and A be as in (2.7) and assume in addition that R

satisfies (9). Then every extension of A by Q is nearly split.

Two more splitting theorems.

Finally, let us show how to combine (2.6)

and (2.7) with the splitting theorems (2.3) and (2.4) to obtain new splitting theorems.

(2.10)

Let Q be a nilpotent group and R a principal ideal domain.

Suppose that A is an RQ-module which is R-torsion-free with finite rank If Ho (Q,A) = 0, then Hn(Q,A) = 0 = H (Q,A) for all

as an R-module. n > 0.

(2.11)

Let Q be a nilpotent group and R a principal ideal domain.

Suppose that A is an RQ-module which is R-divisible with finite total rank as an R-module.

If H° (Q,A) = 0, then Hn(Q,A) = 0 = Hn(Q,A) for all

n > 0.

Proof of (2.10).

We consider only cohomology.

enough to show that H°(Q,A) = 0.

It will, in fact, be

For suppose that this has been proved;

then by (2.6) there is an r / 0 in R such that Hn(Q,A)r = 0. the exact sequence A > multiplication by r.

> A

since Hn(Q,A)r = 0.

>> A/Ar where the left mapping is

We obtain from the cohomology sequence

> Hn(Q,A)

0

Consider

> Hn(Q,A/Ar)

> ...

Now A/Ar is a bounded R-module with finite rank, so

it is a direct sum of finitely many cyclic R-modules.

Hence A/Ar is R-

noetherian and (2.3) may be applied to give H (Q,A/Ar) = 0, and the desired result follows. It remains to show that A

= 0.

Suppose that this is false

and A is a counterexample with minimal rank. zC i 1 from Z(Q/C). Then Ker 6 ^ 0

Let C = CQ(A) and choose

Denote the RQ-endomorphism a \

and A0 has smaller rank than A.

T/A6 be the R-torsion submodule of A/A6. smaller rank than A.

> a(z - 1) by 6.

Thus (A6)^ = 0.

Let

Then (A/Tp = 0 since A/T has

Consequently A^ is a torsion module, which can only

mean that A^ = 0.

Proof of (2.11). that Ho(Q,A) = 0.

We consider only cohomology.

Suppose we have proved

Then by (2.7) there is an r / 0 in R such that

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H (Q,A)r = 0. A[r] >

Applications of cohomology

59

Since A is R-divisible, there is an exact sequence

> A

>> A where A[r] = {a £ A|ar = 0}. Hence, there is an

exact sequence ••••+ Hn(Q,A[r])

> Hn(Q,A)

> 0.

Here A[r] is a finite direct sum of cyclic R-torsion modules, so it is R-artinian.

We deduce from (2.4) that Hn(Q,A[r]) = 0, whence Hn(0,A) = 0. It remains to prove that A^ = 0.

Assume this to be false and

let A be a counterexample with minimal total rank.

Let T be the R-

torsion-submodule of A; then T is R-divisible and hence is a direct summand of A. by (2.6).

It follows that (A/T)

Hence (A/T)

is R-bounded, either directly or

= 0 since A is R-divisible.

Choose z and 0 as in the preceding proof. T

£ 0.

Then T6 ^ T since

Therefore T6 has smaller total rank than A and (T/T o ) o = 0

where To = T n Ker 0. To, then D

If D denotes the maximal R-divisible subgroup of

= 0 because the total rank of D is less than that of A.

Finally, To/D is R-bounded, which shows that A

is R-bounded.

Therefore

V °Extensions with non-abelian "kernel,. To conclude this section we shall describe the classification of extensions with non-abelian kernel by elements of the second cohomology group. of N by Q which have a fixed coupling x element x

:

We shall consider extensions Q "* Out N.

By choosing an

from x A we obtain a function £ : Q -»• Aut N.

is an extension with coupling x*

Suppose that e_i

Then by choosing an appropriate

transversal function we can assume that e_i has an associated pair of functions (£,<j>i).

If e>2 is another extension with an associated pair of

functions (£,fa), let £ = fa - fa • Now equation (4) shows that (x,y)(f>i = (x,y)<j>2 for all x,y in Q.

Hence (x,y)c

E

A = Z(N). It follows

easily from the fact that fa and fa satisfy the factor set condition (3) that c does too.

Thus

C = fa -fae Z 2 (Q,A). Now assume t h a t ei and e_2 are e q u i v a l e n t . n o t a t i o n of ( 2 . 1 ) , we have (x)i|; = 1 and (x)t|; e A. fa - fa £ B2 (Q,A) and fa = i mod B2 (Q,A) .

Then, in the

Therefore

Conversely t h i s congruence

obviously implies the equivalence of e_i and e_2 . Finally, suppose that e_i is a fixed extension with coupling X and that £ in Z2 (Q,A) is allowed to vary.

Then e_(£,i + c) is an

Robinson:

Applications of cohomology

extension with coupling x-

60

The discussion above shows that as

C + B2 (Q,A) varies over H 2 (Q,A), we obtain a complete set of inequivalent extensions. All of this is summarized in the following form.

(2.12)

Let x

:

Q •* Out N be a given homomorphism.

be a function such that x (i)

Let £ : Q -> Aut N

€= x^.

Every extension of N by Q with coupling x has an associated pair of

functions of the form (£>) . (ii)

If ei , e_2 are two such extensions with associated pairs of

functions {KA\) and (£,<j>2), then 2 - <)>i £ Z2 (Q,A) where A = Z(N) . ei and e_2 are equivalent if and only if 2 = cf>i mod B2 (Q,A) .

(iii) (iv)

Let ei be fixed.

Then as £ + B2 (Q,A) t;ar^es otter H2 (Q,A) , the

extensions e_(£,i + c) /ozw? a complete set of inequivalent extensions of N by Q wit/z coupling x«

Consequently there is a bisection between the set of equivalence classes of extensions of N by Q u-£t/z coupling x ^ / sue/z extensions exist) and the group H 2 (Q,Z(N)). Of course, there is no guarantee that any particular coupling will be realized by an extension.

Indeed it is at this point

that the third cohomology group becomes relevant. 3

It can be shown that

X gives rise to an element of H (Q,Z(N)), and that x extension precisely when this element is 0.

3.

is

realized by an

For details see [18], §51.

APPLICATIONS OF SPLITTING AND NEAR SPLITTING In this section it will be shown how the splitting and near

splitting theorems of §2 can be made to yield structural information about infinite soluble groups. I.

Nilpotent supplements

(3.1)

Let G be a group satisfying the minimal condition on normal sub-

groups.

Assume that N < G and that N and G/N are both nilpotent.

there is a nilpotent subgroup X such that G = XN.

Then

If in addition N is

eccentric (i.e.s no G-chief factors of N are central in G)^ then X n N = 1. Proof. f

if G/N

In proving the first part we may suppose that N is abelian. f

1

= (X/N')(N/N ) with X/N

nilpotent, then G = XN and X is nil-

potent by a well-known theorem of P. Hall (see [25], §2.2).

For,

Robinson:

Applications of cohomology

61

With N abelian now, we denote by H the intersection of N with the hypercentre of G.

Put G = G/H and N" = N/H.

Then C-(G) = 1.

Hence

H2 ("G/N",N) = 0 by (2.4) and there is a subgroup Y such that G = YN and Y n N = H.

Now Y/H is a nilpotent group with min-n, so it satisfies min.

Also H satisfies min by a theorem of Baer (see [25], 5.22).

Consequently

Y is a hypercentral group with min, so it is a Cernikov group. Let F be the finite residual of Y. integer i such that E = [F^Y] = [F* i+1 Y] < N. divisible abelian group with min.

Now by min there is an Also E, like F, is a

Since Y/E is nilpotent and E = [E,Y],

we can apply (2.5), (2.7) and equation (7) to show that Y has a subgroup X such that Y = XE and X n E is finite.

Hence G = YN = XEN = XN.

Since

X/X n E * Y/E, the subgroup X is nilpotent. In the second part we assume that N is eccentric.

A simple

induction on j shows that [x n zi+1 (N),.XN] < (Y j+1 (X) n z i + 1 (N))Z i (N). (Here y. -(X) is the (j + l)th term of the lower central series of X.) Since X is nilpotent and G = XN, we have [X n Z.+1(N),rG] < Zi(N) for some r. N.

Thus (X n Z

(N))Z. (N)/Z. (N) is a G-hypercentral factor in

By hypothesis we have X n Z

(N) = X n z (N) for all i.

Since N is

nilpotent, it follows that X n N = 1.

The second supplementation theorem involves a well-known class of infinite soluble groups first considered by Mal'cev [23]. A soluble group with finite total rank is a group which has a series of finite length with abelian factors such that the sum of the p-ranks of the factors taken over all factors and all p = 0 or a prime is finite.

(3.2)

(Zaicev [42], Lennox and Robinson [19]).

Let N <3 G and assume

that N is a soluble group with finite total rank and G/N is nilpotent by finite.

Then there is a nilpotent subgroup X such that |G : XN| is

finite. For example, (3.2) may be applied with N equal to the Fitting subgroup of a soluble group of finite total rank; this is because Mal'cev has shown that such a group is nilpotent by abelian by finite ([23]).

Robinson:

Proof of (3.2).

Applications of cohomology

62

By (1.4) there is a characteristic series with abelian

factors 1 = No < Ni < • • • < N k = N such that each infinite factor has finite rank and is either torsion-free or divisible.

Let k > 0 and proceed by induction on k.

Thus there is a

nilpotent subgroup Y/Ni such that | G : Y N | < °°. Consider first the case where Ni is torsion-free. a positive integer r for which [Nt , Y]/|X , considerations of rank show. group.

There is

Y] is torsion, as

Let A = [Ni , Y ] ; then A/[A,Y] is a torsion

Since Y/A is nilpotent, the final paragraph of the proof of (2.10)

may be applied to give C (Y) = 1 .

It follows from (2.8) that Y nearly

splits over A. Now assume that Ni is divisible.

Since Ni has min, there is

a positive integer r such that A = [Ni , Y] = [Ni , be applied to make Y nearly split over A.

Y ] . Thus (2.9) may

Of course, should Ni be

finite, this statement is trivially true (for any r ) . The foregoing discussion demonstrates the existence of a subgroup Yi of Y such that |Y : -YI A| < « and | Yi n A | < «: for some r.

here A = [Ni , r Y]

Thus Yi is finite by nilpotent, which implies that Yi has a

nilpotent subgroup of finite index, say X.

Then |YiA : XA| < °°, so that

|Y : XA| < - and \c : XN| < «.

In general it is not possible to make X n N = 1.

For more

on this see [19].

II.

Finitely generated soluble groups of finite rank We begin by recalling the definitions of certain classes of

infinite soluble groups.

A soluble group has finite abelian section rank

if each abelian section (i.e., quotient of a subgroup) has finite p-rank for all p = 0 or a prime. Much narrower is the class of soluble minimax groups.

Here

by a minimax group we mean a group with a series whose factors satisfy max or min.

Soluble minimax groups are, therefore, precisely the poly-

(cyclic or quasicyclic) groups.

The spectrum sp(G) of a soluble minimax

group G is the finite set of primes p for which G has a factor of type p°°. Q

A standard example of an abelian minimax group is the additive group of all rationals whose denominators are ir-numbers where IT is a finite

set of primes.

Of course, sp(Q ) = ir.

Robinson:

Applications of cohomology

63

Fig. 1 lists the various classes of soluble groups which, in some sense, have "finite rank".

Here all groups are soluble.

Our object is to prove (3.3)

(Robinson [26])

A finitely generated soluble group G which has

finite abelian section rank is a minimax group. Thus in. Fig. 1 classes (2)-(5) coincide for finitely generated groups. Proof of (3.3).

Let d be the derived length of G.

Of course, when d < 1,

there is nothing to prove; let d > 1 and proceed by induction on d. Writing A = G

, we have that G/A is minimax.

Evidently it suffices

to deal separately with the two cases A torsion-free and A torsion. If A is torsion-free, the proof is straightforward.

For G

is a soluble group with finite total rank; thus by Mal'cev's theorem it is nilpotent by abelian by finite.

If N/Nf is a minimax group, then so

is N by a well-known principle (see [25], §2.2). N is abelian.

A theorem of P

Suppose therefore that

Hall [13] tells us that G satisfies max-n.

Hence the torsion-subgroup of N is finite and we can assume that N is torsion-free.

Let r be the rank of N.

The group G/C-^N) may now be identified with a finitely generated subgroup of GL(r,Q) and hence of GL(r,Q ) where TT is a finite set of primes.

But this means that there is a finite subset S of N such

that the normal closure S of S in G is a minimax group and N/S is torsion.

By max-n this quotient is actually finite and so N is a minimax

group, whence so is G. We turn now to the harder situation where A is a torsion group.

The problem is to exclude the possibility that infinitely many

primes are involved in N.

It is at this point that near splitting is

decisive.

Fig. X

min o<

finite torsion-free rank

(1)

finite abelian section rank

(2)

finite Prufer rank

(3)

finite total rank

(4)

minimax

(5)

>o max

Robinson:

Applications of cohomology

64

Suppose for the moment that G nearly splits over A, and thus |G : XA| < « and |x n A | < «> for some subgroup X.

Then XA is finitely

generated, which implies that XA = < X,Ao> where Ao is a certain finite subgroup of A.

Now the normal closure Ao of Ao in XA surely has finite

exponent, and since r (A) < °° for all p, it follows that Ao is finite. However, X is certainly minimax, whence XA = XAo is minimax and consequently G is minimax. To complete the proof we need another near splitting theorem.

(3.4)

Let Q be a soluble minimax. group and let A be a Q-module. Assume

that A is a (Z-)torsion group and r (A) < <*> for all primes p.

Assume

further that A has no non-trivial ^-elements for any p in sp(Q).

Then

n

H (Q,A) is a torsion group with finite ^p-rank for all p and all n > 0. First observe that (3.4) is sufficient to give the required near splitting.

For in (3.3) we can without loss factor out the sp(G/A)-

component of A since sp(G/A) is finite.

Proof of (3.4). Q.

There is a series 1 = Qo < Qi < • • • <3 Q

/Q. cyclic or quasicyclic.

= Q with

Consider first of all the case m < 1.

If Q is cyclic, there are well-known formulae which exhibit H (Q,A) as a factor of A; the result follows immediately. Suppose that Q is of type p°°. We claim that Hn(Q,A) = 0 if n > 0.

This can be seen in various ways.

Perhaps the simplest is to

observe that H (Qo,A) = 0 for n > 0 and Qo any finite subgroup of Q, by (1.5).

An easy cocycle argument (see [26], Lemma 4.1) then establishes

that H (Q,A) = 0 if n > 1. operates trivially on A.

Now since Aut A is residually finite, Q

Thus H1 (Q,A) = Hom(Q,A) = 0 since p e sp(Q).

This settles the case m < 1.

Let m > 1 and put N = 0

^.

Then H^(N,A) is a torsion group with finite p-rank for all p by induction on m.

Also it has no non-trivial p-elements for any p £ sp(Q).

because the mapping a i

> pa is an automorphism of A and therefore

induces an automorphism of H^(N,A). P

This is

By the case m = 1 we see that

q

H (Q/N,H (N,A)) is a torsion group with finite p-rank for all p.

The

theorem is now a direct consequence of the Lyndon-Hochschild-Serre spectral sequence.

For the convenience of the reader we include a brief

description of this.

Note on the Lyndon-Hoohschild-Serre spectral sequence. N >

> Q

Let

>> P be a group extension and let A be a Q-module.

The LHS-

Robinson:

Applications of cohomology

65

spectral sequence relates the cohomology group H (Q,A) to the mixed cohomology groups H^fPjH^fNjA)) where p + q = n.

There is a series in

Hn(Q,A) 0 = H(0) < H(l) < ••• < H(n + 1} = Hn(Q,A) wherein H(i + 1)/H(i) is isomorphic with a certain factor of H

(P,H (N,A)).

An account of this spectral sequence can be found in

[22].

III.

Nearly maximal subgroups Finally we shall mention without going into full details

another recent application of near splitting.

A nearly maximal subgroup

of a group is a subgroup which is maximal with respect to being of infinite index.

For example, in a finitely generated group each sub-

group of infinite index lies inside a nearly maximal subgroup. It is not difficult to see that in a finite by nilpotent group every nearly maximal subgroup is normal (and so has infinite cyclic quotient group).

The following theorem represents a partial converse of

this statement.

(3.5)

(Lennox and Robinson [20]).

by finite group.

Let G be a finitely generated soluble

If each nearly maximal subgroup has finitely many

conjugates * then G is finite by nilpotent. Thus the finite by nilpotent groups can be recognized among finitely generated soluble by finite groups by the behaviour of their nearly maximal subgroups. Sketch of the proof of (3.5).

We can assume that all proper quotient

groups of G, but not G itself, are finite by nilpotent. Fitting subgroup.

Let A be the

One then uses results of P. Hall [14] to reduce to the

following situation. (i)

A is abelian and is either torsion-free or an elementary abelian

p-group. (ii) R-rank.

Also Q = G/A is nilpotent.

For R = 7L or 2 < t > , the R-module A is R-torsion-free with finite (Here the action of t on A arises from conjugation by an element

in the centre of G/C (A)), (iii)

CA(G) = 1.

(iv)

A is R-rationally irreducible. Given these facts we can apply (2.8) to find a subgroup X

such that \G : XA| < °° and X n A = 1.

There is no loss in supposing that

Robinson:

G = XA.

Applications of cohomology

Then X is nearly maximal in G.

66

Thus \G : Np(X)| < ~ and it

follows that [Ao ,X] = 1 for some Ao with |A : Ao | < °°.

By (iii) this

implies that A = 1, a contradiction.

4.

AUTOMORPHISMS OF GROUP EXTENSIONS We shall consider the problem of constructing an automorphism

of a group extension from automorphisms of the kernel and quotient group of the extension. Consider an extension e_ : N >

> G

>> Q where for

convenience we regard N as a subgroup of G and Q is identified with G/N. The automorphism group of the extension e_ is defined to be the group of automorphisms of G that leave N invariant, that is, Aut e_ = {y € Aut G | N Y = N}. Each y i n Aut e_ induces automorphisms v in N and K in 0.

So

there is a homomorphism 0 : Aut e_ in which yi

> (V,K).

> Aut N x Aut Q An element (V,K) of Aut N x Aut 0 is called

inducible if there is a y G Aut e_ which induces v and K in N and 0 respectively, that is, if (V,K) 6 Im 0.

Obviously a pair (V,K) is

inducible precisely when it is possible to find an endomorphism y of G which makes the diagram below commute: N >——> G —

•i

»

}y

N >——> G —

The kernel of 0.

»

Q

I0

If y e Ker 0, then [y,N,G] = 1 = [N,G,y], whence

[G,y,N] = 1 by the Three Subgroup Lemma; hence [G,y] < A = Z(N). It is easy to verify that the mapping xN • 6

from Q to A.

x i

> x(xN)

y i

> 6

> [x,y] is a well-defined derivation

Conversely, given 6 £ Der(Q,A), the assignment

is an automorphism y. of G which belongs to Ker 0.

and 6 I

Finally

> y. are inverse homomorphisms.

These considerations demonstrate that Ker 0 - D e r ( Q , A ) , so that there is an exact sequence 0

The image of 0.

> Der(Q,A)

> Aut e_ -^—> Aut N x Aut Q.

It is more complicated to describe the image of 0, that

is the set of all inducible pairs.

Robinson:

Applications of cohomology

67

Let T : Q -> G be a transversal function for e_ giving rise to associated functions £ : 0 -»- Aut N and : Q x Q -> N as described in §2. Let v G Aut N and K G Aut Q and suppose that (V,K) is induced by Y e Aut e_. Now T' = K^xy is another transversal function and as such it will give rise to a pair of associated functions £' and c|>f , which we proceed to compute.

Applying y to the equation a

x^' = v" 1 (x K

= x

ax , we obtain

)^v.

(10)

Similarly, starting from x T y T = (xy) (x,y) we find that

(x,yH' = ((xK~\yK"l)40v. Now x T

(ID

= xT(x)i(; for some $ : Q -> N, and x^

= x^(x)if> by (5);

thus we obtain from (10) x^v = vxK^(xK)i|;.

(12)

Also by (6) and (11) we have ((x,y)(j))v = -(x K y K )^ + (xK,yK)(f> + (xK)i^-yK^ + (y K )^. These equations hold for all x,y E Q.

(13)

(Recall that F denotes conjugation

by b in N.)

(4.1)

Let e_ : N >

> G

>> Q be an extension and let E, : Q + Aut N

and <j) : Q x Q -* N be the associated functions arising from a transversal function T.

If v € Aut N and K £ Aut Q, then the pair (V,K) is inducible

if and only if there is a function \\) : Q •> N such that (12) and (13) hold. Proof.

To prove sufficiency we verify that the assignment x T ai

> (xK)T.(xK)i|/-(av), (x e Q, a e N ) ,

is an endomorphism y of G and, in addition, that y induces v in N and K in Q.

All of this is quite routine.

It follows that y is an automorphism.

Thus, once a transversal function has been fixed, there is a bijection between Aut e_ and the set of triples (V,K,I|;) satisfying (12) and (13).

Compatible pairs.

The complexity of the conditions for inducibility makes

them difficult to use directly.

There is, however, a reformulation of the

conditions which is sometimes more convenient.

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Applications of cohomology

68

Define a left action of Aut Q and a right action of Aut N on the set of functions from Q x Q to N by the rules (x,y)K(|> = (xK,yK)c|>

and

(x,y)<|>v = ((x,y)cj>)v

where x,y £ Q, K e Aut Q and v e Aut N. by v in Aut N.

Also define v to be conjugation

Suppose that (£,<J>) is an associated pair of functions for

an extension e_. Applying v and v to (3) and (4) respectively, we see that (£v,<j)v) is also an associated pair of functions for an extension. Furthermore, on replacing (x,y) by (xK ,y K) in (3) and (4) we find that (K£,K4>) is also a pair of associated functions for an extension.

The

conditions (12) and (13) merely assert that these new extensions are equivalent (see (2.1)). Hence we may reformulate (4.1) as

(4.1)*

Let (£,<|>) be an associated pair of functions for

e_ : N >

> G

»

Q.

If v e Aut N and K e Aut Q, then the pair (V,K)

is inducible if and only if the extensions e_(£v,) are equivalent. If X • Q "*• Out N is the coupling of the extension e_, then (12) implies that (14)

XV = K X where \) is now conjugation by v in Out N. on the choice of (£,<(>), but only on x* X if (14) is satisfied.

This condition does not depend

Call (V,K) a compatible pair for

The set of all compatible pairs for x is a

subgroup of Aut N x Aut Q which we shall write Comp(x)•

Therefore,

Im 0 < Comp(x) < Aut N x Aut Q.

A mapping from Comp(x) to H2 (Q,Z(N)) , extension with coupling x a compatible pair for x« for e.

an

Let e_ : N >

> G

d write A for the centre of N.

»

Q be an

Let (V,K) be

Choose an associated pair of functions (£,<|>)

Then E,\> = K S mod(Inn N ) , so we can write

for some function \|j : Q -> N. 1

equivalent to eJ^N),^ ) where

We see from (2.1) that e_(K£,K) is

Robinson:

Applications of cohomology

69

Now there is also an extension e_(£v,<|>v) . It follows from (2.12) that <j)V - (f)1 € Z2 (Q,A) . We may therefore define a function A : Comp(x) •* H 2 (Q,A) by writing ( V , K ) A = (<|>v - (()') + B 2 (Q,A).

Now £(£v,(f>v) and e_(K£,K(|>) are equivalent if and only if £(£v,<J>v) and £(£v,<j)') are equivalent. f

By (2.12) again this happens

2

precisely when <|>v - cf> € B (Q,A), that is, when (V,K) e Ker A. by (4.1)* Ker A = Im 0.

Therefore

Notice that the mapping A is not a homomorphism.

However Ker A has its usual meaning. This completes the proof of the fundamental result on the automorphism group of a group extension. (4.2)

(Wells [41]).

coupling x«

Let £ : N >

> G

» Q be an extension with

Let k be the centre of N regarded as a Q-module via x* Then

there is an exact sequence 0

> Aut e_ — >

> Der(Q,A)

Comp(x) — >

H2 (Q,A) .

We proceed next to examine the Wells sequence in some important special cases. Extensions with abelian kernel. e^ : A >

> G

Consider an extension

» Q where A is abelian:

let it have associated

functions £ = x and <j>. The condition for a compatible pair (\>,K) to be inducible is mod B2 (Q,A), because in this case we may take $ f to be K<|>. Writing A for <j> + B2 (Q,A), the cohomology class of e_, we may express this as Av = K A . A right action of Aut A x Aut Q on H2 (Q,A) may be defined by writing A ( V , K ) = K" 1 (AV) = (K" 1 A)v.

Then

Hence we obtain (4.3) Let e : A >

> G

» Q be an extension with abelian kernel.

Let x and A be the coupling and cohomology class of e_ respectively. there is an exact sequence

Then

Robinson: 0 _ >

Applications of cohomology

Der(Q,A)

— > Aut e — >

70

C ^ ^ C A ) — > 1.

If the extension £ is central, then x = 0 and the compatibility condition is vacuous. 0 _ _ > Hom(Q ab ,A) — >

Thus the above sequence becomes Aut e — >

CAut ^

Q (A)

— >

1.

This exact sequence has been used to investigate groups with finite automorphism group ([29]).

Extensions with injective coupling. extension e^ : N >

> G

Another important case is when the

>> Q has injective coupling x« It is easy to

see that this happens precisely when C f (N) < N, that is C (N) = Z(N) = A. In this situation we consider the mapping 01 : Aut e^

> Aut N

which is formed by restricting automorphisms to N. If y £ Ker 0i , then Y induces a pair (1,K) with K e Aut Q. By the compatibility condition (14) we have x = KX> which implies that K = 1 since x is injective. Therefore, Ker 0i = Ker 0 « Der(Q,A) . Now consider the image of 0i. This contains v if and only if there is a K such that (V,K) is inducible.

The compatibility condition

X\> = <x simply asserts that v normalizes Q X and induces K in Q. Thus

(4.4) Let e_ : N >

> G

» Q be an extension with coupling \. Assume

"G 1 sequences 0

> Der(Q,A)

0

> H1 (Q,A)

^> Aut e_

> N A u t N (Q X )

^> H2 (Q,A)

and > Out e

> N Q u t N (Q X )/Q X

> H2 (Q,A)

where Out e^ = Aut e/Inn G. Proof.

Only the second sequence requires comment.

are the natural ones. terms.

Here all the mappings

It is easy to check exactness at the last three

To establish exactness at H1 (Q,A) one has to show that

6 £ Der(Q,A) corresponds to an inner automorphism Y * precisely when

Robinson:

6 e Inn(Q,A).

If y

c

(xN)

Applications of cohomology

71

is conjugation by g G G, then g e C (N) = A and 0

(.3

= [x,g] for all x in G.

Hence 6 e Inn(Q,A); the converse is clear.

Finally (4.3) and (4.4) may be combined in the following convenient form. V G

N

Aut A ^ ^' x

Let e : A >

then v induces

> G

an

Q be such that A = C n (A) .

If

automorphism in Q through conjugation

call this automorphism v ! .

in Q :

»

If A is the cohomology class of e, we

define

A(QX)-

This is clearly a subgroup of N

(4.5)

Let e_ : A >

> G

»

Q be an extension with abelian kernel and

infective coupling x (that is^ A = C~(A),J. of e_.

Let A be the cohomology class

Then there is an exact sequence 0

5.

> H1 (Q,A)

> Out e_

> N ^

A (Q

X

)/Q X

^> 1.

CONSTRUCTING OUTER AUTOMORPHISMS The object of the final section is to show how the Wells

sequence may be used to construct outer automorphisms of groups.

I.

Automorphisms of free abelianized extensions

Let Q be a finite group and let R >

> F

presentation of Q where F is a non-cyclic free group. and R = R/Rf.

»

0 be a

Write F = F/R1

We shall be interested in the extension

£ : R>

>F

» Q.

In fact such extensions £ are precisely the free objects in the category of extensions by Q with abelian kernel. module.

Of course, R is the relation

It is well-known that Cp-(R) = R (Auslander and Lyndon [1]) and

it follows easily that Ff = Fit(F), the Fitting subgroup of F.

Thus

Aut £ = Aut F and we may hope to apply (4.5) to determine the structure of Out F.

However, first it is necessary to compute some cohomology

groups.

(5.1) Proof.

With the above notation H1 (Q,R) = 0 and H2 (Q,R) = 2 q where q = |o| Let I,, denote the augmentation ideal of the group ring 7LY\.

Recall that there is an exact sequence of Q-modules

Robinson:

Applications of cohomology

R > — > I F /I F I R — »

IQ

(15)

in which the left hand mapping is rRf » hand mapping is induced by IT.

72

> (r - 1) + IpIR

an

d the right-

Moreover the Q-module I^/I I is free (for r

r K

these facts see [12], §3) . Now obviously (IQ) = 0, while Hn(Q,M) = 0 for any free module M if n > 0. Therefore, the cohomology sequence yields 0 > H1 (Q,R) > 0 > H1 (Q,IQ) > H2 (Q,R) Hence H1 (Q,R) = 0 and H2 (Q,R) - H1 (Q,I Q ). sequence to the exact sequence I~ > 0

> (2ZQ)Q

Now apply the cohomology

> TL 0 > H1 (Q,I Q )

> TL

> 0.

»

2 to get > 0,

which shows that H1 (Q,IQ) - ZZ/qZZ. Let A be the cohomology class of the extension e^ : R >

> F

>> Q. Then, in fact, A generates the cyclic group

H2 (Q,R) : this rests on the freeness of the extension e_. Let v e NA

-5-(Q) where Q has been identified with a subgroup of Aut R by

means of the coupling. N

Then av = v'a for all a £ H2 (Q,R).

Hence

—(Q) is identical with 2 KAut R^ n-CQ) = {v e NAut -(Q) I ov = v'a, (Q,R)K J J A R^J ' ' V o ^ H vsc>

(5.2) Let R >

> F

» Q be a presentation of a finite group Q where

F is a non-cyclic free group. Let R = R/R1 and ¥ = F/Rf. Then

This follows directly from (4.5) and (5.1).

Of course it

already indicates that an automorphism of F that acts trivially on R is inner. Out F contains a congruence subgroup. In fact the group Out F is always infinite, as we shall now prove.

Let A = Z(F) = C^-(Q) . If F has rank r,

there is an exact sequence of Q-modules R >

>M

» I

(16)

in which M is free with r generators (and R < M ) ; this is essentially (15). r.

Since 1^ = 0, we conclude that A is a free abelian group of rank

Define K(r,q) to be the kernel of the canonical homomorphism

Robinson: Aut A

Applications of cohomology

73

> Aut(A/Aq) .

So K(r,q) is essentially the congruence subgroup of GL(r,Z£) modulo q. (5.3) Let Q be a finite group of order q and let R >

> F

» Q be a

free abelianized extension with F a free group of rank r > 1. Then Out F has a normal subgroup which is isomorphio with K(r,q). Proof.

I am grateful to Dr. P.J. Webb for several useful ideas in

connection with this proof. Let N be the noriral subgroup of Aut F C

Aut F«>

n C

Aut ? W A »

n C

Aut F ^ A " ) •

If y G N, denote by yo the automorphism induced in A by y. y i

> y 0 is a homomorphism 1 : N

> K(r,q).

Now [R,y,F] = 1 = [y,F,"R], S O [[F,R"],y] = 1. I x of ZZQ. xGQ

The mapping

Suppose that y € Ker I. Let a denote the element

Then

r q E r a mod[F,"R],

(r e R) .

Clearly r° e Z(F) = A, so that R^ < A[F",R] . Since y acts trivially on A and [F,lf|, it acts trivially on R^ and hence on R". N O W H 1 (Q,R) = 0 by (5.1).

Therefore (4.2) implies that y is an inner automorphism of F.

Hence Ker £ < N n Inn F. Conversely any element of N n Inn F acts trivially on A = Z(F) and thus belongs to Ker E.

Therefore

Ker I = N n Inn F, and I m Z = N(Inn F)/Inn F < Out F. To complete the proof we shall show that Z is surjective. Choose any element of K(r,q); this will have the form 1 + qot where a € End A. As before write a for £ x. Then we see from (16) that xGQ Ma < R, so that Ma < A. Consequently v = 1 + aa is a Q-endomorphism of M. Now v operates trivially on M/A, while it operates on A like the automorphism 1 + qa. Hence v is a Q-automorphism of M , and so of R. We wish to show next that v G N°

-^-(Q) . Since v centralizes

Q, this amounts to proving that v acts trivially on H 2 (0,R)• proof of (5.1) we found a natural isomorphism H 2 (Q,R) * H l (Q,I n )

But in the

Robinson:

Applications of cohomology

74

and it is obvious that v acts trivially on H1 ( Q , I Q ) • It follows from (5.2) that v extends to an automorphism y of F which belongs to N; for, of course, y acts on A like 1 + qa.

II.

Outer automorphisms of finite p-groups

As a second application we shall prove a generalization of Gaschutz's celebrated theorem on the existence of outer automorphisms of finite p-groups.

(5.4)

(Gaschiitz

[ 1 1 ] , P . Schmid [ 3 2 ] , U.H.M. Webb [ 4 0 ] ) .

finite

non-abelian ip-group^ then C~

If

G is

a

(Z(G)) contains elements of order

PProof.

Supposing the theorem to be false, we choose G to be a counter-

example of smallest order.

If every proper subgroup of G is abelian, it

is well-known that G is a group of one of the following types (Redei [24]): (i) (ii)

a quaternion group of order 8, m n , n-1 >, (m > 0, n > 1 ) , < x,y|xp = 1 = yP , y x = y 1 + P

(iii)

< x,y,z|z = [x,y], 1 = x P

m

n = yP

= z P = [x,z] = [y,z] >, (m > 0,

n > 0) . However, it is simple to verify that the theorem is true for these groups. Consequently there must exist a non-abelian maximal subgroup M.

Then

M < G and |G : M| = p. Suppose that Z(G) 5C M.

Then G = MZ(G) and M n Z(G) = Z(M>.

By minimality of G there is an outer p-automorphism of M operating trivially on Z(M). But this automorphism can be extended to G by allowing it to act trivially on Z(G). By this contradiction Z(G) < M. Now write C for C n (M). b

If C £ M, then G = CM.

C/C n M - G/M, a cyclic group, and C n M < Z(C). C < Z(G), a contradiction.

But

Hence C is abelian and

Therefore C_(M) = Z(M). b

Next suppose that Z(M) = Z(G). Let z G Z(G) have order p. Write G = < g,M>. Then the assignments g 1

> gz and a 1

> a, (a € M) ,

determine a non-trivial p-automorphism of G that operates trivially on M and hence on Z(G) .

This must be inner, induced by some x in G.

x € C (M) = Z(M) = Z(G), which is impossible.

But then

Therefore Z(G) < Z(M).

Having disposed of these preliminary reductions, let us apply (4.4) to the extension £ : M >

> G

»

Q = G/M.

Now H1 (Q,Z(M)) is a

p-group by (1.5) and derivations from Q to Z(M) give rise to p-auto-

Robinson:

Applications of cohomology

75

morphisms acting trivially on M; therefore, we must have H1 (Q,Z(M)) = 0. Now Q is cyclic here.

The formulae for the cohomology of cyclic groups

yield H2 (Q,Z(M)) = 0. Therefore (4.4) gives Out

£. ~- N O u t *<•<*>'*>

where Q has been identified with a subgroup of Out M via the coupling. This isomorphism tells us that any automorphism of M that normalizes Q will extend to an automorphism of G. By hypothesis the group L = CL Sylow p-subgroups.

(Z(M)) has non-trivial

Now Q acts on L by conjugation, and by Sylow's

Theorem it must fix a Sylow p-subgroup of L. a non-trivial p-element of L, say v(Inn M ) .

Hence Q must centralize Then v extends to a p-

automorphism of G acting trivially on Z(M) and hence on Z(G). This v must be inner, induced by some element of CNM.

However, this implies that

Z(M) = Z(G), a contradiction.

Remark.

Notice that only cohomology of cyclic groups of order p appeared

in the proof.

A totally cohomology-free proof has been given by

U.H.M. Webb [40].

Outer automorphisms of niVpotent groups.

It follows at once from (5.4)

that if G is a finite nilpotent group of order greater than 2, then Out G j- 1.

There has been some recent work attempting to extend this

result to infinite nilpotent groups.

We give a brief account of what is

known. Zalesskii [43] has shown that every infinite niVpotent torsion group has outer automorphisms*

This is basically an easier

theorem than (5.4) since cardinality arguments are often sufficient to demonstrate the existence of outer automorphisms.

However, it is still

an open question whether an infinite nilpotent p-group always has outer p-automorphisms; this is known to be true when the group is countable. For more information on automorphisms of nilpotent p-groups see Buckley and Wiegold [5]. On the other hand, Zalesskii [44] has given an example of a torsion-free nilpotent group with class 2 and rank 3 that has no outer automorphisms.

In the positive direction the analogue of (5.4) for

finitely generated torsion-free nilpotent groups is true; however, this is a much easier result to prove (cf. [38], p. 85).

Robinson:

Applications of cohomology

76

Finally we mention a well-known and apparently very difficult problem:

does every finitely generated infinite nilpotent group possess

an outer automorphism?

This quickly reduces to the case where the Hirsch

length equals 1, but then the difficulties beginwhen the torsion-subgroup has class < 2.

The conjecture is true

For more on this subject see

U.H.M. Webb [39].

III.

Comp lete groups

Recall that a group G is complete if it has trivial centre and trivial outer automorphism group.

We shall show how the exact

sequences of §4 may be combined with splitting and near splitting theorems in §2 to provide significant information about the structure of nonperfect complete groups. Let G be a complete group and suppose that N is a normal subgroup such that C f (N) = Z(N) = A.

Also let Q = G/N be identified with

a subgroup of Out N and regard A as a Q-module.

Then, of course,

H°(Q,A) * CA(G) = Z(G) = 1,

while i t follows from (4.4) that H1(Q,A) = 0. We are therefore led to enquire about the consequences of the conditions H°(Q,A) = 0 = rfCQjA). Ideally one would like H2 (Q,A) to vanish as well:

for (4.4) could then

be applied to give N Q u t N (Q) = 0. This conclusion can be drawn provided that 0 is nilpotent and A has finite total rank.

The basic theorem, which is reminiscent of

results in §2, now follows.

(5.5)

(cf. Robinson [30]).

ideal domain.

Let 0 be a nilpotent group and R a principal

Suppose that A is an ^-module

as an R-module.

which has finite total rank

1

If H°(Q,A) = 0 = H (Q,A), then Hn(Q,A) = 0 = Hn(Q,A)

for all n > 0. Remarks.

(i)

If Q and A are both finite, it is unnecessary to assume

that H° (Q,A) = 0.

The result is essentially due to Gaschiitz in this

case [10] . (ii)

Theorem (5.5) remains true if in the hyptotheses H° and H1 are

replaced by Ho and Hi .

Robinson:

Proof of (5.5).

Applications of cohomology

As usual we discuss only cohomology.

torsion submodule of A. artinian R-module.

> A

»

Let T be the R-

Then, by the assumption about rank, T must be an

Of course, H°(Q,T) - T^ = 0, so (2.4) implies that

H (Q>T) = 0 for all n. T >

77

Applying the cohomology sequence to

A/T, we obtain

0 = A0

> (A/T) Q

> 0

Hence H°(Q,A/T) = 0 = H1 (Q,A/T). theorem for A/T.

> H1 (Q,A)

> H1 (Q,A/T)

> 0.

It therefore suffices to prove the

Assume from now on that A is R-torsion-free.

Because A

= 0, it follows that A Q is a bounded R-module

(either by (2.6) or a direct argument).

If A Q were in fact 0, the theorem

would be an immediate consequence of (2.10). in R for which the p-component of A

Suppose that p is a prime

is non-zero.

Then (A/pA)~ ^ 0.

However, if we apply the cohomology sequence to the exact sequence A >-*-—> A

>> A/pA where the left-hand mapping is multiplication

by p, there results (A/pA)^ = 0. by (2.4) or a direct argument.

This implies that (A/pA) Q = 0, either Thus we have reached a contradiction.

This theorem leads directly to a criterion for completeness.

(5.6)

Let N O G .

Assume that Cr(N) = Z(N) E A has finite total rank

and that Q = G/N is nilpotent. by e. (i)

Denote the extension N >

> G

>> Q

Then Z(G) = 1 = Out e^ if and only if Q is a Carter (i.e.3 self--normalizing nilpotent) subgroup of Out N,

and (ii)

CA(Q) = 1 and A = [A,Q].

Proof.

Assume that Out e = 1 = Z(G). Then H°(Q,A) = 0 = H1 (Q,A) by

(4.4).

Hence H2 (Q,A) = 0 by (5.5).

Q = NQut

N(Q).

It now follows from (4.4) that

Note that Ho(Q,A) = 0 by (5.5); thus A = [A,Q].

Conversely, assume that conditions (i) and (ii) are valid. Then combining (2.4) and (2.10), we see that Hn(Q,A) = 0 for all n. can now apply (4.4) together with (J = L

Classifying complete groups.

We

J Q ) to obtain Out e_ = 1.

For example, if G is a finite complete

abelian by nilpotent group, we can apply (5.6) with N a maximal normal abelian subgroup of G such that Q = G/N is nilpotent.

The problem is

then to describe the Carter subgroups of automorphism groups of finite abelian groups.

This is accomplished in Gagen [8] where it is shown that

these are usually the Sylow 2-normalizers.

This leads to a satisfactory

Robinson: Applications of cohomology

78

classification of finite complete abelian by nilpotent groups. A special case of some interest is when G is metabelian. It turns out that the finite complete metabelian groups of order > 2 are just the direct products of holomorphs of cyclic groups of distinct odd primary orders (Gagen and Robinson [9]). Another possible choice for N is L(Fit G) where L is the nilpotent residual of G. Since the two parts of condition (ii) are equivalent when A is finite, we obtain the following criterion for a finite group to be complete. (5.7) Let L be the nilpotent residual of a finite group G and put N = L(Fit G) . Then G is complete if and only if Q = G/N is a Carter subgroup of Out N and Q acts without fixed points on Z(N). Finite complete metanilpotent groups have been investigated by Hartley and Robinson [15]; here the N in (5.6) is taken to be the Fitting subgroup. Although the structure of such groups can be very complicated, there is a theory of their classification and construction. For more on the recent theory of complete groups see [31].

Robinson:

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79

BIBLIOGRAPHY

1.

14.

M. Auslander § R.C. Lyndon, Commutator subgroups of free groups, Amer. J. Math. (2) 77_ (1955), 929-931. R. Baer, Automorphismen von Erweiterungsgruppen, Hermann, Paris (1935). J.F. Bowers § S.E. Stonehewer, A theorem of Mal'cev on periodic subgroups of soluble groups, Bull. London Math. Soa. 5_ (1973), 323-324. K.S. Brown § E. Dror, The Artin-Rees property and homology, Israel J. Math. 22_ (1975), 93-109. J. Buckley § J. Wiegold, On the number of outer automorphisms of an infinite nilpotent p-group, Arah. Math. (Basel), 3j_ (1978), 321-328. W.G. Dwyer, Vanishing homology over nilpotent groups, Proo. Amer. Math. Soo. 4£ (1975), 8-12. S. Eilenberg $ S. MacLane, Cohomology theory in abstract groups I, II, Ann. Math. (2) £8 (1947), 51-78, 326-341. T.M. Gagen, Some finite solvable groups with no outer automorphisms, J. Algebra 6£ (1980), 84-94. T.M. Gagen § D.J.S. Robinson, Finite metabelian groups with no outer automorphisms, Arah. Math. (Basel), 32_ (1979), 417-423. W. Gaschiitz, Kohomologische Trivialitaten und aussere Automorphismen von p-Gruppen, Math. Z. 88_ (1965), 432-433. W. Gaschiitz, Nichtabelsche p-Gruppen besitzen aussere p-Automorphismen, J. Algebra 4_ (1966), 1-2. K.W. Gruenberg, Cohomological topics in group theory, Lecture Notes in Mathematics, Vol. 143, Springer, Berlin (1970). P. Hall, Finiteness conditions for soluble groups, Proo. London Math. Soo. (3) 4_ (1954), 419-436. P. Hall, On the finiteness of certain soluble groups, Proo. London

15.

Math. Soo. (3) 9^ (1959), 595-622. B. Hartley § D.J.S. Robinson, On finite complete groups, Arch. Math.

2. 3.

4. 5.

6. 7. 8. 9. 10. 11. 12. 13.

16. 17. 18. 19. 20. 21. 22. 23.

24. 25. 26.

(Basel) 35_ (1980), 67-74. P.J. Hilton § U. Stammbach, A course in homological algebra, Springer, New York (1970). 0. Holder, Bildung zusammengesetzer Gruppen, Math. Ann. 46_ (1895), 321-422. A.G. Kuros, The theory of groups, 2nd. ed., 2 Vols., Chelsea, New York (1960). J.C. Lennox § D.J.S. Robinson, Soluble products of nilpotent groups, Rend. Sem. Mat. Unii). Padova 62^ (1980), 261-280. J.C. Lennox § D.J.S. Robinson, Nearly maximal subgroups of finitely generated soluble groups, Arch. Math. (Basel), to appear. S. MacLane, Cohomology theory in abstract groups III, Ann. Math. (2_) 5£ (1949) , 736-761. S. MacLane, Homology, Springer, Berlin (1967). A.I. Mal'cev, On certain classes of infinite soluble groups, Mat. Sb. 2_8 (1951), 567-588 = Amer. Math. Translations (2) 2_ (1956), 1-21. L. Redei, Das schiefe Produkt in der Gruppentheorie, Comm. Math. Helv. 20_ (1947), 225-264. D.J.S. Robinson, Finiteness conditions and generalized soluble groups, 2 Vols., Springer, Berlin (1972). D.J.S. Robinson, On the cohomology of soluble groups of finite rank, J. Pure Appl. Algebra £ (1975), 155-164.

Robinson:

27. 28. 29.

30. 31.

32. 33. 34. 35.

36.

37. 38. 39. 40. 41. 42. 43.

44.

Applications of cohomology

80

D.J.S. Robinson, Splitting theorems for infinite groups, Symposia Math. l]_ (1976), 441-470. D.J.S. Robinson, The vanishing of certain homology and cohomology groups, J. Pure Appl. Algebra 1_ (1976), 145-167. D.J.S. Robinson, A contribution to the theory of groups with finitely many automorphisms, Proc. London Math. Soo. (3) 35_ (1977), 34-54. D.J.S. Robinson, Infinite soluble groups with no outer automorphisms, Rend. Sent. Mat. Univ. Padova 62_ (1980), 281-294. D.J.S. Robinson, Recent results on finite complete groups, in Algebra* Carbondale 1980* Lecture Notes in Mathematics, Vol. 848, Springer, Berlin (1981), 178-185. P. Schmid, Normal p-subgroups in the group of outer automorphisms of a finite p-group, Math. Z. 147_ (1976), 271-277. 0. Schreier, Uber die Erweiterung von Gruppen I, Monatsh. Math. Phys. 34_ (1926), 165-180. 0. Schreier, Uber die Erweiterung von Gruppen II, Abh. Math. Sem. Univ. Hamburg 4_ (1926), 321-346. I. Schur, Uber die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen, J. reine angew. Math. 127 (1904), 20-50. I. Schur, Untersuchungen Uber die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen, J. reine angew. Math. 132 (1907), 85-137. U. Stammbach, Homology in group theory* Lecture Notes in Mathematics, Vol. 359, Springer, Berlin (1973). S.E. Stonehewer, Automorphisms of locally nilpotent FC-groups Math. Z. JAS (1976), 85-88. U.H.M. Webb, Outer automorphisms of some finitely generated nilpotent groups I, J. London Math. Soo. (2_) 2_1_ (1980), 216-224. U.H.M. Webb, An elementary proof of Gaschiitz's theorem, Arch. Math. (Basel) 35_ (1980), 23-26. C. Wells, Automorphisms of group extensions* Trans. Amer. Math. Soo. 155 (1971), 189-194. D.I. Zaicev, Soluble groups of finite rank* Algebra i Logika j ^ (1977)^ 300-312 = Algebra and Logic 16_ (1977), 199-207. A.E. Zalesskii, A nilpotent p-group has an outer automorphism, Dokl. Akad. Nauk. SSSR 196_ (1971), 751-754 = Soviet Math. Dokl, \2_ (1971)^ 227-230. A.E. Zalesskii, An example of a torsion-free nilpotent group having no outer automorphisms, Mat. Zametki \\_ (1972), 21-26 = Math. Notes U_ (1972), 16-19.

81 GROUPS WITH EXPONENT FOUR

S.J. Tobin University College, Galway, Ireland

INTRODUCTION These notes represent the lectures which I gave in St. Andrews under the title "Burnside groups of exponent 4".

Some details which I had

to suppress in the talks as time grew short have survived in these notes. Apart from that and some improvements mentioned below, they correspond pretty well to the ground which I tried to cover. The idea of the course was to survey the present state of knowledge about groups with exponent four, since several outstanding problems have been settled in the last five years - but I wished to provide some insight into these developments by giving some guidance to the history of these problems.

I hoped also to show that this is an

interesting and now accessible branch of group theory and so I have given perhaps more detail than usual at certain points to help clarify new ideas for the non-expert.

The course is self-contained up to a point; complete

proofs of many results are given - on the other hand I have not tried to compress long proofs but I have mentioned instead the ideas involved.

I

have given full references and attributions - correctly, I hope. The plan, corresponding to the five talks, is as follows.

§1

gives a brief resume of work on the Burnside problem and then proves those (few) general results on exponent four groups which were known before 1960. §§2 and 4 deal with the nilpotency and solvability questions, omitting Razmyslov's theorem which is touched on briefly in §5.

For the reader who

is acquainted with the subject the main interest is in §§3 and 5, where most of the results which I know are collected.

Several of these are new;

there are also some new proofs of previous results - and even some proofs of statements which I have never seen proved and was perhaps beginning to doubt.

I will not detail them here - in fact, any unattributed proof or

statement in §§3 and 5 is new (or at least is new to me). I should however mention two developments since the conference:

Theorem 3.4

replaces a much less satisfactory theorem which I gave in my talk, and

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Groups with exponent four

82

Theorem 5.4 confirms a couple of conjectures which I made at St. Andrews. I am happy to express my gratitude to the Mathematisches Institut of the Albert-Ludwigs-Universitat in Freiburg where this rewriting was done; and to University College Galway for the sabbatical leave which enabled me to do it.

1.

BURNSIDE AND EXPONENT FOUR

A)

Burnside groups Let F = F(n) be a free group of rank n and let F

group generated by all k t

powers in F(k > 2 ) .

invariant and the group B(k,n) = F/F exponent k (i.e. x

Then F

be the sub-

is fully

is an n-generator group with

= 1 for every x € B(k,n)) and every group which

satisfies these conditions is a homomorphic image of B(k,n). We use the letter B in B(k,n) to remind us of Burnside who drew attention to such groups in 1902 [8] when he mentioned "a still undecided point" in the theory of groups, asking whether (finitely generated) periodic groups are finite.

He suggested immediately that it

might be easier to consider groups in which the'periods are bounded and so (since the exponent is the l.c.m. of the periods) he asked in effect whether B(k,n) is finite and if so, he asked, what is its order? some answers:

He gave

B(k,l) is cyclic of order k; B(2,n) is elementary abelian

of order 2 n ; B(3,n) is finite with order < 32n~l; B(4,2) is finite with order < 2 1 2 (he claimed = 2 1 2 ) .

Burnside appears to have had no further

success with these groups, although in 1905 he proved that a finitely generated linear group which is finite-dimensional and has a finite exponent is finite. The "Burnside problem" has been the object of much work since then, as also has the "restricted Burnside problem" which asks whether or not there is a largest finite image of B(k,n), say B(k,n) - if so, every finite n-generator group with exponent k is a homomorphic image of B(k,n). At first the results were all positive.

The first success

after Burnside^s was in 1933 when Levi and van der Waerden [32] showed that B(3,n) has order 3

where t = n + \1\ +

!J and nilpotency class 3

(so that it is metabelian); and in 1940 Sanov [41] proved B(4,n) finite but his method gave an extremely high bound for the order.

Then in 1956

P. Hall and G. Higman [26] showed the existence of B(6,n) and gave its

Tobin:

Groups with exponent four

83

order as 2 a 3° where a = 1 + ( n - l ^ , c = b + m 2

+

+ \bJ , b = 1 + (n-l)2n

_ . Of course this is a solvable group, but even

more - its derived length is 3.

Kostrikin (1955) and Higman (1956)

established the existence of B(5,2) and B(5,n) respectively; Marshall Hall Jr. in 1958 [22] showed that B(6,n) is finite (and hence is B(6,n)) and (also in 1958) Kostrikin [30] determined the existence of B(p,n) for all primes p. This far things were going splendidly, although R.H. Bruck in his review of Hall's paper referred to a "heroic piece of calculation" and the same might well be said with reference to the Lie ring methods used for B(p,n).

Something new happened in 1964: Golod [12] produced

finitely generated infinite p-groups, thus answering Burnsidefs initial query - although his groups were not "Burnside" since the periods were not bounded.

Soon afterwards in 1968 (after an earlier false start)

Novikov and Adian showed that B(k,n) is infinite for n > 2 and k any odd integer > 4581; since then the lower bound for k has been reduced to 665 (see Adian [1]). Of course this leaves as many questions as it answers, and puts particular emphasis on the groups B(2 r ,2); thus the group B(8,2) is an object of considerable interest just now. The groups which satisfy the exponent law x

= 1 form a

("Burnside") variety B, , and the group B(k,n) is the (relatively) free group of rank n in this variety.

Thus many of the problems to be

discussed here could be given a varietal colouring - but I intend just to use the notation occasionally for convenience. if all the groups in B, are solvable.

B, is said to be solvable

The fact that B(k,n) has a derived

length which is independent of n when k = 2,3 and 6 means that B2 , B3 and Be are solvable (they have derived lengths 1, 2 and 3 respectively). During the decade 1970-80 Bachmuth and Mochizuki have shown that Bs is not solvable, but Razmyslov has disposed of the problem completely by showing that there are non-solvable groups of exponent k for k = 4,9 and any prime greater than 3; thus only B2 , B3 and Be are solvable. To round off this brief survey I mention two reduction theorems. Hall and Higman [26] showed the existence of a maximum finite solvable n-generator group of exponent pt

l

p2

2

... p

t

provided that such a group

exists for each of the distinct prime-power exponents p. -1 for every n. A much simpler argument is used by Gupta and Newman [18] to show that k k B(p ,n) is finite if and only if all groups of exponent p generated by finitely many elements of period p are finite.

Tobin:

Groups with exponent four

84

Brief accounts of the earlier results may be found in such texts as Hall [23], Coxeter and Moser [9], Magnus, Karrass and Solitar [34] or Robinson (Finiteness conditions in groups).

A nice paper by

M. Hall Jr. [24] deals carefully with the results known at that date. M.F. Newman [38] compiled a list of problems of Burnside type for the 1977 Bielefeld conference; furthermore he compiled a bibliography. There are two other general references I mention now.

There

is a growing volume of computer work on groups with exponent 4, and we shall need certain computed results later on, but the work itself is outside the scope of these talks.

The interested reader should consult

an article (also in the Bielefeld Proceedings) by Havas and Newman [27]. Secondly I wish to refer to the very fine monograph [15] written by Narain Gupta.

Apart from a wealth of results on commutator calculus in

metabelian groups, Engel groups and groups with exponent p, he gives a full and well-organized account of the work which led to the upper bound 3n-2 for the class of B(4,n).

B)

Commutators For convenience we tabulate here the standard results needed

for work with commutators.

They are rather few, and most are readily

verified (an exception is (v)(c) which requires some care).

Notation and definitions. [a,b] = a"1 a

In any group G we define a

= a ^ b ^ a b , Va,b G G.

= b

ab and

Simple (left-normed) commutators are

defined inductively [ai,a2,...,an+1] = [[ a i ,a 2 ,...,a n ], a ^ ] , n > 1. It is convenient to regard ai ,a2 , etc. as commutators of weight 1; then [ai ,. . . ,a ] is of weight 1 in each member of the entry set {ai ,. . . ,a } and of total weight n.

Weight is relative to the entry set:

thus if

a = [x,y] and b = [y,z,x] the commutator [a,b] is of weight 2 in the entry set {a,b} but relative to the set {x,y,z} it has weight 2 in x, 2 in y, and 1 in z giving a total weight 5. We will write [[a,b],[c,d]] as [a,b;c,d] and use left-norming, thus [a,b; c,d,e; u,v] is [[[a>b]> [c>d,e]],[u,v]] and so on. For subgroups A and B of G we define the subgroup [A,B] = < [a,b] | V a E A, b ^ B > i.e. the subgroup generated by all the commutators [a,b].

[A,B,C] means [[A,B],C] i.e. the subgroup

generated by all the elements [x,y] where x £ [A,B] and y £ C.

Tobin:

Groups with exponent four

85

We define two fully invariant descending series (or chains) in G as follows: lower central: derived:

Yi G = G, Y i+ 1 G = [ Y ^ G ] , i > 1;

D1 G = DG = [G,G], D 1+ 1 G = D(D 1 G), i > 1.

Thus y 2 G = DG, the commutator or derived subgroup of G, y 3 G = [G,G,G], Y 4 G = [G,G,G,G] and so on.

(Notations frequently encountered in the

literature (not in these notes) are: X

G. or r. (G) for y.G, <5.(G) or G^1J

2

for D G, G' for DG and G" for D G.) We say that G is nilpotent if 3t such that Y t G = 1; G has class c if c is the least integer such that Y

,G = 1; G is solvable if

3s such that D G = 1; G has derived length d if d is the least integer such that D G = 1. G is [n ->- k] if every n-generator subgroup has class at most k. Consequences of the definitions. (i)

If a 6 End(G) then [a,b] a = [a a ,b a ]; thus A a < A, B a < B =* [A,B] a <

[A,B].

In particular, if A and B are normal subgroups of G then so also

is [A,B], and in addition [A,B] < A n B in this case.

(ii) [x,y] = [ y ^ ] " 1 = [y^" 1 ] X = [y"1 , x ] y , xy = yx[x,y] and y"1 x = xtx^Jy"1 . (ii)' [A,B] = [B,A]. ( i i i ) [x,ab] = [x,b][x,a] b = [x,b][x,a][x,a,b], [ab,x] = [a,x] b [b,x] = [a,x][a,x,b][b,x]. These identities are in constant use. ( i i i ) ' If A,B,X are normal subgroups then [AB,X] = [A,X][B,X]. (iv) [a,b,c a ][c,a,b C ][b,c,a ] = 1: Witt identity. Note that by ( i i i ) [a,b,c a ] = [ a , b ; c , a ] [ a , b , c ] [ a , b , c ; c , a ] . (iv)f

Three subgroup lemma.

If A,B,C are normal subgroups then

[A,B,C] < [B,C,A][C,A.,B]. (v)

Generators of commutator groups.

Although [a,b,c] e [A,B,C] for all

a £ A, b £ B, c £ C these elements are not enough, in general, to generate the group [A,B,C].

But they are sufficient in one important case, namely

when the subgroups are normal. (a)

Let w(a,b,...,x) be a commutator form with entries a,b,...,x.

Then

if A,B,...,X are normal subgroups the group w(A,B,...,X) is normal and is generated by all the commutators w(a,b,...,x) with a £ A, b £ B, x € X. This lemma says, for example, that [G,G;G,G] (i.e. D 2 G) is generated by the elements [x,y;u,v] for all x,y,u,v in G. say that fxi ,x2 ;x3 ,XA ] is a generic word for D2G.

Note:

we will

Tobin:

(b)

Groups with exponent four

86

We can do b e t t e r than t h i s in some special cases.

generating set

If X i s a normal

for G ( i . e . x^ < X for a l l x e X, g G G) we can show that

D2G = < [xi ,x2 ;x3 ,X4] | Vx. e X > and similarly for DrG. (c)

Again i f Y i s any generating set for G, then ynG i s generated modulo

Yn+1G by simple commutators of weight n in elements from Y. (vi)

Properties

r , s > 1.

of the lower

central

series.

[ Y

G

> Y

G

] ^ Y

G, a l l

More generally, any commutator of weight n in group elements is v

contained in ynG. C)

In particular, D G < Ysj IK G .

Groups with exponent four:

the Sanov theorem

From now on we consider only the variety B4 for which we will write E; B(4,n) becomes B(n) and we write "G e B" or "G is a group with exponent 4" in a rather random fashion. the finiteness of B(n).

For completeness we establish

A proof quite different from Sanov's is offered

in [15]; we give the usual form here. Theorem ([41]). Proof.

B(n) is finite for all n.

B(l) has order 4; we induct on n.

We regard B(n) = < aj ,a2 ,. . . ,a >,

assumed finite, as embedded in B(n+1) = < a 1> a 2 ,... > a ,a A(n) = < B(n),a

2

> and then consider B(n+1) = < A(n) ,a

is proved if we can show the following:-

>.

Let

>. The theorem

in any group G e B, if a e G and

H is a finite subgroup such that a2 € H then < H,a > is finite. If w £ < H,a > we have w = hl ah2 ah3 . . .ah ah

.. where each

h. £ H ; if |H| = N we will show that when t > N the length of w can be reduced, and hence < H,a > is finite. The law g4 = 1 now gives aha = h""1 a"1 h"1 a"1 h"1 = h"1 ah*ah-1 where h* = a 2 h~ x a 2 £ H.

We use this to write several equivalent

representations for the word w, giving special attention to the h2-position. Thus hi ah2 ah3 ah4 ah5 a. .. = hi ah2 (h^1 ah^ah^1 )h4 ahs a. .. = hi ah2 ah3 (h^1 ahuahu1 )hs a.. . = hi ah2 (\u\\31 ahahAha"1 )hTahi"1 hs a. . . . We see that the second position may be occupied by h2 , h.2 haT1 hjluhs^hs"1 and so on.

When h.

will contain t - 1 elements of H.

(or possibly h l)

f i r s t appears, the l i s t

If now t = N+l, either

the l i s t contains

N d i s t i n c t elements including 1 and so the length was actually reduced in one of the moves, or there i s a repetition. h.-position can be eliminated:

In the second case a different

to see t h i s consider a simple example:

suppose h2h4h3'1 = h 2 h 4 h 6 h7 1 hf 1 }^ 1 .

This t e l l s us that hfih^hs"1 = 1; taking

Tobin:

87

Groups with exponent four

this in the form hshThe1 = 1 we see that by two changes as above on the right hand side of hs we could replace h5 by 1.

Corollary.

This concludes our proof.

B(n) is a 2-groupj nilpotent and solvable. Naturally values were sought for the order, the class and the

derived length of B(n). We can now (1981) answer the second two queries exactly and give a fairly good estimate for the order, see Fig. 1.

D)

Groups with exponent four:

the Tobin theorem

In the early nineteen fifties the solvability properties of the varieties B2 and B3 were known; for 06 there was a strong feeling that things would turn out similarly - which they did.

There was no reason to

suppose that groups of exponent 4 would be much different in this respect. I worked on them for some time, but they doggedly refused to become solvable.

I noticed that B(2) has derived length exactly three - it was

20 years later when a group was found with derived length greater than three.

However, I found an unexpected connection between solvability and

nilpotency, which I published in 1956 [43]. This was the only other general result apart from Sanov's, before the very fine theorems of Wright in 1960-61 opened up a new phase in the study of these groups.

I give the

theoream as it appeared in [43].

Theorem 1.1.

If a finitely generated group G has exponent 4 and ^-length X-2 X-2

X3 then the class of (G) is at least 2 " and at most 5 In the next few lines I went on to say that the novelty here was in the upper bound, and that the interest lay not in the bound so much as in the fact that it is independent of the number of generators of the group in question. Fig. 1

n

Here I was using <j>(G) for the Frattini subgroup of G;

B(n) class

2

5

3

3

7

3

4

10

4

n>2

3n- 2

order

derived length (d)

2d"1

< 3n-2 < 2 d

2

|B(n)

12

Determined 1954

26

9

1972-73

24

22

1975

< k.exp2 ((4+2/2)n)

1977-81

Tobin:

Groups with exponent four

88

I defined the Frattini series and the Frattini length exactly like the derived series and the derived length.

This is a good example of how not

to write a theorem; I was dealing with an important series of subgroups of G, but the fact that they were Frattini subgroups - with all the connotations of non-generators and so on - was totally irrelevant and concealed the real nature of the theorem.

The point is that for a 2-group

P (such as our finitely generated G) the subgroup (|>(P) is actually P2 . Furthermore if P is generated by elements of period 2 (as is our <(>(G)) then in fact <J>(P) = P2 = DP.

Hence if we decode the message in Theorem

1.1 we may write it plainly:

Theorem l.l' . For a finitely generated group G £ B

Y , ? (G2) < D ^ V ) . 5X+1 (The other part was merely the remark that if 1 / D 1 f y

2

_(G ).)

(G2) then certainly

We have here a result which (as I had indicated) is

2x-z

independent of the number of generators, and really has nothing to do with the finiteness of G. Theorem 1.2.

We thus get the interesting qualitative result:

Let G be any group of exponent 4.

Then G is solvable only

if G2 is nilpotent. Since G2 > DG we may write: Corollary.

For a group G of exponent 4 the following statements are

equivalent: G is solvable; G2 is nilpotent; DG is nilpotent. The crucial step in the proof of Theorem 1.1 (l.l' ) is the following: Lemma 1.3.

Let K be a group of exponent 4 with [K2,K2] = 1.

Let TLiK

be the group ring of K over the integers modulo 2 and let J be the ideal generated by the ring elements {(1+a)3 |a e K}. Then (1+x) (1+y) (1+u) (l+v)e j for all x,y,u and v in K2 . This is proved in §3.2 of [43]; this formulation is due to Gupta [15].

In the original paper the result is not formulated as a

lemma since it is used directly to prove Theorem 1.4 below.

Also in the

original the proof used a special notation which I thought helped conciseness.

It has been urged upon me that it does not help comprehension.

Although not convinced, I transcribe the original here in a more acceptable notation.

Tobin:

Proof of 1.3.

Groups with exponent four

89

Working modulo J we have (1+a)3 = l + a + a2 + a 3 2

2

3

Replace a by ax, where x £ K , giving 1 + ax + a [x,a] + a x x2 £ (K2)

=0

=0. (since

= 1, and axa2 [x,a] = a3x[x,a] = a3 x ) . Adding these equations

gives a(l+x) + a2 (1 + [x,a]) + a3 (l+xa) = 0, which gives (1+x) + a(l + [x,a]) + a 2 (l+x a ) = 0.

(*)

Repeating this procedure with a -> ay and finally a -+ az where y and z € K2 we get (l+z)(l + [y,a])(l+xa) = 0

Vx,y,z e K2 , Va e K.

We may replace x a by u, where u is any element of K2 . So (1+z)(1+u)(1 + [y,a]) = 0.

If we now multiply (*) by (1+z)(1+u) and use

the last relation we get (1+x) (1 + z) (1+u) + a2 (1+z) (l+u)(l+xa) = 0. Now (l+xa) = (1 + x[x,a]) = (1+x) + (1 + [x,a]) + (1+x)(1 + [x,a]) so we have (1+x) (1+z) (1+u) (1+a2) = 0. 2

Again (l+ai 2 a 2 2 ) = (1+ai2) + (l+a22) +

2

(1+ai ) (l+a2 ) etc., so we get finally (1+x)(1+z)(1+u)(1+w) = 0, Vx,z,u,w€ K2 .

To apply this result, let G ^ B abelian subgroup of G. automorphism of A.

and let A be a normal elementary

Then G acts on A by conjugation:

a -»• a

is an

Let CLA be the centralizer of A in G; then every element

of g = gCpA induces the same map a -> a g = ag say.

These maps lie in the

endomorphism ring of A which has characteristic 2; here gi + g2 is defined in the usual way.

Obviously also gi g2 = gi g2 .

If we consider the group

ring 2Z2 (G/CJV) and define gi +g2 to be gi +g2 we get a ring homomorphism into End(A).

Now 1 = (ag"1)

= a.ag.ag .ag

= a(l + g + g2 + g 3 )

means that (1+g)3 = 0 for all g e G i.e. if we let K = G/CGA then the ideal J in the lemma is in the kernel of the homomorphism. 2

2

2

2

the lemma provided [K ,K ] = 1 i.e. [G ,G ] < CGA.

If this is so, we have

(l+x)(l+y) (l+u)(l+v) = 0 for x,y,u,v € K2 , which means for all a £ A that [a,x,y,u,v] = 1 for x,y,u,v e G 2 ,

Now we can use

Tobin:

Groups with exponent four

and t h i s gives [A,G2 ,G2 ,G2 ,G2 ] = 1.

90

Indeed i f N i s any normal subgroup

of G we could provide such a s i t u a t i o n by p u t t i n g [G 2 ,G 2 ] in C^N and making N elementary abelian (by p u t t i n g N2 = 1 ) , hence as a consequence of 1.3 we can s a y : If G is in B and N < G then [N,G 2 ,G 2 ,G 2 ,G 2 ] < [G2,G2,N]N2 . To r e t u r n t o [ 4 3 ] , I applied Lemma 1.3 i n t h e simple case G e B, A = [G 2 ,G 2 ] and DA = 1, i . e . D2 (G2) = 1.

Then A2 = 1 and CfiA > A = [G2 ,G 2 ] so we have

immediately: Theorem 1.4.

Let G be any group of exponent 4.

Then

Y6(G2) < D 2 (G 2 ). The r e s u l t in 1.1 now followed, by using: Lemma 1.5.

Let G be any finitely

generated group with prime power

exponent p .

Suppose there exists

Y , (D G) < D

+

S+1

a positive

G for every non-negative

integer s such that

integer

t {here D°G means G) then

H+l

Y j G < D s +1

G for any positive

integer

d.

However, Edmunds and Gupta [11] noticed that the proof of Lemma 1.3 goes through p r a c t i c a l l y unchanged i f the group ring i s taken to be 2Z4K; the d e t a i l s are given in [15]. Thus i t i s only necessary to provide an abelian normal subgroup of G having [G2,G2] in i t s c e n t r a l i z e r ; hence they stated the following lemma in [11]: Lemma 1.6.

Let G ^ B and let N < G.

Then

[N,G 2 ,G 2 ,G J ,G 2 ] < [G2 ,GS ,N] [N,N] . I used this result to get a much better relation between the derived length and the class of G2 , in [44]: Theorem 1.7. If G is a group with exponent 4 D (G2) > Y t (G 2 ) where t = 1 + y(4 X -l). Proof.

Starting with N = [G2,G2] in 1.5, we get inductively

Y, + ([G2,G2]) > y

(G2) for s > 1. Now assume the theorem is true for

X = k (it is clearly true when X = 1). Then D k + 1 (G 2 ) = D k ([G 2 ,G 2 ]) but now [G2,G2] = (G 2 ) 2 so we may apply our hypothesis to get D ([G2,G2]) > Y ([G2,G2]) where n = 1 + -r<4 -1), but this group in turn contains Y (G2)

Tobin:

Groups with exponent four

91

fl k 1 1 k+1 where m = 2 + 4-j— (4 -1) > = 1 + — (4 -1) hence the theorem is true for all X > 1.

This is more direct than the deduction of 1.1 from 1.4.

At

the time it appeared, the old result 1.1 (or rather the qualitative version 1.2) was being used by Bachmuth, Mochizuki and others in an effort to establish unsolvability for B.

They were not successful, but

now that we know B is in fact non-solvable we can put our results here in reverse and make statements such as

Theorem 1.8.

Let F be a free group of infinite rank.

Then F2/F4 is not

nilpotent.

2.

COMMUTATOR LAWS IN GROUPS WITH EXPONENT FOUR Let G be any group with exponent 4.

In studying the nilpotency

of subgroups we examine relations among commutators in G; we can simplify matters greatly be considering only commutators of a given weight k or less, i.e. by working modulo y,

(G), which we shall sometimes (but not

consistently!) denote just by YT.+ 1 • We will consider the expansions of (ab)4 , (abc)4 and so on modulo y, +1 ; as did Wright in his pioneering papers on the nilpotency classes of groups in B.

His results were very good indeed; in [47] (1960)

he showed that an n-generator group with generators of period 2 has class < n+1, and conjectured that this result is best possible. confirmed by Gupta, Mochizuki and Weston [16] in 1974.

This was

In [48] (1961)

Wright proved that the class of B(n) is at most 3n-l if n > 1, and conjectured that this is "not too far from the true class". was extremely close:

His bound

Gupta and Newman improved it to 3n-2 for n > 2, and

we know now that this is in fact the true class. In this section I should like to sketch briefly the development of these results, giving some details where the arguments can be put concisely.

A)

Basic congruences We commence with the expansion of (ab)4 , or rather, following

Newell and Dark [37] we expand (xa"1)

using a"1 x = x[x,a]a~* . We work

modulo y 4 G, collecting x-terms to the left and a"1 terms to the right.

Tobin:

Groups with exponent four

92

This gives us 1 = (xa" 1 )

= x4 [x,a] [ x , a , a ] [ x , a , x ] a' 4 mod y4

thus [x,a]

E y4 for a l l x and a in G.

(*)

This is the simplest and also the most fundamental relation in groups in B.

It gives:

Theorem 2.1. Proof.

(Y k G) 2 < Y k + 2 G for

k >

2

'

The simple argument which is required here might normally be

summarized thus:

(*) is true, hence 2.1.

However I want to use it to

illustrate some principles which are essential in this work.

To start we

remark that if {u,v,...} is a set of generators for y^G, then (Y^G)2 is generated modulo [y,G,y.G] by the set {u 2 ,v 2 ,...}. we may take the generators of y,G

to be

For present purposes

the commutators [gt ,g2 ,. . . ,g,] ,

for arbitrary elements g. in G; since k > 2 we have

So to prove 2.1 i t i s enough to show t h a t the elements [gl ,g 2 , . . . >g,]2 are in y,

From (*) we have [ x , a ] 2 = "JT a), where each OJ. i s a commutator

OG.

X

of weight > 4 in G.

We don't know the OJ. , but clearly we could find values

for them if we carried out the full expansion for (xa"1) ; and clearly then each w. is a commutator in x and a, of positive weight in each. Furthermore this is a group law:

x and a may take any values in G.

Let

us then replace x by [aj ,a^ ,. .. ,a, .,] where the a. are arbitrary elements K—X

of G.

X

Since x appears at least once in OJ. the weight of OK increases by

at least k-2 and becomes > 4+k-2 = k+2. 2

[at ,a^ ,... ,a, , , a ] G Y k + 2 Comment.

G

Thus we have

and the theorem is proved.

Suppose that we were given only the relation (*) as a law in a

certain group G; this might very well reflect a law [x,a]2 = ~fT a), where other variables appeared in the commutators a). . Some a), might fail to have x as an entry, and the proof of 2.1 would fail. principle, described below, is very useful. derive another identity [x,a]2 = ~[~f v positive weight in x and in a. ab initio.

But here a simple

It tells us that we could

with each v

of weight > 4 and of

Thus we might as well assume this situation

The idea is due to G. Higman (see Tobin [43], also [25] and

Tobin:

Groups with exponent four

93

[15] for various versions).

Lemma 2.2.

Let en a2 .. .ar = bib2 . . .b

where en ,... ,a

are all of positive weight in each of a set of entry-

variables xi ,x2 ,.. . ,x . did2 ...d

be a group identity in a group G,

We may replace the product bib2 . . .b

by a product

where each d. is a commutator in the b .-elements and is of

positive weight in each x, . Proof.

Suppose that for some i there is a b. which does not have x i as

an entry.

We may move all such b. to the left using uv = vu[u,v] where u

and hence [u,v] has x. as an entry. b....b

C1C2...C

is a group identity: Ci c2 . . . c .

Now we have ai a2 ...a

=

where x. appears in each c-term and in no b-term. put x. = 1.

We get 1 = b....b

This

and so aia2...a

=

By carrying out this process at most n times we get en a2 . . .a

= d!d2...dt.

Corollary 1.

Suppose that 001 w2 ...OK E 1 mod y. nG is an identity where

each OK is a commutator of weight k with entries from a set of variables {xi ,...,x }, where r < k and at least one of the OJ. has positive weight in all the variables.

Then we may suppress any terms which do not involve

all r variables3 leaving an identity to -.-wQ E 1 mod y, ..G in which every variable x. appears in each commutator.

Since [u,v]2 £ y4 we have [u,v] = [v,u] mod Y4 ond

Corollary 2.

[u,v,xi ,. . . ,x, ] E [v,u,xi ,. .. ,x, ] mod Y 4 + u • Usually, however, these congruences will be expressed modulo Y3 and Y ? ^

res

P e c tively.

We note also that

2

[x,b ] E [x,b,b] mod Y 4 G.

(1)

Thus the commutator on the left, which has weight 2 in the entry set {x,b 2 }, has effectively weight 3 in the entry set {x,b}.

In general

[ai ,a2 ,. . .r,a ,b ,Ci ,. .s . ,c ] = [ai ,.. . ,a ,b,b,ci 3,. . . ,cx'*"b" ] t"o mod Y_. _ .-r^* (I 1 ) r A very clear tabulation of important congruences is given in [11] Lemma 1, also in [45]. In addition [11], Lemmas 2 and 3, gives a useful compilation of rearrangement congruences which allow us to regroup entries in certain commutators.

Our requirements are more modest, and we

now derive the relations which we need.

Tobin: Lemma 2.3.

Groups with exponent four

94

In any group G of exponent 4 the following congruences hold.

[x,a] 2 = [x,a,a,a] [x,a,a,x] [x,a,x,x] mod [Y2G,y3G] {and so mod ys G) .

(2)

Jacobi:

(3)

[a,b,c] [b,c,a] [c,a,b] = 1 mod [Y2G,y2G] {and so mod Y4G).

[x; a,b] = [x,a,b] [x,b,a] mod y4G.

(3')

[x,y,z 2 ] [y,z,x 2 ][z,x,y 2 ] [x,y;x,z][y,x;y,z] [z,x;z,y] = 1 mod y5G. (4) Wright: [x,a;b,c][x,b;c,a][x,c;a,b] = 1 mod y5G. (5)

Hall:

Wright:

[x,a;y,b;c][x,b;y,c;a][x,c;y,a;b] = 1 mody6G.

(6)

1

Proof. (2) comes by continuing the expansion of (xa" ) . (3) comes from the Witt identity, and is valid in all groups. (3' ) comes from (3) and Corollary 2 above. For (4) expansion of (xyz)2 gives (xyz)2 = x 2 y 2 z 2 [y,x] [z,y] [z,x]o) where u) ۥ y3G. Square again, work modulo ys G and note that since all commutators now occurring will have weight 4 we may ignore those which do not involve all three variables x, y and z. For (5) (xabc)2 = x2 a2b2 c2 [a,x] [b,x] [c,x] [b,a] [c,a] [c,b]w where u) e y3G, again as in (4). The expansion of (xaybc)4 , using (5) and (3), gives (6). These are the basic relations on which the examination of the lower central series depends. We will develop a few consequences which are needed for the general case and which are sufficient to determine the class of B(2) quite easily. Corollaries

to Lemma 2,3. [x,a,b,a 2 ] = [x,a2 ,b,a] modyeG.

(7)

In congruence (6) put y = a = c, giving [x,a;a,b;a] = 1 which by (3') becomes [ [x,a,a,b] [x,a,b,a] ,a] = 1 which gives (7). [x,b,b,b,a,a] = 1 = [x,b,b,a,a,a] mod yy.

(8)

In congruence (4) put y = b, z = a and replace x by [x,b], giving [x,b,b,a2 ] [x,b,a,b2 ] [x,b,b;a,b] [x,b,a;a,b] = 1 mod y6 E [x,b,b,a2]{[x,b,a,b2][x,b2,a,b]}[x,b2,b,a][x,b,a;a,b] E [x,b,b,a2 ] [x,b2 ,b,a] [x,b,a2 ,b] [x,b,a,b,a]. Replacing a here by a2 gives, modulo y 7 , [x,b2,b,a2] = 1 which is the first part of (8). Again, commutating with a gives

Tobin:

Groups with exponent four

95

[x,b,b,a 2 ,a] [x,b2 ,b,a,a] [x,b,a 2 ,b,a] [x,b,a,b,a 2 ] = 1. The product of the last two factors is = 1 by (7), and we have just seen that the second term is = 1 and so [x,b,b,a2,a] = 1 mod Y7 • Hall:

[x,y,z,a,a,a] = 1 = [x,y,a,a,a,z] mod Y7 •

(9)

2

In the congruence [x,b ,a,a,a] = 1 let b = yz and we get immediately [x,[y,z],a,a,a] = 1 which is [y,z,x,a,a,a] = 1.

Returning

now to (2) we see that [x,y,a]2 = [x,y,a,a,a] mod ye and so [x,y,a,a,a,z] = [[x,y,a]2,z] = [x,y,a,z]2 E [x,y,z,a]2 [x,y;a,z]2

= [x,y,z,a]2 =

[x,y,z,a,a,a] = 1.

B)

The group B(2) We can now prove:

Theorem 2.4. Proof.

The class of B(2) is at most 5.

For this it suffices to show that y6 (B(2)) = y7(B(2)) since B(2)

is nilpotent.

So we assume that B is a group of exponent 4 on generators

a,b with Y7 (B) = 1 and we wish to show that every simple commutator of weight 6 in a and b is trivial.

There are 2s formally different

commutators; the number is halved since [a,b,,,,] = [b,a,,,,] and halved again by interchanging a with b.

Hence there are only eight cases to

examine, namely the commutators of form [a,b,a,*,*,*].

Let us do so.

By

(9) [a,b,a,a,a,a] = 1, [a,b,a,a,a,b] = 1, [a,b,a,b,b,b] = 1. By (8) [a,b,a,a,b,b] (= [b,a,a,a,b,b]) = 1.

For the others we need (7)

which says (with x = b) that [a,b,a,b,a] E [a,b,b,a,a] mod Y6 • Hence [a,b,a,b,a,a] = [a,b,b,a,a,a] = 1; [a,b,a,b,a,b] = [a,b2,a2,b] = [a,b,a2,b2] = 1 above; [a,b,a,a,b,a] = [a,b,a,b,a,a] = 1 above; [a,b,a,b,b,a] = [a,b2,a,b,a] = [b,a,b,a,b,a] = 1 above. Thus B(2) has class at most 5.

This is in fact the precise class of B(2); it is a simple matter, knowing that Ye (B(2)) = 1, to derive the commutator structure of B(2) (see [25]) and to show that |B(2)| < 2 1 2 . reference that

Let us just note for future

Tobin:

Groups with exponent four

96

[b,a,a;b,a] = [b,a,a,a,a] = [b,a,a]2 (10)

[b,a,b;b,a] = [b,a,b,b,b] = [b,a,b]2 and these two elements generate YsB(2).

Incidentally, this shows that

YsB(2) = [YsB(2),YaB(2)] = (Y 3 B(2)) 2 . Burnside [8] claimed that |B(2)| = 2 1 2 ; he actually proved |B(2)| < 2

12

and there was some doubt about the true value.

B(2), which is of order 2

12

I constructed

, in my thesis [42]; I did not publish it but

the information was recorded in Coxeter and Moser [9]. They also gave the following presentation: (ab)

4

= (a-'b)

4

2

= (a b)

4

B(2) = < a,b > with defining relations a4 = b4 = = (ab 2 ) 4

2

subgroup < a ,b > has order 2

6

= (a^b^ab)4

= (a^bab) 4

=1.

The

6

and index 2 .

My work used an unsophisticated version of what is now termed a "power-commutator presentation" [27].

Gupta [15] gives a construction

for B(2) which exhibits clearly the commutator structure. be the free nilpotent of class 5 group on 2 generators.

Let G = gp<"a,b> We list the basic

commutators of weight up to 5: a,b; [b,a] = ci ; [b,a,a] = c2 , [b,a,b] = c3 ; [b,a,a,a] = o* , [b,a,a,b] = cs , [b,a,b,b] = ce ; [b,a,a,a,a] = c7 , [b,a,a,a,b] = c8 , [b,a,a,b,b] = c9 , [b,a,b,b,b] = ci o , [b,a,a;b,a] = Ci i , [b,a,b;b,a] = Ci 2 . Define Ni

= < C7

, Cs

,. . . , ci 2 , C7 Ci 1 , C9 ci 1 , Ci 0 ci 2 , Cs Ci 1 Ci 2 > ,

N2

= < Ni , C 4 2 , c s 2 , C 6 2 > , N 3

= < N2,C22C7,c32Cio

>,

N4 = < N3 ,Ci 2 C4C S c 6 Ci2 >, Ns = < N4 ,a4 ,b4 > We can readily verify that each N^ < G and is consistent in the sense that it does not interfere with the structure of N. -. of class exactly 5 with order 2

12

Let H = G/N5 . Now H is

and the only thing necessary is to check

that it has exponent four.

C)

The groups B(n), n > 2 Having discussed |B(2)| I should like to mention the remarkable

claim made by Wright in [48], where he says "a little direct computation shows that the order (of B(3)) is at most 2 7 0 " .

Remember that this was

in 1961 - the upper bound 2 7 2 was proved by M. Hall Jr. [25] in 1973; and in 1974 the precise order was finally given [6] as 2 6 9 .

Tobin:

Groups with exponent four

97

We now take a brief look at the nilpotency class problem for B(n).

The congruences (8) and (9) show that the occurrence of three

consecutive equal entries in a (simple) commutator of weight 6 or more has a strong impact.

The next step is to exploit this fact by developing

congruences which allow entries to be rearranged, so as to bring equal entries together or almost together.

This was the procedure in [48],

subsequently refined by Gupta and some of his co-workers; it leads to Wright's quadruple entry congruence.

Lemma 2.5.

Let G £ B, let k > 6, and in [xi ,x2 ,. . . ,x, ] let 4 (or wore)

of the entries xi ,. . . ,x, be the same. [Xl,x2,...,xk] e

Then

Yk+1G.

(This result has been strengthened recently:

see Lemma 5.7.) B

An immediate consequence is y~ +1 ( n ) ^ Yg + ? B ( n )> t n u s t n e class of B(n) is at most 3n. However, with a little extra work, and still exploiting the triple entry congruences, Wright was able to show: Theorem 2.6.

The class of B(n) is at most 3n-l.

(For n=2 of course this is exact.) For this last stage an alternative proof [15] includes the following: Lemma 2.7.

Let c be a commutator of weight 3n-l, having n-1 triple entries

x22 ,...,x ,. .. ,xn and one double entry xi ; and let n > 3. Then modulo c = 1 or c = v n = [x2 ,xi2 ,x3 ,x 2 2 ,X4 ,x 3 2

Corollary.

y 7 -is generated modulo y- by commutators of the form v . on— 1 on n It is clear now that y- must be generated by elements of the

form [v ,xi]; manipulation of this finally produces commutators all with consecutive triple entries and so [vn,Xi ] = 1. In 1974 Gupta and Newman published an improvement of the bound in Theorem 2.6, starting from the result ygB(3) = 1 just established on the computer by Bayes, Kautsky and Wamsley [6]. In fact they had calculated this result already by hand (as is shown in [15]).

The proof depends on

Lemma 2.7 and on a closer examination of certain congruences which we have previously seen.

To take an easy example, consider again the relation (8)

Tobin:

Groups with exponent four

98

in the form [x,y2,z2,z] = 1 mod Y7 • We regard this as an element in B(3) and we know that Y9B(3) = 1.

So [x,y2,z2,z] is a product of commutators

of weight 7 or 8 and each must have positive weight in x. replace x by [x,y]:

Now suppose we

all terms previously of weight 8 and those of weight

7 with more than one entry x now have weight 9 and are trivial.

But a term

of weight 7 with just one entry x must have y three times and the insertion of one more y sends the commutator into 1 by Lemma 2.5.

Hence

[x,y,y2 ,z 2 ,z] = 1

i s a law in groups with exponent 4.

The Jacobi i d e n t i t y gives [ x , y , y 2 ] =

2

[x,y ,y]a) where a) e ys , w = J J c. with c. a simple commutator of length i > 5.

Now [x,y,y 2 ,z 2 ,z] = [x,y 2 , y , z 2 ,z] [u>,z2 ,z] and [w,z2 ,z] = f j [ c ^ z 2 ,z]

= 1 by ( 9 ) .

x

Thus 2

2

[ x , y , y , z , z ] = 1.

(11)

Again, c l o s e r s c r u t i n y of congruences (4) and (5) gives [ x , y 2 , y , z 2 ] E [ x 2 , y , z 2 , y 2 ] mod y8 .

(12) 2

2

2

Now by Lemma 2.7 the generators of ysB(3) have the form [y,x ,z,y ,z ] which is [x 2 ,y,z,y\x 2 ] = [x2 ,y,z\y 2 ,z] = [x,y2 ,y,z2 ,z] = 1, by (7), (12) and (11). Hence ysB(3) = 1; this gives a basis for an induction which yields the final result ([17]).

Theorem 2.8.

3.

The class of B(n) is at most 3n-2, where n > 3.

COMMUTATOR STRUCTURE UNDER ADDITIONAL CONSTRAINTS We will now look at some commutator calculations of a simpler

kind - simpler, that is, because of additional conditions imposed on the exponent-four group.

The conditions will be of two kinds:

(A) a general

condition on commutators (as in metabelian groups) or (B) special constraints on the generators (as for instance in Wright's 1960 paper).

A)

Commutator Conditions A good opening gambit is to make the commutator subgroup

abelian, thus in a sense "linearizing" the calculations.

In 1967 Narain

Gupta and I published a joint paper [20] giving precise results for metabelian groups with exponent four:

Tobin:

Theorem 3.1.

(a)

Groups with exponent four

99

Let M(n) = B(n)/D 2 B(n), i.e. the free metabelian group

of exponent 4 on n generators.

The class of M(n) is n+2 if n = 2 or 3 and

is n+1 if n > 4. (b)

1 / G -£s any metabelian group of exponent four then (Y3G)2 = 1. Recently Newell and Dark [37] have shown that the results of

Theorem 3.1 hold under the weaker condition [Y2G,Y3G] = 1 .

Of course this

is not a weaker condition when G = B(2) because in any 2-generator group F the quotient y2 F/YS F is cyclic and so [Y2F,y2F] = [ Y 2 F , Y S F ] . with n > 2, then [Y2G,Y2G] i [Y2G,Y3G].

If G = B(n) ,

In fact Newell and Dark examine

a more general situation, and get the following results:

Theorem 3.2.

Let G be a finite group such that [Y2G,Y3G] = 1.

Let a be

an endomorphism of G such that a a = a4 on a subset T of G for which |T| > 15|G|/16.

Then

(i)

a0 = a4 for all a e G;

(ii)

G is a direct product of a 2-group* a 3-group which is nilpotent of

class at most 3 and an abelian group; (iii)

if G has n generators G is nilpotent of class at most n+2 if n = 2

or 3, and n+1 if n > 4; (iv)

if G is a 2-group then ( Y S G ) 2 = 1. Because of Theorem 3.1 the results here in (iii) and (iv) are

best possible.

The non-trivial a requires careful handling, but the

commutator calculations which yield (iii) and (iv) are essentially in an exponent-four context (here with induction on |G| available); the same crucial commutator identities which hold in the metabelian case are shown to hold here for the 2-group component, with of course the same consequences. We may compare this result, for trivial a, with a theorem in [20] which generalizes 3.1:

Theorem 3.3.

If H is a group with exponent 4 and if k is an integer such

that Y^H lies in the centre of Y2H then (i)

( Y k + 1 H ) 2 = 1, and

(ii)

if H is generated by n elements the class of H is at most n+k for

n > 2, and, is at most n+k-1 if n > max(4,k-l) . For k=2 this result is best possible, but 3.2 shows that an improvement is possible when k=3; and so the next theorem (which is new) is of interest in that it effects a substantial improvement for all k > 3.

Tobin:

Groups with exponent four

100

For k=3 it includes the results of 3.2 as applied to groups with exponent four, but without the restriction to finite groups.

The result for large

n ( > 2k-3) is particularly interesting.

Theorem 3.4.

Let H be a group of exponent four such that [YI,H,Y2H] = 1

for some integer k > 2.

Then

(i)

(Y k H) 2 = 1 if k > 3 {and consequently (y 3 H) 2 = 1 when k=2);

(ii)

if H is generated by n elements the class of H is bounded as follows: if k = 2 or 3 the class is at most n+1 if n > 4; and is at most n+2 if n = 2 or 3 {these bounds are all best possible);

£/ k > 3 the class is at most n+1 if n > 2k-3 (tHs found -£s fcest possible) and when n < 2k-3 the class is bounded by min(3n-2,n+k-l,2k-2).

Remark 1.

For k > 3 the result (i) is true under the weaker condition

[Y,H,YJH] = 1, as we show later (Theorem 5.4). This suggests the following consideration.

For each positive integer k let X(k) be the least positive

integer such that (Ywi^G) 2

^ [Y2G,y,G] for every group G of exponent four.

Then {X(k)} is a monotonic increasing sequence, and X(k) < k; for the initial values we have A(l) = 2 since for all G £ B we know (Y2G)2 < Y4G, the example of M(2) already quoted shows that A(2) = 3 hence also A(3) = 3, and the B(3) tables [6] show that X(4) = 4.

We can say more:

X(k) •> «

as k -* 00 because, as we will see later in connection with Theorems 5.4 and 5.10, for every positive integer s there exist groups X £ B such that (Y X ) 2 * y 2S 2S

X and hence (Y X ) 2 * [y X,Y - X] which shows that +1 2S 2 - 1

X(2 S+1 -1) > 2 s for every s.

Remark 2.

Let a(n,k) be the least upper (i.e. best possible) bound for the

class of n-generator groups H in B which satisfy [Y2H,YJH] = 1; we may take n > 1.

Then the double sequence {tf(n,k)} is monotonic increasing in each

argument; for fixed n it takes the constant value 3n-2 for k > 3n-3, and for fixed k each term cr(n,k) is > n+1 and becomes n+1 for n > 2k-3.

The

bound given in (ii) for k > 3, n < 2k-3 is not a(n,k) in general, but may be fairly close when n and k are small.

Thus for instance the upper bound

given for n=4, k=4 is 2k-2 = 6 and certainly a(4,4) > 5.

The B(3) tables

show readily enough that a(3,4) = 5 and a(3,5) = 6 whereas the bounds in (ii) are 6 and 7 respectively; the tables for B(4) [2] show (not so readily) that a(4,7) = 8 and a(4,8) = 9 while in each case the bound in

Tobin:

Groups with exponent four

101

(ii) is 10. Proof of Theorem 3.4.

We need two preliminary lemmas which are true in

all groups (in B or not).

The first is a simple technical one which we

will need again later. Lemma 3.5. In any group G a commutator of the form [uv,ai ,a2 ,. .. ,a ] may be expressed as the product of two commutators w1

u for suitable Proof.

w1

. ai

elements

w2 ,a2

w .,a n

nn r

-,

] [v,ai ,a2 ,... ,anJ

wi ,w2 , . . . , w

in G.

This comes simply by iteration of the fundamental commutator

relation [uv,ai] = [u,ai ] V [v,ai ] and so [uv,ai,a 2 ] = [ [u,ai ] V [v,ai ] ,a2 ] = [ [u,ai ]V,a2 ] *-v>ai -I [v,ai ,a2 ] and the pattern is clear. The second lemma concerns conditions under which we may permute certain entries in a (simple) commutator without changing the value. Lemma 3 . 6 .

Let k be a positive

integer

> 2 and let G be any group

such

that [Y2G,YkG] = 1. Then (i)

if c is a simple commutator in elements of G, of length at least k+2,

the entries after the k-th entry of c may be permuted in any way without affecting the value of c in G; (ii) if c is a simple commutator> of length at least 2k-1, the entries after the second entry of c may be permuted without affecting the value of c. Proof.

(i)

Let d e Y k (G); i f a,b E G we have [d,ab] = [d,baw] = [d,ba]

where w £ Y2G and expansion shows t h a t [ d , a , b ] = [ d , b , a ] . two consecutive e n t r i e s after

Hence in c , any

the k - t h entry may be interchanged; t h i s

gives the statement. (ii)

We may assume t h a t k > 3.

Let v e Y t G where 2 < t < k; by ( i ) we

may w r i t e [ v , a , b ] = y [ v , b , a ] for a r b i t r a r y a,b G G and for a s u i t a b l e yG

[Y2G,YtG].

Hence (with Xi , . . . , x R e G) [v,a,b,x 4 , . . . ,x^] =

u [ v , b , a , x i , . . . , x ] by Lemma 3.5 where for s u i t a b l e wi ,w2 , . . . , w have wx wt w u = [y ,xi , - - - , x n ]

G

[YtG,Y2G,nG].

in G we

Tobin:

Groups with exponent four

102

(Here we will use the notation [A,0G] = A, [A,(s+1)G] = [A,sG,G] which defines [A,nG] for all non-negative integers n, for any subgroup A of G.) Using the "three subgroup lemma" an easy induction gives n

i r t[Yt ( 5,iG],[Y2 G, (n-i)G]]

n

*

i=0

i=0

If t+2+n > 2k-1 then for each i here t+i > k or 2+n-i > k and so u=l above. Thus i f c i s a simple commutator of length at least 2k-1 any two consecutive entries after the second may be interchanged, and the statement follows. We continue now with the proof of Theorem 3.4. Let G £ B and let a,b €= G. We have 1 = (ab2 ) 4 = a2 [a,b 2 ]a2 [a,b 2 ] = a4 [a,b 2 ] [a,b2 ,a2][a,b2] giving the group law [a,b 2 ] 2 = [ a , b 2 , a 2 ] .

(1)

Replacing a here by ba and expanding we get, using y6B(2) = 1, [a,b 2 ] 2 = [ a ,b 2 ,(b 2 a 2 [a,b][a,b,a])] = [a,b2 , (b2 a2 [a,b]) ] = [a,b2,b2][a,b2,a2][a,b2;a,b]. But 1 = [a,b 4 ] = [a,b 2 ] 2 [a,b 2 ,b2 ] shows that [a,b 2 ,b 2 ] = [a,b 2 ,a 2 ]

(2)

and has period 2, by (1). Thus we get [a,b 2 ] 2 = [a,b 2 ;a,b] e [Y3G,y2G]

(3)

Now we are ready to prove part (i) of our theorem. If k > 4 we have (by Theorem 5.4) (YjG)2 < ^kG>YkG-' w ^ i c ^ i m P l i e s (i) f ° r k > 3. Suppose now that k=3 and l e t H = G/[Y2G,Y3G]. Then for any x,y e H we have, from (3), [x,y 2 ] 2 = 1.

(4)

Replace y here by yv, where v is an arbitrary element of H, and expand: this gives ([x,y 2 ][x,v 2 ] Wl [x,[v,y] ] 2 ) 2

= 1 for certain wi ,w2 € H, and the

commutators here are all in Y » H which is abelian so by (4) ([x,[v,y] ] 2 ) 2 = 1 which is equivalent to [v,y,z]2 = 1, Vv,y,z e H and so (Y3H)2 = 1. For part (ii) we continue initially with k=3 and n=2. 2

2

2

By (4),

2

(1) and (2) we have 1 = [x,y ,y ] = [x,y ,x ]; since commutators of weight 6 in x and y are trivial and [x,y]2 € Y4H we may rewrite the last equations as 1 = [x,y,y,y,y] = [x,y,y,x,x].

Tobin:

Groups with exponent four

103

In each of these commutators we may interchange the first two entries without altering the value, since [x,y] = [y,x]3 = [y,x] modulo Y4H; and by Lemma 3.6 (ii) we may permute the entries after the second.

Hence we

see that ys< x,y > = 1, and so for n=2, k=3 (hence also for k=2) we have established the upper bound for the class.

That this is best possible

stems from the examples constructed in [20], as does every best possible result claimed in the theorem.

For n=2, k > 3 the condition [y2H,y,H] = 1

is vacuous. Next we take k > 3, H = G / [ Y 2 G , Y J G ] with G ^ B , and we let w e y,

H, x G H.

Then by part (i) [w,x]2 = 1.

The full expansion for

[a,b]2 where a,b e G is [a,b]2 = [a,b,b,b][a,b,b,a][a,b,a,a][a,b,a;a,b]

so in H we have the result 1 = [w,x]2 = [w,x,x,x] and again by part (i) we may write this as 1 = [w,x,x 2 ].

(5)

Replacing x here by xy we get 1 = [w,xy,x2y2d] = [w,xy,x 2 y 2 ] since d G y2H. In the expansion note that, for example, [w,x,y,x 2 ] equals [w,x,x 2 ,y] by Lemma 3.6(i) and equals 1 by (5). We obtain 1 = [w,x,y 2 ][w,x 2 ,y]

(6)

which we may rewrite as [w,x,y,y] = [w,x,x,y].

(6' )

Finally, in (6) replace y by yv and expand, using [w^ 2 ,)^] = 1 (by (6)); this gives 1 = [w,y,v,x,x].

(7)

This is the basic commutator relation from which all else follows; taken in conjunction with (6' ) it shows that a simple commutator of length > k+3 with a repeated entry after the (k-l)th place equals 1. First we look at the special case k=3, n=3. must consider commutators of weight at least 6.

To apply (7) we

A simple commutator of

weight 6 in three variables must have a repeated entry in the last four places, therefore such a commutator must be 1.

This means that the class

of a 3-generator group H for which k=3 is at most 5, as claimed. We now consider groups H for which k=3, n > 4 or k > 3, n > 2k-3; if H is such a group we claim that Y assume that Y

H +3

= !•

H +2

=

*

anc

*

s0 we

ma

y

A simple commutator of length n+2 is of length

Tobin:

Groups with exponent four

104

> min(2k-l,k+3) and we may apply relation (7) and Lemma 3.6(ii).

Thus we

need to consider only commutators of the form w = [x,y,a,x,y,b,...,c] where the entries a,x,y,b,...,c are the n distinct generators of H.

But

now using the Jacobi relation [x,y,a] = [a,x,y][a,y,x] modulo Y4H, and so, using (7) again, w = [a,x,y,y,x,b,...,c][a,y,x,x,y,b,...,c] = 1. Hence y +Ai = 1, as claimed. Finally we must consider the case k > 3, 2 < n < 2k-3.

For

fixed k, the class of H is obviously bounded by the value for n = 2k-3, that is 2k-2.

Again, a simple commutator of length n+k in n generators

must have a repeated entry after the (k-l)th place and so is trivial by (7).

Thus the class of H is at most n+k-1; as a last resort we note that

for given n the class cannot exceed 3n-2.

This completes the proof of

Theorem 3.4.

Corollary 3.4.1. for some k > 2. Proof.

Let G be a group with exponent 4 such that [yiGyj^G] = 1 Then y3 (G2) = 1 if k=2, and Y k (G 2 ) =

For k = 2 or 3 consider [a 2 ,b 2 ,c 2 ].

H / k > 3 .

This is a commutator of

weight 6 in three variables hence by Theorem 3.4 its value is 1.

For

k > 4 we have various bounds for the class of an n-generator subgroup of G, but in all cases the class is at most n+k-1.

So if we consider

[ai 2 ,a22 ,. .. ,a, 2 ] we have a commutator in k variables which must be 1.

We may express the result for k=2 more vividly:

Corollary 3.4.2.

For all groups G of exponent 4

[G 2 ,G 2 ] > D2 (G) > [G2 ,G2 , G 2 ] . These subgroups are distinct in, for example, B(2). By way of contrast we note that in general D2 (G) does not contain [G 2 ,G 2 ,G]. We remark finally that the groups considered in this section A ) , i.e. G G B with ^ G ^ G ]

= 1, are necessarily solvable of length at

most 1 + Iog2k.

B)

Conditions on the Generators In 1960 Wright published the following result [47]:

Tobin:

Groups with exponent four

105

Theorem 3.7. A group of exponent four which is generated by n elements each of period two has class at most n+1 if n > 3; for n=2 the class is at most 2. It is clear that if G G B, x and y e G and x2 = y2 = 1 then 2 [x,y] = 1 and (by induction on the weight) every commutator in G with entries of period two is i t s e l f of period 2. Again 1 = [x,y 2 ] = [x,y] 2 [x,y,y] giving 1 = [x,y,y] = [y,x,y]. This establishes the result for n=2.

(8) Now in the Hall congruence §2(4)

replace x by [x,y] and put x2 = y2 = z2 = 1 .

Using §2(3) and (8) above

we get easily [x,y,z,y,z] = 1 modulo yeG and we also see that this is enough to establish the result for n=3.

This gives a basis for induction

for n > 3; for details I refer the reader to Gupta [15], where the organization of congruence procedures in groups from B has the effect that Theorem 3.7 is now an easy consequence of the general discussion of the class of B(n). The dihedral group of order eight shows that the bound given for n=2 is attained.

Wright conjectured that his result for n > 3 was

best possible; he was of course right, as was shown in 1974 by Gupta, Mochizuki and Weston [16]. We can however get a lower bound than Wright's if we make the stronger assumption that the generators are actually squares (Tobin 1975 [44]):

Theorem 3.8.

In any group with exponent four a subgroup generated by n

squares has class at most n. Proof,

This result is best possible for all n.

It is enough to show that, in a group G £ B, any simple commutator

in which all the entries are squares is trivial if the same entry occurs twice.

Consider [u2,...,v2,a2,x2,...,y2,a2]

where all the entries are squares. Since for any b,c £ G we have [b 2 ,c 2 ] = (b 2 c 2 ) 2 i t follows that [u2 ,. . . ,v2 ] = w2 for some w e G; and since [w 2 ,a 2 ] = [a2 ,w2 ] i t suffices to prove that [a2 ,xi 2 ,. . . ,x t 2 ,a2 ] = 1 for a l l t > 1. The case t=l (n=2) is clear, as in Theorem 3.7. For t > 1, the original proof depended on a group law derived from relations in the B(3) tables; but we can in fact write down a more suitable law, from which the result follows immediately:

Tobin:

Lemma 3.9.

Groups with exponent four

106

Groups of exponent four satisfy the laws [x2 ,y,ai ,a2 ,. .. >ag,x2 ] = 1 for all s > 1. I am indebted to M. Vaughan-Lee, who pointed this out to me

in St. Andrews.

I give here a direct proof in the spirit of the original

[44]. Examination of the B(3) tables [6] shows that in any G G B, for all x,y,a in G, 1 = [x 2 ,y,a,x 2 ]. a

s

by

a

s a s+l'

ex

Pand

anci

This is the case s=l; now replace

induct on s, and the result is clear.

The statement that the bound is best possible in Theorem 3.8 was not included in the 1975 statement of the theorem - with good reason. It is easily seen that the bound is not attained for some particular n=N if and only if 1 = [ai2 ,a 2 2 ,. . . ,a 2 ] is a law in every G e B, which is true if and only if G2 has class less than N, which in turn is true (see Theorem 1.2) if and only if G is solvable of derived length bounded by a fixed function of N.

So the question of the goodness of this bound for

all n in 3.8 was equivalent to the question of the solvability of groups with exponent four.

In the meantime however Razmyslov [40] has constructed

a non-solvable group in B; and so the bound is best possible. For the discussion of the next theorem we will need a consequence of 3.8 which we record as:

Corollary to 3.8.

In any group G ^ B

let the elements bi ,bi ,...,b ,...

be squares and let A be the subgroup < bi ,b2 ,.. . ,b t ,... >.

Then the

normal closure in A of each generator b^ is abelian. Proof.

For fixed i the normal closure of b. is generated by the set

{b. | x £ A } ; b.

can be expressed in terms of b. and [b. ,Ci ,c2 ,. . . , c ]

where s > 1 and the entries c. are all from the set {bi ,b 2 ,...}, and as a consequence of 3.8 these all commute in pairs.

We remark now that groups analogous to those in Theorems 3.7 and 3.8 but with an infinite set of generators are not always solvable (otherwise G e B => G2 solvable => G solvable). By the end of the 1960fs we knew a few important general facts about the variety B - but very few.

We knew that B(n) is a finite 2-group,

that its class is at most 3n-l and at least n+1 (or 5 if n = 2,3). This latter fact of course showed that B is not a nilpotent variety; on the other hand the groups {G2 |G e B} might turn out to be nilpotent and we knew that this is equivalent to the solvability of B.

The only concrete

Tobin:

Groups with exponent four

107

information available was that B(2) has class 5 and is not metabelian: thus 3 is a lower bound for the derived length of B(n), and it even seemed possible that it would prove to be the actual value. At this juncture Gupta and Weston (1971) ushered in the new decade with a paper [21] devoted in the main to the solvability problem. They established new criteria for the solvability of groups with exponent four; and they proved incidentally that third-Engel groups with exponent four are solvable. Their first theorem (3.10 below) was a very subtle observation at the time, and they proved it using Theorem 3.7. We will however use the Corollary to 3.8, and the result becomes almost obvious. Theorem 3.10. Let F be a free group of infinite countable rank3 let G = F/F4 and let {ai ,a2 ,a3 ,...} be the generators of G corresponding to a free basis for F. Let H be the group generated by a set {xi,x2,X3 ,...} satisfying only the following conditions and their consequences: for each i = 1,2,3,... (i)

x.2 = 1 ;

(ii)

the normal closure of x^ in H is abelian;

(iii)

[xi,h,h,h] = 1 for all h in H. Then G is solvable if and only if H is solvable.

Proof.

It is easily seen that (i) and (ii) together imply that (hx^)4 =

[x.,h,h,h]; since every element in H is of the form x. x. ...x. we see I

1

2

that H may be defined equivalently by (i), (ii) and (iii)*: all h in H.

t

h4 = 1 for

Thus H is a homomorphic image of G, and if G is solvable so is H.

Now the subgroup G2 is generated by the (countable) set of all squares

in G, and each of these generators has, by the Corollary to 3.8, abelian normal closure in G2.

Thus in fact G2 is a homomorphic image of H, and G2

contains D(G); thus if H is solvable so also is G. The next result seemed to offer another way to attack the solvability problem; we follow the ideas in [21] but work with G2 rather than with H: k~ Theorem 3.11. If for some integer k, with G as in 3.10, D G < y , G then 2+1 G is solvable with derived length at most k+1. Proof.

Let G* be the subgroup of G2 generated by the subset

{xi,x2,...,x ,} where x. = g.2 and g. is an arbitrary element of G, z^K

1

1

1

Tobin:

1 < i < 2 .

Groups with exponent four

Then y ,

108

G* = 1 since any simple commutator with repeated

2+1 entry must be 1 by Theorem 3.8. But G* is a homomorphic image of G, so k k D G* < y , G* = 1. In particular the generic commutator word for D G*

2+1 (see §1) with the entire list {xi,...,x 2 ,} entered (for instance, in that order) equals 1; but {g2 |g € G} is a normal generating set for G2 and so we have D (G2) = 1 giving D

+1

G = 1 as claimed.

In the light of Razmyslov's result, we may add:

Corollary.

k~ D G ^C y

G, for all integers k > 1.

2+1 Gupta and Weston went on to prove that if the full third-Engel condition is introduced into the group H the resulting quotient group, H say, is centre-by-metabelian, and this enabled them to show that a thirdEngel group of exponent four is solvable with derived length at most 5. We can do slightly better:

since as in the proof of 3.10 G2 is a homo-

morphic image of H we may state:

Lemma.

If G is a third-Engel group in B then (? is centre-by-metabelian

(and G has derived length at most A), This result may be found in [15], proved by a different method (using Lemma 4.3). However it has been superseded by Theorem 5.8 (VaughanLee 1979) which says that under these assumptions G itself is centre-bymetabelian.

Examples,

In [21] an example is given, due to C.K. Gupta, of a matrix

group which is a homomorphic image of the group H above and so is centreby-metabelian; it is not metabelian, in fact its derived length is precisely three, and it is not nilpotent.

I shall close this section with a brief

mention of some analogous groups, also due to C.K. Gupta, which are exhibited in [15]. These are groups of lower triangular nxn matrices over a commutative ring P, where n = 2, 3 or 4, and consequently they are solvable of derived length at most 3; and they lend themselves to straightforward calculation. Let F be a free group of countable rank on free generators xi ,x2 ,x3 ,... and let 7L2 be the field of integers modulo 2.

Let R be the

group ring of F/F2 over Z 2 and let P be the polynomial ring P = R[A] where

Tobin: A = {X. .

1 1,1-1

Groups with exponent four

109

| i - 2 , 3 , 4 , . . . ; k = 1 , 2 , 3 , . . . } i s a s e t of independent

associative and commuting variables. 1

<J>4 0 0

For each x, define

0

0

0

X«

x,

0

0

o

x£'

I

o

o and l e t 2 (x, ) , fa (x, ) be r e s p e c t i v e l y the leading 2-rowed submatrix and the leading 3-rowed submatrix in <J>4 (x, ) , where for w G F we w r i t e w_ = wF2 . For each t = 2 , 3 , 4 l e t * denote the m u l t i p l i c a t i v e group generated by {<j> (x, ) |k > 1 } ; then $ i s a homomorphic image of F under the mapping induced by <> j : x, «-> <j>f(x, ) , a l l k.

I t i s easy t o check t h a t these groups

have exponent four, and t h a t they are n o n - n i l p o t e n t (for i n s t a n c e an easy computation shows t h a t the (2,1) e n t r y of the matrix (j>4 ([xi ,X2 , . . . ,x ]) i s non-zero for every n ) . Perhaps the main feature of these groups i s t h a t $ i s an example of a [n -> n+t-1] group ( i . e . a group in which every n-generator subgroup has c l a s s at most n+t-1) which i s not a l s o a [n -> n+t-2] group, n > 2 for t = 2,3 and n > 3 for t = 4 .

In addition we may note t h a t $2 has

t r i v i a l c e n t r e , i s metabelian and in fact ($2 2 ) 2 = 1; $3 i s third-Engel but not metabelian; $4 i s [2 -*• 4] but not [2 -> 3] and, i f for the moment we w r i t e yi f o r y - C ^ ) , i > 1, then [Yr>Ys >Yt>Yi >Yi > • • • >Yi ] i s never t h e i d e n t i t y subgroup. 4.

In p a r t i c u l a r , y

has c l a s s e x a c t l y 3 for a l l t > 2.

THE CLASS OF B(n) AND SOLVABILITY

I shall return later to the Gupta-Weston 1971 paper, but first I wish to describe results obtained by N. Gupta and others in a series of papers which followed i t rapidly.

First came the Gupta-Gupta

paper [14] in 1972; this gave the first evidence of a direct connection between the nilpotency class of B(n) and the general solvability problem. Theorem. integer)

If the class of B(n) is less than [5n/2] for all n > N (a fixed then groups of exponent 4 are solvable of bounded length. The proof is short, requiring only the theorems of Wright on

the class of B(n) and on groups with generators of period 2, as well as some properties of the Gupta-Weston group H.

It has only historic interest

however in view of the fundamental result of Gupta and Quintana [19] which

Tobin:

Groups with exponent four

110

also appeared in 1972:

Theorem 4.1.

If for some n the class of B(n) is less than 3n-2 then groups

of exponent four are solvable. The details of the proof are changed and made clearer in [15] although the idea is the same.

Lemma 4.2.

The proof depends on two lemmas»

Let k be a fixed integer > 2 and let G be a [k -*• 3k-3] group

of exponent 4.

Then for every m > 2 the group G is [m •*• 2m+k].

The proof in [15] is brisk, depending on the rearrangement congruences developed there for commutators in B(n).

Lemma 4.3.

If G is a group of exponent 4 satisfying the law [v2 ,x,y2 ,x2]

= 1 then G satisfies the law [x2 ,y2 ;u2 ,v2 ;z2 ] = 1. This is a stronger version of a central lemma in [14] (a corollary of the main result there was used in [19]). 2

2

I give the proof.

2

In the given identity replace x by x z u ; we are now dealing with a commutator of effective weight 5 in 5 squares; expansion using Theorem 3.8 gives 1 = [v J , x 2 z 2 u 2 , y 2 , [z 2 ,x 2 ] [z 2 ,u 2 ] [u 2 ,x 2 ] ] = [v 2 , x 2 y

;z 2 ,u 2 ] [v 2 , z 2 ,y 2 ;u 2 ,x 2 ] [v 2 ,u 2 ,y 2 ;z 2 ,x 2 ]

and the result comes quickly by interchanging x and y and using the Jacobi and Wright congruences (§2(3) and §2(5)). Now let us suppose that for some k > 2 the class of B(k) is < 3k-3.

Let G = F/F4 where F is free on the free generators Xi ,x2 ,x3 ,...

Then G is a [k •> 3k-3] group and so by Lemma 4.2 G is a [m -*• 2m+k] group for all m > 2.

In particular G satisfies the law

1 = [xi2 ,x2 ,xa2 ,x 2 2 ;x42 ,xs ,X62 ,x 5 2 ;.. •ix3k+i»x3k+2'x3k+3>x3k+2-' since this is a commutator of weight 7(k+l) in 3(k+l) variables.

By

Lemma 4.3, if W is the subgroup of G generated by all elements {[a2 ,b,c2 ,b2] | a,b,c € G} D3 (G2) < W.

we have [G2 ,G2 ;G2 ,G2 ;G2 ] < W and so certainly

But now we also have y.

(W) = 1 so W is solvable, hence G2 ,

and hence G. Soon after that came the discovery by Edmunds and Gupta [11] in 1973 that if the class of B(n) should fall below 3n-2 for any n then in fact it would fall for all n to n + X(n) where X(n) if not constant must

Tobin:

Groups with exponent four

be less than a fixed constant K.

Theorem 4.4. most r.

111

This follows from Theorems 1.2, 4.2 and:

Let G be a group of exponent 4 with G2 nilpotent of class at

Then G is [n -*- n+k] w£t/z k=r if n > 2r+2 arc^ k = 3 + [4(r+l)/3]

if n < 2r+2. The proof (for which we may assume G = < ai ,a2 , .. . ,a involves considerable manipulation, modulo y

>)

+ -.G, of simple commutators

of positive weight in each of the generators and of total weight m = n+k+1 for the appropriate value of k.

The idea is to show that every such

commutator is congruent to a product of commutators of weight at least r+1 in G 2 ; as usual the triple entry, quadruple entry and rearrangement congruences mentioned in §2 are very much in evidence.

Remark.

The groups * , t = 2,3,4 described in §3 show that for r = 1,2,3

respectively the bound given in Theorem 4.4 is best possible (when n > 2r+2).

Corollary to 4.4.

If for some k the class of B(k) is less than 3k-2 then

for all n the class of B(n) is n + X(n) where X(n) < K and K is constant. At that stage nobody knew the precise class of B(3); if it should turn out to be less than 7 then the big questions about class and solvability in B would be answered.

But in 1974 the class of B(3) was

found to be 7 by Bayes, Kautsky and Warns ley [6] - and also by Gupta and Newman [17] who made use of it to reduce the bound for the class of B(n) to 3n-2 if n > 3.

This made the problems extremely tantalizing - and again

perhaps the computer work going on in Australia might settle them by showing a class less than 10 for B(4). It did not, of course, and this is a good point at which to turn our attention to another line of development which was initiated by the Gupta-Weston paper.

I refer to the last section of the paper, in which

they exploited their group H (see Theorem 3.10) to give a ring-theoretic equivalent to the solvability problem.

This paper really marks a turning-

point in the development of the theory of groups with exponent 4:

not

alone because it was followed by a spate of fresh results but also because (although the authors were carefully neutral, and even proved that thirdEngel groups are solvable) from about this time on the conjecture of solvability began to fade, so to speak, and be replaced by the conjecture of non-solvability - which so far as I know appeared in print for the first time in 1972 in Bachmuth [3], and was repeated in 1973 in the paper of

Tobin:

Groups with exponent four

112

Bachmuth, Mochizuki and Weston [5] on "A group of exponent 4 with derived length at least 4". such a group.

Note the title:

this was the first known example of

They also showed, in the notation of Theorem 3.10, that

D3G 5S Y 9 G . Soon afterwards, still in 1973, Bachmuth and Mochizuki [4] gave a ring-theoretic criterion for non-solvability in B, and showed that Ys (DG) ^ 1.

In 1978 these results were considerably improved on, by Doyle,

Mandelberg and Vaughan-Lee [10], who showed that the criterion given by Bachmuth and Mochizuki works both ways, and also showed that [D3 G,D2G,D2 G] ^ 1.

Since this implies that ys(DG) ^ 1 they presented it as further

evidence for the existence of non-solvable groups. more startling evidence appeared.

In the same year however

This was the paper by Razmyslov [40],

on what he calls the Hall-Higman problem, in which he constructs a nonsolvable group of exponent four.

This naturally overshadows the series of

papers which I have just mentioned, but it does not detract from the interesting work in them - and I now want to discuss them together. The basic idea in [21] and [5] (in [4] and [10]) was to use the new Theorem 3.10 (and the old Theorem 1.2) to tackle solvability by examining the solvability (nilpotency) of DH (for the rest of this section H means the group defined in 3.10); and to do so by embedding H (or suitable images) in an associative ring in such a way that problems on group commutators are replaced by equivalent problems on Lie ring commutators.

This is rather like the Magnus embedding of a free group in

an associative ring; as a reference for the construction of a multiplicative group with an exponent in an associative ring Bachmuth, Mochizuki and Weston cite the 1963 Canberra Lecture Notes of Bruck [7]. I will give first of all the Gupta-Weston equivalence as rewritten in [15], but with a little more explanatory detail than appears there.

We commence by renumbering

the generators of H as {xo ,Xi ,X2 , . . . } , just for later notational convenience.

Definitions.

Consider now a free associative ring R with unit element 1,

freely generated over the field 7L 2 by non-commuting indeterminates {ao ,ai ,a2 ,a3 , . . . } .

Let Ji be the ideal in R generated by all monomials

a. a. ...a. having a repeated term (i.e. i. = i, for some j £ k).

Let

J2 = idealD{((l+a. ) . . . (1+a. )-l)3 |k > 1}; J = idealD{Jt ,J2 }; K ll 1, K

Tobin:

Groups with exponent four

113

R = R/J; G = gp < (l+a±) | a ± = SL + J, i = 0,1,2,... >. We work in R.

Then (1+a.)2 = 1 since Ji C J; hence also

[(l+ii),(l+ij),...,(l+ik)] = 1 + (a.,a.,...,^3

(1)

where (a. ,a.) is the ring commutator a. a. + a.a. and we use the leftnorming convention here also. is a repeated entry. abelian.

Again Ji C j => (a. ,a.,. . . , L ) = 0 if there i_ J K

Thus the normal closure in G of each generator is

But for any g e G J2 C J => (g-1)3 = 0 => (g-1)4 = 0 =* g4 = 1 .

Hence G is a homomorphic image of the group H, and a commutator identity in H will induce a commutator identity in R. By a rather neat trick we can go in the opposite direction also.

Let A be the normal closure of Xo in H; since A is abelian, generated

by conjugates of Xo, it is an elementary abelian 2-group and we note that the condition [xo,h,h,h] = 1 for all h e H implies [x,h,h,h] = 1 for every element x in A.

Now for each x. in H with i > 1 define an endomorphism of

A p(x.): x -> xp(x.) = [x,x.] for all x £ A.

These mappings generate a

subring R* of End (A) under the operations xCPipa) = (xpi)pi and x(pi+p2) = (xpi) (xp2) . C l e a r l y R* has c h a r a c t e r i s t i c 2; p(x. ) . . . p ( x . 11

and f i n a l l y

( 1 + pVx. ) ) ( 1+ p ( x . ) ) . . . ( 1 /

ll

but ( p ( h ) ) 3 = 0 for any h e H.

1

t

) = 0 i f i . = i, ^

for k ^ j ;

+ p ( x . ) ) - 1 = p ( x . x. . . . x .)

12

1.

ll

12

Hence R* i s a homomorphic image of R.

a l s o have, for x e A, x(p (xi) ,p (x2 )) = x ( p ( x i ) p ( x 2 )

1-

We

+ p(x2)p(xi)) =

[x,xi , x 2 ] [x,x 2 ,xi ] = [ x ; x i , x 2 ] by the Jacobi i d e n t i t y ; thus (p (xi) ,p (x 2 )) = p([xi , x 2 ] ) . Now we e a s i l y get (with an obvious n o t a t i o n ) t h a t 1 = Ym(DH) implies 0 = (ai ,a 2 ;a 3 ,a4 ; . . . ; a 2 m _ 1 , a 2 m )

(2)

in R, which implies the corresponding identity in R* giving Y m ( D H ) ^ Z(H). Hence:

Theorem 4.5.

With R and a. as already defined, groups of exponent four

are solvable if and only if R satisfies the relation (2). Actually the solvability criterion in this form has not been used.

Bachmuth, Mochizuki and Weston used the relevant ideas in the

following way.

Define the polynomial A(l,2,3,4,5,6,7,8) in the variables

Tobin:

Groups with exponent four

114

ai,a2,...,ag (i.e. the third derived Lie commutator) as follows, where ar,...,a8 are generators in the ring R: [ [ [1 + ai ,l+a2],[l+a3 , 1+a* ] ]; [ [1+as ,l+a« ], [l+a7 , l+a8 ] ] ] E 1 + A(1,2,...,8) modulo J as in (1). Then A(l,...,8) is a homogeneous polynomial of degree 8, each term having weight 1 in a., 1 < i < 8. It is shown in [5] - with some computer assistance - that A(l,2,...,8) ^ J and so the commutator above in G is not 1. Thus: With G as defined in the ring R, D3 G ^ 1 and so D 3 H / 1.

Theorem 4.6.

Corollary 1. D 3 H ^ y 9 H. Consider the endomorphism <j> of H defined by x.<J> = x. if 1 < i < 8, x.<|> = 1 otherwise.

Then y9 (H*) = 1 but D3 (H) ± 1.

Corollary 2. D 3 H ^ ZH. This was proved by using a matrix group G* which we will now describe.

This kind of group was introduced by Gupta, Mochizuki and

Weston in proving Wright's conjecture [16], but although their paper was submitted in 1971 its publication was accidentally delayed until 1974. Using the group G* Bachmuth and Mochizuki [4] derived a simpler ringtheoretic condition for non-solvability of H. Still using R, let G* denote the group of 2x2 matrices over R generated by the set Co

" Li

ij

Ci

" L\

l

\

Then each C. has period two and any element of G* has the form M =

with uGGandrf = [ v(u » + £ +u+1) and M2 = | ^

b

M t 2 = M22 = 1

J] = l-

^ 1 where a2 = b 2 = 0 s o t h a t then

A ain if M

* = [^ a j ]

8

( 1 + a ) 2 = (1+b) 2 = 1 and

[Mi ,Mi ] = (MjM 2 ) 2 = | 1 + ( a j b : )

L (a,b) same type if (a,b)2 = 0, for instance if b = a..

° 1 which i s a g a i n c

lj Thus for example

.. and

Tobin:

Groups with exponent four

[Ci,C 2 ,C 3 ] = p + ^ i , a 2 , a 3 ) L Cai,a 2 ,a 3 )

01

I f we

115

i e t C(v) = P

we have C(v) 2 = I always, C(l) = Co ; [C(v),Mi] = ["^

?1 where v €E R j l = N say and

[C 0 ,N] = I . These r e l a t i o n s show immediately t h a t the generators C o , C i , C 2 , . . . of G* s a t i s f y a l l the conditions p r e s c r i b e d for the generators of H; hence G* i s a homomorphic image of H. Now we have the c r i t e r i o n ( [ 1 5 ] ; the "only if" p a r t comes from [4]): Theorem 4 . 7 .

y

+1

(DH) = 1 if and only

if

0 = ai (a2 , a 3 ) (a4 , a s ) . . . ( a 2 m , a 2 m + 1 ) in R. Proof.

[xo ,xi ;x 2 ,x 3 ; . . . ; x 2 m , x 2 m + 1 ] = 1 i n H implies

[C0,Ci ;C 2 ,C 3 ; . . . ; C 2 m , C 2 m + 1 ] = I in G*, t h a t i s

ii(a 2 ,i 3 )...(a 2m ,a 2m+1 ) This implies at (a2 , a 3 ) . .. ( a 2 m > a 2 m + 1 ) = 0 in R, so p (xi )p([x2 ,x3 ]).. . p([x« .x^.,]) = 0 in R* giving [xo ,Xi :xa ,X3 :... :xo ,x^ ., 1 = 1 in H.

For what follows we need to review the definitions of the ideals Ji and J2 in the ring R, and recall that R = R/J where J = Ji + J2 . Bachmuth and Mochizuki attempted to show that the polynomial P

m = a i ^ a2 ' £ l 3 ^ a 4 ' a s ^ * *' (a2m'a2m+P i s n o t i n J f o r a n y m> a n d f o u n d i t : expedient to calculate not in R but in a larger ring Ri which has R as a quotient. Let us for a moment work in R modulo Ji , so that we can ignore any monomial with a repeated entry, and now let us dissect the generators { ( l + a i i ) O a . i ) . . . O a . ) - I}3 ,

of the ideal J2 .

t> 1

Clearly this expression is = 0 when t = 1 or 2. Let

T3 (ai ,a2 ,a3 ,. .. ,a ) be the homogeneous component of degree one in each of the generators ai,a2,...,a , and so of total degree n, in the expansion of {(1+ai) (1+ai)... (1+a ) - I}3 where n > 3.

If, again for the moment, we

write ai = a, a2 = b, a3 = c and a4 = d we see that T3 (a,b,c) must come from (a+b+c)3 and is abc + acb + bac + bca + cab + cba but what is more to

Tobin:

Groups with exponent four

116

the point here is that {(1+a)(1+b)(1+c)(1+d) - I}3 must be congruent to T3(a,b,c) + T3(a,b,d) + T3 (a,c,d) + T3(b,c,d) + T3 (a,b,c,d) and so on for larger values of n. This means that if we let T be the ideal of R generated by all T3 (a. ,a. ,...,a. ) where t > 3 we will have T E J2 modulo Ji so ii

12

it

J = Ji+J2 = Ji+T.

The polynomial P

entry a., and so if P

is homogeneous, of degree one in each

is in J i.e. in Ji+T it must lie in T.

Let K be

the ideal generated by all the terms a.3 , i > 0 in R; then K C Jt n j 2 . The idea in [4] is to show that P

£ (T+K) and so £ T.

Let Ri = R/(T+K);

and let S(k) for k > 2 denote the additive subgroup of Ri generated by the monomials of degree 3 in each of ai ,a2 ,. . . ,a,.

The elements of S(k)

are homogeneous polynomials of total weight 3k.

(For convenience we write

a. now instead of a. for the generators of Ri.)

Bachmuth and Mochizuki

now prove:

Theorem 4.8A.

Each S(k), k > 2, as a vector space over Z 2

has dimension

at most one; every element of S(k) is a multiple of the monomial M(k) = a k _ 1 a k _ 2 . ..a2 Their proof depends on identities relating to monomials of degrees 5,6,7 and 8 in five variables; these identities were obtained by the authors and K. Weston with the aid of computers.

Theorem 4.8B.

In S(2j+1), j = 1,2,3,... M(2j+1) may be written in the form

Thus if S(2j + 1) ± {0} there is a group G. with exponent four such that Y-+ I ( D G O

i 1-

They were able to prove, for a start, that

S(2) i {0} and S(3) t {0}; this in itself merely says that D2 G t 1 which is not new (G here means as in 3.10 the free group of infinite rank in B) but from the fact that M(2) and M(3) are not in T+K the authors were able to deduce that ai (ai ,a2) (a2 ,a3) (ai ,a3) (a2 ,a3) £ T+K and hence ai (a2 ,a3) (a4 ,as) (a6 ,a7) (as ,a9) £ T+K which shows that ys (DG) ^ 1; and they claimed that similarly [D3G,DG] ^ 1. The idea used is to consider, in the ring R looked at as a vector space, the subspaces T+K and 5(n) where £(n)

is

spanned by all the

monomials of degree 3 in each of ai,a2,...,an (total weight 3n). S(n) is the corresponding subspace in R, = R/(T+K) and S(n) » $(n)/(S(n) n (T+K)).

Tobin:

Groups with exponent four

117

The computer is used to examine a spanning set for $(n) n (T+K) and (hopefully) to show that it does not span 5(n). In 1978 Doyle, Mandelberg and Vaughan-Lee [10] used the same technique but commenced by effecting a very substantial reduction in the size of the spanning set to be tested.

This enabled them to prove that

S(4) ± {0} and S(5) ± {0} and again to deduce from this (by computer manipulation of polynomials):

Theorem 4.7A.

[D3 G,D2 G,D2 G] ^ 1; Ys (DG) t 1.

They also claim that if S(n) = {0} it follows that [x,ui ,u2 , . .. >u 3n ] e Y 3 n + 2 G where IL e {yl ,y2 ,. . . > y n h 1 < i < 3n and where each y. appears as an entry exactly three times.

From a criterion stated

by Gupta and Newman [17] they deduce that G is then solvable:-

Theorem 4.7B.

If S(j) = {0} for some j then groups of exponent four are

solvable. I will conclude with a little sidelight on the computer requirements.

For the 1973 paper [4] an IBM-360 machine computed

S(3) ^ {0} in one hour of central processor time.

For the 1978 work a

UNIVAC 90/80 limbered up by recalculating (with some shortcuts) the S(3) result in 25 seconds; it determined S(4) in 20 minutes - but it needed 20 hours to decide S(5). Perhaps the time was ripe for Razmyslov, after all.

5.

RECENT DEVELOPMENTS Recent here means comparatively recent - the last five years

or so.

In this time the non-solvability of B (and thus the class of B(n)),

the derived length of B(n) and reasonable bounds for the order of B(n) have been determined.

For these I can give only very brief descriptions.

On

the other hand there are several interesting results which can be demonstrated readily. In the 1970fs while computers in the U.S. were seeking groups with longer derived series, work was proceeding in Australia to establish precise details of the finite groups B(3) and B(4). B(3) was presented by Bayes, Kautsky and Warns ley [6] in the form of about 5 printed pages giving a minimal list of commutators of weight i generating y^B modulo Y i + 1 B f ° r each i, such that each group element has a canonical expression as a product of these basis elements (each to the power 0 or 1) in order; it

Tobin:

Groups with exponent four

118

gives the canonical form for the square of each basis member and for the commutator of any pair.

B(4) was calculated by Alford, Havas and Newman

[2] but its presentation runs to over 180 pages of tables. I wish to talk about some use which M. Vaughan-Lee and I have (independently) made of these results.

Using both tables he has (as we

will see) drawn some sophisticated conclusions from them; I have made some simple observations from B(3).

[I should interpolate here that I had not

seen the B(4) tables prior to the St. Andrews meeting; since then I have received a microfiche copy from M.F. Newman which I acknowledge with thanks:

it has thrown some light on the "observed" Lemmas 5.5 and 5.6.] Let G = F/F4 where F is free of infinite rank. k

G £or some k

3.11, if D G < y v 2+1 1 , , "I

>

then

°

k+1

(G) = 1. Now D G < y V G for 2k

T,

G = D(D G) < D(y G) < (y V G ) 2 . 2k 2k should happen that

all k and so D

X

G)2 < Y

(Y 2r

then G is solvable. groups (y f G)

2

By Theorem

k

Thus if for any r it

G r+1 2 r +1

(1)

With this in mind it seemed useful to examine the

and see if one could improve the well-known fact (Theorem

2.1) that (y G ) 2 < y

G +2

for all n > 2.

Examination of B(2) shows already

that (Y2G)2 fC Y S G and (Y3G)2 5C y6 G but I was able to show ([44], 1975):

(Y4G)2 < y 8 G, and so (Y G ) 2 < y

Theorem 5.1.

.G for n > 4; and for

n=4 this is best possible. This was best possible because of the proof in [5] that D3 G 5£ Y9G, from which (YO G ) 2 f£ Y £

2

G.

The proof, based on an identity

5

2' +l [ a , b , c , d ] 2 = 1 mod YsG; but M.L. Newell

from the B(3) t a b l e s , showed t h a t

has remarked in discussion t h a t i t uses only [Y4G,Y4G] = 1 and so ( Y4 G) 2 < [Y4G,Y4G], t h u s : Theorem 5.1A.

(Y4G)2 =

[Y 4 G,Y4G].

Naturally the question of extending t h i s a r i s e s ; I thought t h a t perhaps (y G)2 < y result

Lemma.

G might be t r u e for n > 4 because I had the

(unpublished).

[YkG,Y2G]2 < Y 2k+4 G, for

all

k > 2.

following

Tobin:

Groups with exponent four

119

This comes from the group identity [x,y,a]2 = [x,y,a,a,a][x,y,a,a;x,y] which is also a consequence of the B(3) tables. I have now found that the conjectures arising from these results (and the knowledge that (1) can never be satisfied) are all true; Theorem 5.4 contains the result, which is the strongest possible, but first we need preliminary lemmas.

Lemma 5.2A.

For all m,n > 1 the following is an identity in groups with

exponent 4 [a2 ,xi ,x2 ,. .. ,xn;a2 ,yi ,y2 ,.. .,ym] = 1. Proof.

For n=l this is Lemma 2 of [44], where it is deduced by induction

on m from the identity [a2,x;a2,y] = 1 obtained from the B(3) tables. Now in the same way if we assume the result true for a given n, by x b where b is any group element; write u = [a2 ,xi ,.. .,x

replace x v=

2

[a ,yi , . . . , y m ] .

..],

Then [u,xRb;v] = [[u,b][U,XR][u,xn,b];v] = [cd;v]

where c = [u,b][u,xn], d = [u,x^,b].

Thus 1 = [c,v] [d,v] but [c,v] = 1

since v commutes with [u,b] and [u,x n ].

Thus 1 = [u,xn,b;v] as required.

Lemma 5.2A is the form we need to use, but it is a special case of a general statement which we can prove quite simply.

Lemma 5.2B. 2

an entry a Proof.

Any commutator* simple or compound* of weight > 6 in which appears twice equals 1 in groups with exponent four.

We verify this for simple commutators; then the truth for complex

commutators follows from the corollary to the next lemma. [a2,xi,...,x ,a2] = 1 for n > 2.

Lemma 3.9 shows

The other possibilities are checked in

much the same way from the B(3)-identities 1 = [x,a2,y,a2] = [x,a2 ,a2 ,y] = [a2 ,x,a2 ,y] = [x,y,a 2 ,a 2 ]. For instance, in the first of these we may replace x by [xi ,...,xn] and induct from the y-term to get [xi ,... ,x ,a2 ,yi ,. .. ,y ,a2 ] = 1.

Remark.

This lemma should be compared with the quadruple entry identity

in Lemma 5.7.

They appear to be related but they are independent results.

Our next lemma could be expressed more generally, but it suits us in the form given.

Tobin:

Lemma 5.3.

Groups with exponent four

Let B(n) = < ai ,. . .,a

120

>; let w be any compound commutator of

weight one in each generator a^, thus of total weight n.

Then w is a

product of simple commutators (of weight > n)j each being of positive weight in each generator a^.

Corollary.

Let v be a compound commutator in any group G in V>; let the

entries in v be xi ,. . . ,x, and let x. occur r. times in v, 1 < i < k. Then v is a product of simple commutators in the entry set {xi ,... ,x, } and each commutator is of weight at least r. in x. , 1 < i < k. Proof.

Since B(n) is nilpotent and w £ y B(n), w = ~| f c. where each i

commutator c. is simple of weight > n in the generating set {ai ,...,a }. This equation is an identity in B(n), so by the Higman argument (see Lemma 2.2) we may remove any c. which does not have positive weight in every a., while at the same time introducing new complex commutators of c-elements, getting w = ~| f b. ~| |" d, where the b. are those c. which J

J

k

meet our requirements and the d, are (compound) commutators in c's (and so are of weight > 2n) and have each positive weight in every a..

Now

each d, is a product of simple commutators, h^ say, in the entry set {ai,...,a } with weight > 2n. So w = ] [b. ] [ h and again this is an n J £ j £ identity in B(n); but now the removal of unsatisfactory h-terms produces no new elements j- 1, since the class of B(n) is < 3n, and so the process stops. For the corollary, if v £ G has weight n, we relabel each entry, starting from the left-most one, as yi,y2 ,...,yR in order; the lemma now guarantees an expression for v as a product of simple commutators of positive weight in each y..

This is an identity in the

variables y., and the result is clear. Remark.

This Corollary has an implication for the quadruple entry Lemmas

2.5 and 5.7.

Lemma 5.2C.

Groups with exponent 4 satisfy the laws [a 2 ,b 1 ,b 2 ,...,b n ] 2

Proof.

=l/orn>2.

For n=2 this is a consequence of the B(3) tables.

Standard

induction, by expansion on the b -entry, and use of Leirma 5.2A gives the

Tobin:

Groups with exponent four

121

result. Theorem 5.4. Let G be any group of exponent 4. Then (y G ) 2 = [y G,y G] for all n > 4. T H s £s best possible in the sense that (y G ) 2 ^ y

G

for any n, where G ts F/F4 and F is a free group of infinite rank. Proof.

[a2 ,bi ,. . . ,bt] is an element in yt+2 G *

Let w =

[x2 a2 ,bi ,. .. ,b ] .

Then by Lemma 3.5, for certain group elements v.,y,c.

w = [[[x2,b!]

,ba]

... b t ] t [a 2 ,b 1 ,...,b t ]

= [y2 ,ci ,c2 ,. .. ,ct] [a2 ,bi ,. . . , b t ] . Hence by Lemma 5.2C, i f t > 2, w2 = [y2 ,ci , . . . , c t ; a 2 ,bi , . . . , b t ] G [y

2G,Yt+2 3

G

]-

I t i-s d e a r now t h a t we can prove [h,bi , . . . ,b ]* £ D(y

i f h e G , t > 2.

-G-

In p a r t i c u l a r , l e t h = [gi ,g 2 ] ; then [gi ,g 2 ,bi , . . . ,b ] 2

G D Y

^ t+2 G ' ) i f t > 2* T h i s i m P l i e s so (y G)2 = [y G,y G] when n > 4.

that

^t+2G)2 ^

D Y

^ t+2 G ^

i£

t >

2

For the r e s t , suppose t h a t for some k (y v G) 2 < y , .G.

and

Let

ai ,a2 , . . . ,a, , bi ,b2 , . . . ,b, , x,y be 2k+2 d i s t i n c t generators of G corresponding to elements of the free b a s i s of F.

Then

[ai , . . . , a R ; b i , . . . , b k ] = where each c. is a (simple) commutator of weight 2k+l or 2k+2 in the generators of G and w is a word in the generators of G which lies in Y 2 k + -G.

This must be an identity in G and so (i) we may assume that each

c. has positive weight in each of ai,...,a, , bi ,. . . ,b, and (ii) we may substitute [x,ai ] for ai and [y,bi ] for bi : this gives [x,ai ,...,ak;y,bi,...,t and hence [Y k + 1 G,Y k + 1 G] < Y 2 k + 3 G .

Now by induction [ytG,YtG] < Y 2 t + 1 G ,

all t > k. Take any t such that k < t = 2 for an integer r. Then D r + 1 G = D(D r G) < D(y G) < y 2 2

G; +1

this implies that G is solvable, by Theorem 3.11, which is contradicted by Razmyslov's Theorem 5.11. Thus (y^G)2 ^C y , result in Theorem 5.4 is best possible.

G for any k, and the

Tobin: Corollary,

Groups with exponent four

122

(a) If G is a group of exponent 4;

(i)

\G^7kG

(ii)

if t > 4 then yfG £s dbelian if and only if it is elementary

(b)

^s e^emeni;ary

obelian for all k ^ 3; abelian;

YiB(n) is elementary abelian when 2k > 3n-2 and n > 2.

Remark. The B(3) and B(4) tables show (see Lemma 5.6) that (y3B(3))2 t 1 and ( Ys B(4)) 2 = Y 10 B ( 4 ) * 1We generalize this in the following: Conjecture (a). (y^Bfn))2 t 1 when 2k < 3n-2, n > 2. This is certainly true if Conjecture (c) after Lemma 5.6 is true; we give some positive evidence after Theorem 5.8. The tables also show that if B(4) = < ai ,a2 ,a 3 ,a4 > then [ a i , a 2 ] 2 , [ a i , a 2 , a 3 ] 2 and [ai ,a2 ,a 3 ,a4 ] 2 are all t 1. This suggests: Conjecture (b). [ai ,a2 , . . . , a ] 2 ^ 1 where {a. | 1 < i < n} are the (relatively) free generators of B(n). This would of course give the weaker conclusion that (y B(n)) 2 ^ 1. We can give a simple result in this direction: Lemma 5.4 A.

For every k > 2 (y 2k B(2k)) 2 t 1.

Proof. Consider [ai 2 ,a 2 2 , . . . ,a^2 ; bi 2 ,b 2 2 ,. .. ,b^2 ] = w say, where the entries a . , b . are the distinct free generators of B(2k). The relation w = l would imply that y, (G2 ) is abelian, where G is defined as in Theorem 5.4; but this is not possible since G (and so G2) is not solvable. But [ai 2 ,a 2 2 , . . . ,a, 2 ] € y 2 ,B(2k), hence Y ^ T ^ C 2 ^ * S n o t abelian, which gives us our conclusion. Corollary.

If k > 2, (Y 2] B(2k+l)) 2 ^ 1.

The next two lemmas are minor curiosities gleaned from the B(3) (before B(4)!) and B(4) tables. Lemma 5.5. As in 5.4 let G be the free group of infinite variety B. Then (i) ZG > YS (< x,y >) for all elements x and y in G, (ii) G has a non-trivial centre, and ( i i i ) G is centre-by-~[2 -• 4],

rank in the

Tobin:

Proof,

Groups with exponent four

123

(i) The B(3) tables show that [y,x,y,y,y,g] = 1 is an identity in

G, i.e. ZG contains [y,x,y,y,y] and [x,y,x,x,x] but these two elements generate ys (< x,y > ) . G.

For (ii) let a,b be two of the free generators of

Then < a,b > = B(2) and ysB(2) t 1.

explanation.

(iii) is stated in [15] without

It is a consequence of (i).

Corollary ([15]).

(by 5.11)

In groups of exponent 4 [2 ->• 4] does not

imply solvability. The B(4) tables show that the analogous result for three variables is not true, i.e. if a,b,c are free generators of G then Y7 (< a,b,c >) is not contained in ZG.

Lemma 5.6.

(i) In B(3), writing y. for y.B(3), we have

Ye = [YS>YI3 = [Y4»Y2] = [Y3*Ya]; (ii) Y9

In B(4), writing again y. for Y - B ( 4 ) we have V

= [Yi»Yjl

(iii)

Y? = [Ye >YI ] = [Ys ,Y2 ] = [Y4 >Ys ] •

Y9B(4)

Comment,

i'3 with

i

+

3 = 9;

y10 = [Y^YJ]

V i,j with

i + j =10.

= [D3B(4),

(iii) is used in [45], where it is stated as a deduction from

the B(4) tables. In B(2), [ Y 2 , Y 3 ] = Y5 but this is not true for B(3). The similarity of (i) and (ii) suggests the following: Conjecture (c). For B(n), with n > 3, [Y-»Y«] = Y- + - whenever i + j = 3n-2. Conjecture (d). For B(n), with n > 3, [Y->Y-] = Y- + - whenever i + j = 3n-3. While in this mood of easy speculation, perhaps I may digress for just a moment to mention some computer results reported by Havas and Newman [27].

Let us write

L(n;k) = < xl ,... ,x | exponent 4 and xi2 = . .. = x,2 = 1 >. n KThese groups are intermediate between B(n) = L(n;0) and Wrightfs special groups L(n;n).

They were introduced by Leech [31], and studied by

Macdonald [33], and by Havas and Newman who have determined the classes and the orders in a number of cases.

Let us look at their results on the

classes (values for B(n) and L(n,n) are of course previously known):-

Tobin:

B(3)

Groups with exponent four

: class 7

B(4)

: class 10

L(3;l) :

7

L(4;l) :

10

L(3;2) :

5

L(4;2) :

8

L(3;3) :

4

L(4;3) :

6

L(4:4) :

5

124

L(5;4) : class 7 L(5;5) :

6

For general n, the class must jump from n+1 to 3n-2 in n steps.

What happens?

Let us guess:

Would it be - going upwards - one

jump of 1, then n-2 jumps of 2, and no change at the final step i.e. L(n;l) and B(n) always have the same class? We will next consider results of Vaughan-Lee [45] who has made very effective use of the B(3) and B(4) tables, in consequence of which he has determined the exact derived length of B(n) (see Theorem 5.8). Apart from getting specific relationships among elements of B(3) and B(4), he used the tables to get more detailed information about the basic congruences in §2 (and about two other useful congruences which he derived from the tables).

He noticed that with one exception these are all what

we may term special congruences.

I will use the exception to explain the

rule. We have the very useful congruence [a,b2] = [a,b,b] mod Y4 which however tells us only about the 3-weight part of the following full relation: [a,b2] = [a,b]2[a,b,b] = [a.b.b] [b.a.a.a] [b.a.a.b] [b,a,b,b][b,a,b,b,b]. So far as [a,b2] is concerned b in effect appears twice, a once, and the element is in y 3 ; and [a,b,b] satisfies the same criteria.

On the right

hand side of the final relation there are four terms of weight greater than three; in three of these b occurs at least twice and a at least once: they have "higher special weight" than [a,b2] or [a,b,b].

But [b,a,a,a]

has b only once as an entry, and its presence prevents the congruence from being special. But something can be salvaged:

the B(3) tables show that the

2

congruence [a,b,c ] = [a,b,c,c] mod ys is special.

Vaughan-Lee then

remarked that the proof of Wright's pivotal result, on quadruple entries, used only special congruences and therefore is itself a special congruence. Let us recall (Lemma 2.5) Wright's congruence.

Let G e B, n > 6.

Ifw

is "a commutator of length n in G" (i.e. a simple commutator of weight n) and of weight 4 or more in one element a of the entry set then w € y + .G.

Tobin: Groups with exponent four

125

Now we apply the observations in [45]. Since w = 1 mod y +. is a special congruence, w is a product of higher weight commutators each of which must have the entry a at least 4 times and is therefore by Wright's result a product of special commutators of still higher weight, and so on. But w lies in a finitely generated subgroup of G, so w = 1. Hence ([45], Lemma 2): Lemma 5.7. Let G be a group of exponent 4 and let u be a commutator of weight n with entries from G. If n > 6, and if u is of weight 4 or more in some element of G, then u = 1. Remark 1. Wright's proof (as his statement suggests) establishes his q.e. congruence for simple commutators only; hence the same is true for Vaughan-Lee's identity. However Lemma 5.3 now shows that "commutator" in 5.7 may be interpreted as meaning both simple and complex commutators. 2. In [15] Gupta conjectured (correctly) that Wright's quadruple entry commutators are trivial. He asked the related question "If G has exponent four and x £ G is the normal closure < x > nilpotent, and if so what is its class?" We can now answer this easily: Corollary 1. I / G 6 B and x e G then < x > is most 4, and this bound is best possible. Proof. The subgroup < x > is generated by the commutator of weight five in these elements has has total weight at least six, hence it is 1 by hand, in B(2) the element [b,a,a,a,a] ± 1.

nilpotent of class at set {x,[x,g] | Vg e G}; any x appearing five times and Remark 1. On the other

Corollary 2. The Gupta-Weston group H (see 3.10) is a [2 •* 4] group. This result is stated in [15], without explanation. We can give an easy proof using 5.7. It is enough to show that [a,b,b,b,b] = 1 for all elements a,b in H. We recall that H is generated by elements Xi,x2,... of period 2, each of which has an abelian normal closure in H. Let a = "| f v- » where each y. is an x.. From the expansion of [yi y2 ,b] we have for suitable elements v^ in H

J-b] = T T [y^b^ 1 = T T c[y.,b][y.,b,v ]) i

1

Thus again for suitable w^

i

1

i

x

1

x

Tobin: Groups with exponent four [a.b.b] = T T [[yj.bJty^b.v.J.b] i=TJ

126 ([yi,b,b][yi,b)v.,b])

i

because the normal closure of each generator y. in H is abelian; so [a.b.b] = T T ([y i ,b,b][y i) b,b ) w i ][y.,b ) v i ,b][y i ,b,v i) b,w i ]). i It is clear now that [a,b,b,b,b] may be expressed as a product of simple commutators all of which have weight > 5 and have b as a quadruple entry. Those of weight > 5 are 1 by 5.7; those of weight 5 must have the form [y.,b,b,b,b] which is 1 by the definition of H. Corollary 3. ([46]) (i) B(2) is not in the derived group of any group of exponent 4, (because) (ii) DG is [2 -*- 4] being contained in G2 which is an image of H, (and so) (iii) the non-solvable Bumside variety B has a proper non-solvable subvariety {generated by DG). Again in [45] using the idea of special congruences to improve some results from [19], Vaughan-Lee states the following lemma; first we need some notation: for each n > 3 define a commutator V n = [Xi ,X2 ,X3 ,X 2 2 ,X4 ,X 3 2 ,X5 ,X4 2 ,...,X n _ 2 2 ,X n ,X n _ 2 1 ,X n 2 ]

which lies in y_ ^>(ji) (this was introduced in [17]). For any group G € B let v (G) be the verbal subgroup of G generated by all the images of v in G. Let k > 2 be a positive integer. Lemma 5.7A. If w is a commutator in G which is of weight 3 or more in each of k distinct variables (as well as^ possibly^ involving other variables) then w e ^ + 1 ^ ) . Since in y- 2 B ( n ) a non-trivial simple commutator has either n-1 or n-2 triple-entries we may state: Corollary 1. Y 3 n - 2 B t n 5 < v n _i B ( n ) if n> 4. From this, using Theorem 4.1, we may deduce: Corollary 2 ([17]). If for some positive integer n ( > 3) the identity v n = 1 is iea law in groups of exponent 4 then groups of exponent 4 are solvable.

Tobin:

Groups with exponent four

12 7

And now since there are non-solvable groups of exponent four we get the useful information:

Corollary 3.

v (B(n)) ^ 1,

(n > 3 ) .

This result is important for the determination of d(n), the derived length of B(n), which we will now consider. first approximation by a simple argument: groups in B would have derived length < X.

We can easily get a

D B(2 ) ^ 1 since otherwise all Thus for 2 < n < 2

we find

X+l < d(n) < X+3; and in fact d(n) = X+l or X+2 if also 3n-2 < 2

X+2

,

Since B(2) is non-metabelian and of class 5 we have d(2) = 3. The exact value of d(n) for n > 2 was announced in [45] (in which a slightly less precise result was established).

I am thankful to M. Vaughan-Lee for

permission to quote from his (unpublished) proof; it has the same startingpoint, namely Corollary 3 above, but the argument proceeds quite differently from that in [45].

Theorem 5.8.

Let d(n) be the derived length of B(n).

Then if n > 2, and

2 k ~ 1 < 3n-2 < 2 k , d(n) = k. Proof (outline). k

The first step is to show that if 2 k < 3n-2 then k

D B(n) $ 1; the idea is to find a suitable element in D B(n) which is of weight 3n-2 in the generators of B(n), having one generator appear once as the first entry - and the other n-1 each appearing three times, and which can eventually be reduced to the form v

(and so is not 1).

Thus for example to show that D2B(3) / 1 we might note that in [zi ,Z2 ;za ,Z4 ] the substitutions zi = [xi,xa], Z2 = X3 , z3 = X22 , z4 = x 3 2 give us [xi ,x2 ,x3 ;x22 ,x32 ] which by §2 (3) and §2 (8) becomes [Xi ,X2 ,X3 ,X22 ,X 3 2 ] = V3 7* 1. Although this simple example illustrates the idea, the actual proof that such an element exists goes in a different fashion.

We may

write a generator of D B(n) in left-normed form as [Zi ,Z2 ,[Z3 ,Z4],[[Z5 ,Z6],[Z7 ,Z8 ]] ,...,Z ,]]...]. 2k Now if we let t = 3n-2 - 2

(*)

and write zi = [xi ,yi ,y2 ,... ,y ] we may k relabel each subsequent z. as y. . giving us an element in D B(n) which

has weight 3n-2, namely

Tobin:

Groups with exponent four

128

The proof shows that it is possible to choose the entries yi >y2 > • •. ,y3

?

so

tnat

eacn

°f

tne

elements X2,x3,...,x

occurs three

times in the list, and the element reduces to the form v

- for this, a

number of identities are given which are specially adapted to unravelling a complex commutator of this kind in which certain patterns of repeated entries occur. k

k

The second step is to show that D B(n) = 1 if 3n-2 = 2 case 3n-2 > 2

being obvious).

(the

Here the element (*) is rewritten by

repeated use of the following identity (in weight 3n-2) [x,...,[a,b,c],...] = [x,...,a,c,b,...][x,...,b,c,a...] where the entries (if any) represented by dots remain unchanged.

This

identity comes from the Jacobi identity §2 (3) which gives [w,[a,b,c]] = [w,[a,b],c][w,c;a,b] and the Wright congruence §2 (5) which gives [w,c;a,b] = [w,a;b,c][w,b;c,a] both congruences being modulo terms of higher weight. The element (*) reduces to a product of terms of the form [wi ,wa ,U2 ,w3 ,us ,. ..,u

JJW ] where each w i is a commutator [ z i n y zif2}-'

in the elements z. and each u. is one of the z..

The main part of the

work now lies in proving the following interesting result:

Lemma 5.8 A.

For n > 3 the relation

[xi 2 ,x 2 2 ,y2 ,x 3 2 ,y3 ,X42 ,.. .,y n _ 1 ,x n 2 ] = 1 is a low in B (n). The proof proceeds in two stages:

(a) to prove that in order

to establish this law it suffices to show that 1 = [xi ,x2 ,xi(2;),X3

'xi(3)'-"»xi(n_1)'xn 1

where {xi,X2,...,x } is a set of free generators for B(n) and each x.f, . n H.KJ is a member of this set; (b) to prove that in fact all such words are equal to 1.

The proof in each stage is quite difficult and requires some

detailed case-by-case analysis using certain identities from B(4). To conclude we note that since the commutator in Lemma S.8A is of weight 3n-2 in B(n) we again get a commutator which equals 1 if we replace any x.2 by a2b2 , a 2 b 2 c 2 etc., and hence again if we replace x.2 Clearly we can replace all the x.2 in this way, and thus 1 v finally we obtain D B(n) = 1.

by [a,b].

Tobin:

Remark.

Groups with exponent four

129

In support of Conjecture (a), namely that (y,B(n))2 £ 1 whenever

2k < 3n-2, we can now cite some evidence. of integers:

m

Consider the following sequence

= 4, n. - = 4(n.-l), 1 < i.

Now for all i we have

+ 1

3n.-4 = 2 and hence by Theorem 5.8 we have D + 1B(n.) ^ 1. This means 1 1 2i that D B(n.) is not abelian, consequently y «. B(n.) is not abelian or l

_zi

equivalently (y ? . B(n.))2 ^ 1. 1 2 x (y,B(n.))2 f 1 when 2k < 3n.-2. K

1

l

This implies that, for all i > 1,

1

We will next consider nth-Engel conditions and some closely related conditions in groups with exponent four; we gather the results in the omnibus Theorem 5.9 below.

By way of introduction we might mention

that Marshall Hall remarked in [24] that all groups in B are 5th-Engel, i.e. satisfy the law [x,y,y,y,y,y] = 1. R. Baer pointed out in 1940 that 3rd-Engel groups in general are not nilpotent - the example he gave (see [15]) was in effect C2wr (C2 x C2 XC2 x ...) where C2 is a group of order 2.

A rather nice example

which is also easy to compute is due to K. Weston and may be found in [29], page 132.

Each of these examples is a group G which is an extension of an

elementary abelian 2-group by another elementary abelian 2-group; hence G has exponent 4 and satisfies [G2,G2] = 1 (which as we point out in 5.9 is stronger than the 3rd-Engel condition on G ) . We could also produce a group with [G2,G2] = 1 and having an infinite properly descending lower

G6B

central chain, by noting that for each n > 2 there is a group G(n) € B satisfying [G(n)2,G(n)2] = 1 with nilpotency class precisely n+1 (see Lemma 5.9B), and by taking G to be their direct product. Again, it is known that in general a 3rd-Engel group is 2metabelian (i.e. every 2-generator subgroup is metabelian) and hence is [2 -*• 4]; furthermore, 3rd-Engel groups are not in general solvable.

It is

also known (§3) that 3rd-Engel groups in B are solvable, so it is of interest to note that 2-metabelian groups in B are not. In order to emphasize them we single out two of the results in Theorem 5.9 as separate lemmas.

We showed in §3 that a 3rd-Engel group

G € B has derived length at most 4 ([15]), but the next result, due to Vaughan-Lee [45] gives us much more precise information:

Lemma 5.9 A. 2

A 3rd-Engel group G with exponent four satisfies the law

2

[a ,b ,c] = 1. Proof.

In [a,c,c,c] = 1 where a and c are arbitrary elements of G the

substitution a -»• ab gives immediately [a,c,b,c,c] = 1 mod y 6 G.

The

Tobin:

Groups with exponenet four

130

substitution c -^ be in this second relation gives (using §2 (7)) [ a , b , b , c , c ] [ a , c , b , b , c ] [ a , c , b , c , b ] = lmody 6 G. Expansion of [a,c,be,be,be] = 1 gives [ a , c , b , b , c ] [ a , c , b , c , b ] [ a , c , c , b , b ] = 1 mod y6G. Combining these results we get [a,b,b,c,c][a,c,c,b,b] = 1 mod yeG i . e . [a,b 2 ,c 2 ] [a,c 2 ,b 2 ] = 1 mod y6G. By the Jacobi relation this gives [b 2 ,c 2 ,a] = 1 mod y6G. Now the B(3) tables are used to show that if G is a three-generator group, say G = < a,b,c >, then yeG = 1. Thus finally [a 2 ,b 2 ,c] = 1. Corollary. If G is a 3rd-Engel group in B then (i) [G2 ,G2 ,G] = 1, thus G is centre-by-metabelian, D3G = 1 and this bound on the derived length of G is best possible; ( i i ) G2 is nilpotent of class < 2, and this bound is best possible. Proof. Clearly [G2,G2,G2] = 1; i f [G2 ,G2 ] = 1 then G is metabelian, but the group $3 (in §3) i s a (non-nilpotent) 3rd-Engel non-metabelian group with exponent 4; thus also D2$3 ^ 1. Lemma 5.9 B. (a) If G is a group with exponent four then [G2,G2] = 1 if and only if G is a [2 •* 3] group. (b) If G e B and [G2 ,G2] = 1 then G is [n + n+1], and this result is best possibles for every n > 2. Proof. If G is [2 + 3] then G satisfies the law [x2 , / ] = 1, thus [G2,G2] = 1. If [G2,G2] = 1, since G2 is generated by elements of period 2 we see that G2 is elementary abelian. Thus G is metabelian and for a l l x,y,z inG, 1= [x,y] 2 = [x2 ,y2 ] = [x,y,z 2 ]; hence 1 = [x,y,y,y] = [x,y,y,x] = [ x , y , z , z ] . I t is clear (see the proof of Theorem 5.9 below) that < x,y > has class at most 3. Any commutator of length n+2 which has a repetition after the 2nd entry i s t r i v i a l (since G is metabelian, the repeated entries may be brought together); if there is no repetition the element may be written [ x , y , y , x , . . . ] and i s again = 1. If F is the free group on n free generators, n > 2, the quotient F/(F 2 ) 2 has exponent 4, satisfies the condition [G2,G2] = 1 and has class exactly n+1 (see [42], or [20] Lemma 5). Corollary.

If G e B and is [2 + 3] then G is [n + n+1] for every n > 2.

Tobin:

Remark.

Groups with exponent four

This result, but with n ^ 3, is Theorem 3.8.2 of [15].

131

In [15]

also it is shown that a 3rd-Engel group G in S is [n •+ n+6] for all n > 14; and the better result that G is [n •* n+2] for large n is attributed to M. Newman (unpublished). We show now that Newman's result is in fact true for all n > 2, and is exact:

Lemma 5.9 C.

A 3rd-Engel group G with exponent four is an [n -> n+2]

group, and this result is best possible, for all n > 2. Proof.

Lemmas 5.9 A and 5.9 B together show that G is a centre-by-

[n •* n+1] group for every n > 2; hence G is [n -*• n+2] for all n > 2. The group $3 already referred to is 3rd-Engel and, for each n > 2, is [n •* n+2] but not [n -»• n+1] .

Corollary (to the proof).

I / G G B and is [3 + 4] then G is [n -> n+2]

for all n > 2. As we are considering conditions of the form [n •> n+k] it may be worthwhile formalizing the following simple observation:

Lemma 5.9 D.

I / G G B and is [n •+ n+k] for fixed k, for all n > N, then

G is solvable (with a bounded derived length). Proof.

Let X = max[N, k+1]; then [gi2 ,g22 ,... ,gx2 ] is of weight 2X > X+k

in X > N variables and so is 1, which implies that G2 is nilpotent of class < X; hence G2 is solvable of bounded derived length, and so G has this property also.

Remark.

This is in some sense a converse to Theorem 4,4 which says that

if G2 has class r then G is [n + n+r] for n > 2r+2; by Theorem 1.2 G2 has class r, for some r, if and only if G is solvable. We now write the general theorem on nth-Engel conditions.

Theorem 5.9. (a) All groups in B are 5th-Engel, and in fact are [2 •* 5]. (b) For a group G in B the following are sets of equivalent conditions: (a) (i) G is 4th-Engel (ii) G is [2 •* 4] ( i i i ) G is 2-metabelian (6) (i) G is [2 - 3] (ii) [G2,G2] = 1 (y) (i) G is 2nd-Engel (ii) [G2,G] = 1, i.e. G is centre-by-elementary abelian ( i i i ) G is [2 + 2] (iv) G has class at most 2.

Tobin:

Groups with exponent four

132

(c) For groups G with exponent four the following conditions acre successively (i) (iv)

weaker, and

G is 2nd-Engel 2

2

[G ,G ,G] = 1

distinct:

(ii) G is [2 -> 3] (v) G is

( i i i ) G is 3rd-Engel

4th-Engel.

Of these conditions only the first

implies that G is

nilpotent,

and all but the last imply that G is solvable. (d) For G e B, the condition

[G2 ,G2 ,G] = 1 implies that G is

centre-by-

metabelian, but the converse is not true.

Proof. Several of these statements are proved by examining B(2); we recall relevant details here, writing y. for y.B(2) and using a,b for the generators of B(2). y6 = 1, Ys i s generated by [b,a,a,a,a] and [a,b,b,b,b]; Y4 i s generated mod ys by [ b , a , a , a ] , [b,a,a,b] and [a,b,b,b]; Y3 i s generated modulo Y4 by [b,a,a] and [a,b,b]. Also [y2 ,yi ] = [Y3 >Y2 ] = Ys ; and [b,a,a,a,a] = [b,a 2 ,b 2 ] = [b2 ,a2 ,b] by the Jacobi relation. Thus (a) and (b)(a) are true, and G i s 2nd-Engel i f and only if G i s [2 -v 2 ] ; also i f [G2 ,G2 ,G] = 1 we see that G i s 4th-Engel. Clearly the condition [G2,G2,G] applied to B(2) can only force ys = 1 and so i s s t r i c t l y weaker than the 3rd-Engel condition, which gives also [a,b,b,b] = [b,a,a,a] = 1 in Y4• We could also show that the 3rd-Engel condition on B(2) does not force [b,a,a,b] = 1 and so i s s t r i c t l y weaker than the condition [2 ->• 3 ] , but we know this already from the non-metabelian 3rdEngel group $ 3 . I t i s known that every 2nd-Engel group satisfies the law 3 [x,y,z] = 1; hence a 2nd-Engel group in B satisfies [x,y,z] = 1 so i t has class < 2, and conversely. Furthermore, i f G i s such a group, [x,y 2 ] = [x,y] 2 [x,y,y] = 1 and so we have [G2 ,G] = 1; conversely [G2 ,G] = 1 =* [x,y] G centre of G => G has class at most 2 (and so (b) (Y) i s true) . Lemma 5.9 B shows that this i s a s t r i c t l y weaker condition than [G2,G2] = 1. We remarked in Lemma 5.5 that groups in B are centre-by-[2 •+• 4 ] , Hence i f [2 -> 4] groups were solvable, a l l groups of exponent 4 would be solvable, which i s not the case. Hence 4th-Engel i s a s t r i c t l y weaker condition than [G2 ,G2 ,G] = 1. We have already remarked that [G2 ,G2 ] = 1 does not imply nilpotency; Lemma 5.9 A completes the proof of (c), and 5.9 B gives (b) (3). For (d), i t suffices to notice that B(2) is centreby- metabelian, and so [G2,G2,G] = 1 i s a s t r i c t l y stronger condition. This completes our proof of Theorem 5.9. Continuing the process of getting more exact information on B(n), A.J.S. Mann has examined the quotients y.B(n)/y.+^B(n), which are

Tobin:

Groups with exponent four

133

elementary abelian for i > 2, in order to get reasonably good bounds for the order of B(n) when n > 5.

I am thankful to him for sending me a

preprint of his paper [35] on the subject.

This is what one might now

call quite classical in its technique, using all the standard congruences to establish normal forms for generating sets for certain subgroups of Y./y-+i (f° r the present discussion we will write B for B(n)

which is

generated by {xi,X2,...,x }, and y^ for Y ^ B ) • If we let r(t) be the rank of Y t /Y t + 1 > t > 2, and if we let st(k) be the rank of the subgroup generated modulo y

. by the simple

commutators of positive weight in each of xi,X2,...,x, (and in these only) and of total weight t, then we have

3n-2 logjB| = 2n

n

r(t)

<

t=2

2n

H

r

il

" k=2 1*'>

3k

"2

I

t=k

s. i

The central idea is to look at the dimension of the subgroup generated, modulo Y t + 1 > by all the commutators in {xi,...,x,} which have a specified weight vector (wi,...,w.) (i.e. have weight w. in x. , 1 < i < k, where 1 < w. and Ew. = t ) . l

l

J

As a simple example of the kind of result which is developed for this purpose I quote the following: Lemma 5.10 A. group G G B.

Let c be a commutator with entry set {xi,...,x, } in a If c has weight 3 in x. and weight 2 in x. (i ^ j) then c

is a product of commutators with the same entry set in which the x. entries are adjacent. The main part of the paper is concerned with a bound for the rank of the group G, modulo y, .. where G, is the subgroup generated by the k! commutators [x

,x2 ,...,x, ] where a is any permutation on k letters.

If we denote this rank by f(k) the following proposition is established.

Lemma 5.10 B.

f(k) < 2f(k-l) + f(k-2) + 3.

This gives f(k) < C(l + /2)

for some constant C.

Since

f(2) = 1, f(3) = 2 and f(4) = 5 (from the tables) we may take C = j here. The ranks of the groups generated by vectors with different entry weights in {xi ,... ,x, } are bounded in terms of the function f (k); K 3k-2 this leads to an upper bound for £ s (k), and this in turn yields an t=k r upper bound for |B|. The final result is:

Tobin:

Theorem 5.10.

Groups with exponent four

134

There is a constant K such that < K exp2 ((4 + 2/2)n) .

This confirms a conjecture of G. Higman [28] to the effect that the sequence {— log log|B(n)|} is bounded.

As a lower bound for

|B(n)| the estimate Ki exp2 (4 ) is given, where Ki is a constant very close to 1.

This improves the lower bound exp2(l+n + 2 (n-1)) given previously

in 1947 by Sanov in his paper "On Burnside's problem, Dokl. Akad. Nauk, SSSR (N.S.)" (and also independently in 1960 by Tobin in "Simple bounds for Bumside p-groups, Proc. Amer. Math. S o c " ) . Finally we come to Razmyslovfs Theorem, which has had a decisive effect on the theory of groups with exponent four.

Theorem 5.11 ([40]).

There exists a non-solvable group of exponent four.

The idea is to construct a group of exponent 4, with generators of period 2, inside a quotient ring of an associative ring of polynomials in infinitely many non-commuting variables, very much like the group G described in the earlier part of §4.

The difference is that Razmyslov

constructs a non-(Lie-) solvable quotient algebra in which the group is embedded.

The details do not lend themselves to a brief exposition. As we have already seen, due to work of Gupta, Newman and

Quintana, this result has the important corollary.

Corollary.

The class of B(n) is 3n-2, for n > 2. We have used the result in Theorem 5.11 in a number of places.

To conclude, we mention a consequence of a somewhat different kind, given in [46] (where a similar result is established for certain other exponents also) :

Theorem 5.12.

There exists a countable group of exponent four which has

no maximal subgroups. REFERENCES

1. S.I. Adi an, The Bumside problem and identities in groups * English translation by J.C. Lennox 5 J. Wiegold, Springer-Verlag, Berlin (1979). 2. W.A. Alford, G. Havas § M.F. Newman, Groups of exponent four, Notices Amer. Math. Soc. 22 (1975), 301.

Tobin:

Groups with exponent four

135

3. S. Bachmuth, Exceptional primes in a variety, in Parkside Conference 1972, Lecture Notes in Mathematics, Vol. 319, Springer-Verlag, Berlin (1973), 19-25. 4. S. Bachmuth § H.Y. Mochizuki, A criterion for non-solvability of exponent 4 groups, Cornm."PureAppl. Math. 26_ (1973), 601-608. 5. S. Bachmuth, H.Y. Mochizuki § K. Weston, A group of exponent 4 with derived length at least 4, Proo. Amer. Math. Soc. 3£ (1973), 228-234. 6. A.J. Bayes, J. Kautsky § J.W. Warns ley, Computation in nilpotent groups (application), in Proo. Second Internat. Conf. Theory of Groups, Springer-Verlag, Berlin (1974), 82-89. 7. R.H. Bruck, Engel conditions in groups and related questions. Lecture Notes, Austral. Math. Soc., Canberra (1963). 8. W. Burnside, On an unsettled question in the theory of discontinuous groups, Quart. J. Pure Appl. Math. 33^ (1902), 230-238. 9. H.S.M. Coxeter § W.O.J. Moser, Generators and relations for discrete groups, 2nd ed., Springer-Verlag, Berlin (1972). 10. J.K. Doyle, K.I. Mandelberg § M.R. Vaughan-Lee, On solvability of groups of exponent four, J. London Math. Soc. j^8 (1978), 234-242. 11. C.C. Edmunds $ N.D. Gupta, On groups of exponent four IV, in Parkside Conference 1972, Lecture Notes in Mathematics, Vol. 319, Springer-Verlag, Berlin (1973), 57-70. 12. E.S. Golod, On nil-algebras and residually finite groups, Izv. Akad. Nauk USSR Ser. Math. 2%_ (1964), 273-276. 13. F.J. Grunewald, G. Havas, J.L. Mennicke $ M.F. Newman, Groups of exponent eight, Bull. Austral. Math. Soc. 20_ (1979), 7-16. 14. C.K. Gupta § N.D. Gupta, On groups of exponent four II, Proc. Amer. Math. Soc. 31 (1972), 360-362. 15. N.D. Gupta, Burnside"groups and related topics, University of Manitoba (1976). 16. N.D. Gupta, H.Y. Mochizuki $ K.W. Weston, On groups of exponent four with generators of order two, Bull. Austral. Math. Soc. IO_ (1974), 135-142. 17. N.D. Gupta § M.F. Newman, The nilpotency class of finitely generated groups of exponent four, in Proc. Second Internat. Conf. Theory of Groups, Springer-Verlag, Berlin (1974), 330-332. 18. N D. Gupta § M.F. Newman, Groups of finite exponent, Bull. Austral. Math. Soc. \2_ (1975), 99. 19. N D. Gupta $ R.B. Quintana Jr., On groups of exponent four III, Proc. Amer. Math. Soc. 33. (1972), 15-19. 20. N D. Gupta $ S.J. Tobin, On certain groups with exponent four, Math. Z. 102 (1967), 216-226. 2 1 . N D. Gupta $ K.W. Weston, On groups of exponent four, J. Algebra 17_ (1971), 59-66. 2 2 . M Hall Jr., Solution of the Burnside problem for exponent six, Illinois J. Math. 2_ (1958), 764-785. 2 3 . M Hall Jr., The theory of groups, Macmillan, New York (1959). 24. M Hall Jr., Generators and relations in groups - the Burnside problem, Lectures on Modern Mathematics, Vol. II, Wiley, New York (1964), 42-92. 25. M Hall Jr., Notes on groups of exponent four, in Parkside Conference 1972, Lecture Notes in Mathematics, Vol. 319, Springer-Verlag, Berlin (1973), 91-118. 26. P Hall § G. Higman, On the p-length of p-soluble groups and reduction theorems for Burnside's problem, Proc. London Math. Soc. 6_ (1956), 1-42.

Tobin:

Groups with exponent four

136

27. G. Havas § M.F. Newman, Applications of computers to questions like those of Bumside, in Burnside groups9 Lecture Notes in Mathematics, Vol. 806, Springer-Verlag, Berlin (1980), 211-230. 28. G. Higman, The orders of relatively free groups, in Proc. Int. Conf. Theory of Groups Canberra 1965* Gordon and Breach, New York (1967), 153-165. 29. M.I. Kargapolov § Ju.I. Merzljakov, Fundamentals of the theory of groups* Springer-Verlag, Berlin (1979). 30. A.I. Kostrikin, On Burnside's problem, Dokl. Akad. Nauk SSSR 119 (1958), 1081-1084. 31. J. Leech, Coset enumeration on digital computers, Proc. Cambridge Philos. Soo. 5£ (1963), 257-267. 32. F.W. Levi § B.L. van der Waerden, Uber eine besondere Klasse von Gruppen, Abh. Math. Sem. Univ. Hamburg 9_ (1933), 154-158. 33. I.D. Macdonald, Computer results on Burnside groups, Bull. Austral. Math. Soc. £ (1973), 433-438. 34. W. Magnus, A. Karrass $ D. Solitar, Combinatorial group theory* Interscience, New York (1966). 35. A.J.S. Mann, On the orders of groups of exponent four, submitted for publication. 36. H.Y. Mochizuki, On groups of exponent four: a criterion for nonsolvability, in Proa. Second Intemat. Conf. Theory of Groups* Springer-Verlag, Berlin (1974), 499-503. 37. M.L. Newell § R.S. Dark, On certain groups with a fourth-power endomorphism, Proc. Roy. Irish Acad. 80A (1980), 167-172. 38. M.F. Newman, Bibliography, in Burnside groups* Lecture Notes in Mathematics, Vol. 806, Springer-Verlag, Berlin (1980), 255-274. 39. P.S. Novikov $ S.I. Adian, Infinite periodic groups I, II, III, Izv. Akad. Nauk SSSR Ser. Mat. 32: (1968), 212-244, 251-524, 709-731. 40. Ju.P. Razmyslov, On the Hall-Higman problem, Izv. Akad. Nauk SSSR Ser. Mat. 42 (1978), 833-847.

41. I.N. Sanov, Solution of Burnside's problem for exponent four, Leningrad Gos. Univ. Ped. Inst. Uc. Zap. Mat. Ser. 2£ (1940), 166-170. 42. S.J. Tobin, On groups with exponent 4, Ph.D. thesis, University of Manchester (1954). 43. S.J. Tobin, On a theorem of Baer and Higman, Canad. J. Math. 8_ (1956), 263-270. 44. S.J. Tobin, On groups with exponent four, Proc. Roy. Irish Acad. 75A (1975), 115-120. 45. M.R. Vaughan-Lee, Derived lengths of Burnside groups of exponent 4, Quart. J. Math. 30 (1979), 495-504. 46. M.R. Vaughan-Lee § J. Wiegold, Countable locally nilpotent groups of finite exponent without maximal subgroups, Bull. London Math. Soc. JU5 (1981), 45-46. 47. C.R.B. Wright, On groups of exponent four with generators of order two, Pacific J. Math. 1£ (1960), 1097-1105. 48. C.R.B. Wright, On the nilpotency class of a group of exponent four, Pacific J. Math. 11 (1961), 387-394.

137 THE SCHUR MULTIPLIER:

AN ELEMENTARY APPROACH

J. Wiegold University College, Cardiff, CF1 1XL, Wales

1. HISTORICAL INTRODUCTION AND APOLOGIA The Schur multiplier was born in the important paper [35]. Schur talked about "fractional linear substitutions" of a finite group G, but we think in terms of projective representations p :G

> GL(n,(C)/Z,

where Z is the group of scalar matrices in GL(n,(E), that is, the centre. For each a in G, p(a) is a coset mod Z so that we can choose a matrix A(a) e p(a), one for each a in G. Then as p(ab) = p(a)p(b), for all a,b € G, we have A(a)A(b) = r , A(ab) for some complex number r , € ( [ * . a,D a,D Associativity of matrix multiplication gives r

a,b r ab,c

= r

a,bc r b,c

fl)

for all a,b,c in G. Thus r is a 2-cocycle, but it won't worry us. Conversely, for any r : G x G

> (E* satisfying (1), there is

a projective representation of degree |G| = m giving rise to it, like this.

For each u in G, let A(u) be the mxm matrix with rows and columns

indexed by the elements of G and having entry r

_i

in row p, column

u-V Suppose now that we had made a different choice B(a) e p ( a ) , so that B(a)B(b) = s K B(ab) for suitable s ,. Then A(a) = d B(a) for a, D a, D a some d € ([*, and we get A(a)ACb) = r a>b A(ab) =

r^d

dadbB(a)B(b) = d a d b s a ) b B ( a b ) Thus for all a,b in G,

Wiegold:

The Schur multiplier

Two functions r,s : G x G d : G

138

> (C* are equivalent if there exists a function

> (C* such that (2) holds.

This is clearly an equivalence

relation, and the set of all equivalence classes [r] of functions satisfying (1) is a group under the obvious multiplication [ri][r2] =

[r.r,]. In fact it is merely the factor-group of the group of 2cocycles by the group of 2-coboundaries (those 2-cocycles r such that r

, = d d, /d , for some d : G a,b

a D

> (C*). J

ab

This group is the "Multiplikator" M(b) of G; we call it the multvpViev of G. follows.

It is commutative, and Schur proved it finite, as

Consider [r] e M(G). We have a representation of degree

m = |G| associated with r, and as above, with mxm A(a) etc. y A(a) A(b) = r , A(ab) a, u so det A(a) det A(b) = r m , det A(ab). a, D Let 6

a

stand for any m-th root of det A(a), 6 = det A(a); and set a

sa,b =-^±-i 6

ra,b'

ab

Thus s

, = 1. But [r] = [s], so that every r is equivalent to s such a, D that s , is an m-th root of 1. It follows that a,b 2

|M(G)| < m

.

This was the best-known bound for |M(G)| until the 1950's, when Walter Ledermann and B.H. Neumann improved on it a little [29] and they stimulated J.A. Green to get a much better answer [10]. Later we shall see a somewhat shorter proof than Green's, which depended on spectral sequences. Here's one way in which protective representations arise naturally.

Let p* : H

> GL(n, (C) be an ordinary irreducible

representation of H, and A a central subgroup of H.

By Schurfs Lemma,

for each a e A we have p*(a) = j(a)I , where j (a) is an n-th root of 1. We get a projective representation of G = H/A by defining p : H/A

> GL(n, (C)/Z in the obvious way: p(h A) = p*(h)Z.

Wiegold: The Schur multiplier

139

Schur proved the following converse. If G is any finite group, there exists a group H and a central subgroup A such that H/A a G and every irreducible protective representation of G comes from an ordinary representation of H in the way just described. For given G, the H of smallest order are called "Darstellungsgruppen11 (covering groups in modern English); though they may be non-isomorphic in general, the subgroup A involved is always isomorphic with M(G). Schur proved that different covering groups of G have isomorphic derived groups; Jones and I [27] proved them to be isoclinia, and we'll see the proof later. I thank Professor Joachim Neubuser for pointing out that this result was proved already by P. Hall in [15]. Another property of covering groups discovered by Schur, and one we shall concentrate on, is this. For any group G, a defining pair (after W. Haebich) for G is a pair (H,A) such that (i)

A C Z(H) n H»,

(ii) H/A a G. It will be seen that orders of first members H of defining pairs for G are bounded. The A going with H of maximum order are all isomorphic to M(G). What we shall do is to prove them isomorphic and use this to define M(G). The method was evolved by M.R. Jones (see [23]), though I am sure plenty of other people have had the same idea. This type of approach is the one that has been found useful in getting simple proofs of results on the multiplier. As a quick example we work out M(Q 8 ), using the well-known result. Theorem 1.1. For any group X, Xf n Z(X) is omissible. Write Q8 = < a,b| a2 =b2 = [a,b] >, and suppose that H/A = Q 8 with A C Z(H) n H 1 . Then H = < a,3 > where aA = a and 3A = b, and of course a,3 satisfy the following congruences modulo A: a2 = 32 = [a,3]. As H is central, it follows at once that H is nil-2 ([a,3] commutes with a and 6) and so 1 = [a2 ,3] = [a,3]2 and Hf is of order 2. But A C H ! and H/A = Q8 ; and the only conclusion is A = l so that M(Q 8 ) = 1. Not very exciting, but it illustrates the strategy. Finally, here's another way of looking at the multiplier (Hopf [18]). Write your finite group as G = F/R, where F is free, of rank k say. Then F/[F,R]/R/[F,R] a G;

Wiegold:

The Schur multiplier

140

as R/[F,R] is central and of finite index in F/[F,R], i t follows that F f /[F,R] is f i n i t e . A fortiori, (Ff H R ) / [ F , R ] is f i n i t e . Now R/[F,R]/(F f n R)/[F,R] a R/Ff HR s* F'R/F f , which is a subgroup of finite index in the free abelian group F/Ff of rank k, so that i t is i t s e l f of rank k. That i s , R/[F,R] = C(F» n R )/[F,R])xT, where (Ff n R)/[F,R] is the finite part of R/[F,R] and T is free abelian of rank k. The beautiful result is that (Ff H R ) / [ F , R ] is always M(G), whatever presentation you give for G. Suppose that G has a presentation on k generators and r relations, G = F/R where F is free of rank k and R = < xt ,...,x > F . With bars denoting images mod [F,R], _

_

_

R = < Xj ,... ,x r >

F"

—

B

-

< Xj ,... ,x r >.

— = d(M(G)) + k, where d means minimum number But —R = M(G) x TLk, so d(R) of generators. Thus r > k + d(M(G)). Corollary 1.2. If G is a finite group, then in any presentation on k generators, at least k + d(M(G)) relations are necessary. So for example, cyclic groups and groups like , i m , n r , -. >. < a,b|a =b = [a,b] > have trivial multipliers; dihedral groups < a,b|an =b 2 = (ab)2 = 1 > have cyclic multipliers, etc. In [34], B.H. Neumann asked whether a group with trivial multiplier always has a k-generator k-relator presentation. To ease the next bit of discussion, define: Definition 1.3. For any finitely presented group G, the deficiency def(G) is the minimum of r(£) - gOP) taken over all presentations ¥ of G on g(P) generators and r(P) relations. Swan [37] answered Neumann's question by making groups with trivial multipliers and arbitrarily large deficiency. His groups require large numbers of generators, so I thought it would be amusing to do it with small generating numbers (see §5 for details). Groups with zero deficiency are exceptionally difficult and rare: see the nice survey of Johnson and Robertson [22]. Very recently Campbell and Robertson [6]

Wiegold:

The Schur multiplier

141

have given a clever 2-generator 2-relation presentation for SL(2,p), p a prime. For a survey on presentations in general, see Warns ley [40]. It will be obvious that I have made no attempt to be encyclopaedic in these lectures, so that no doubt many relevant references are absent. The aim was to take a l i t t l e corner of the study of multipliers, namely the specifically "group-theoretical" corner, and talk about that in as self-contained a manner as possible. For a wider viewpoint see the forthcoming book by Beyl and Tappe [5]. I thank the many people at the conference at St. Andrews who gave advice, criticism eto.9 especially Rudolf Beyl. I also thank Dr. Colin Campbell and Dr. Edmund Robertson for their unfailing polite and kindly assistance with the preparation of this a r t i c l e : at numerous times they have altered, added, checked and corrected without a qualm. 2.

TRANSFER AND GROWS WITH FINITE

CENTRAL FACTOR-GROUPS

The only result on the transfer that we need is the following: Theorem 2 . 1 . Let K he a subgroup of index n in a group G,, and T the n transfer sfer iinto an abelian factor-group K/K. Then x(g) = g K for all Z(G).

A proof can be found in Huppert [19] and in lots of other Huppert is a good general reference for facts not proved here. To get our definition of M(G), we need the following result

places. of Schur.

Lemma 2 . 2 . finite Proof.

Let G be any group with G/Z(G) finite

of order q.

Then Gf

is

of order hounded in terms of q. Write G = Z(G)ri U . . . U Z(G)r , so t h a t G* i s generated by

elements of the form [ z r . , w r . ] with z,w € Z(G) = Z, t h a t i s , by the I q ( q - l ) elements [ r ^ r . ] ,

1 < i < j < q.

But (G/Z) f = G ' Z / Z s G'/G'OZ

i s f i n i t e of order a t most q, so by S c h r e i e r f s formula Gf n z can be generated by a q-bounded number of elements.

All we need do i s prove t h a t

G'HZ has q-bounded order; and to do t h a t , i t i s enough t o prove t h a t every element has q-bounded order. of G i n t o Z.

That i s easy. q

Let T be the t r a n s f e r

Then for x £ G'HZ, T(X) = x by Theorem 2 . 1 .

abelian and x e G f , so T(X) = 1.

But Z i s

Thus x q = 1.

The answer given here ( i m p l i c i t l y ! ) for | G f | in terms of q i s awful.

We s h a l l find the r i g h t answer l a t e r .

Wiegold: The Schur multiplier

142

3. MULTIPLIERS VIA PRESENTATIONS AND DEFINING PAIRS Definition 3.1. For any finite group G, a defining pair is a pair (H,L) of groups suoh that L c Z(H) n H 1 and H/L SB G. Observe that |H/Z(H)| < |G|, S O that |H'| is bounded in terms of |G|, by Lemma 2.2; so |L| is, so |H| is. Thus we can speak of maximal defining pairs; we shall prove that the second members of maximal defining pairs for G are all isomorphic. It is all summed-up in: Lemma 3.2 (M.R. Jones [23]). Let F be a free group and R a normal subgroup such that F/R a G is finite. Put B = R/[F,R], C = F/[F,R], D = (F1 nR)/[F,R], Then (i) D is the torsion part of B and is finite; (ii) if (H,L) is a maximal defining pair for G., then |D| < |L|; (iii) if (K,N) is any defining pair for Q3 then N is an epimorphio image of D. Proof. As in §1, R/[F,R] is central and of finite index in F/[F,R], so Ff/[F,R] is finite and D is finite. But B/D a F'R/F' is free abelian so B = D x E for some E. Then (C/E, B/E) is a defining pair for G, and we have done (i), (ii). Part (iii) is slightly more complicated, though not very complicated. Let X be a free basis for F, and IT : F » G the natural epimorphism with kernel R, so that G = < XTT >. Let a be the natural epimorphism of K onto G with kernel N. We should have at least one picture, so here it is:

For each x in X, choose an element k in K such that k^a = XTT. We have G = < Y^o : x e X > so then K = N< k x : x € X > and finally K = < k x : x e X > as N is omissible. Thus the homomorphism p : F > K defined by xp = kx is an epimorphism, and pa = TT as xpa = kxa = XTT for all X.

Next, i t i s c l e a r t h a t Rp = N as f € R <=> fir = 1 «•* fpa = 1 «-> fp € N.

Wiegold:

The Schur multiplier

143

Thus [F,R]p = [Fp,Rp] = [K,N] = 1, and p induces an epimorphism p* : F/[F,R] > K, (f[F,R])p* = fp for all f in F. We are going to show that p* maps D onto N. To see this we f i r s t observe that (Ff HR)p = F'pHRp : y 6 F'pHRp => y = f p = rp with obvious notation, while ffTr = f'pa = rpa = nr = 1, so that ff e F» O R and y e (Ff H R ) P . Thus finally Dp* = (F1 H R ) p = F'p n Rp = Kf O N = N. Theorem 3.3. With the notation and assumptions of Lemma 2.23 the second members of maximal defining pairs are all isomorphio to (Ff H R ) / [ F , R ] . The proof is obvious and we can define the multiplier M(G) of G to be any group isomorphic to the second member of a maximal defining pair for G. The first member of a maximal defining pair is a covering group for G. As a corollary to the proof of Lemma 3.2, we have: Corollary 3.4 ([27], see also [15]). Covering groups for a given finite group are mutually isoclinic. Proof. In the set-up of the proof of Lemma 3.2, suppose that K is a covering group, so that D = N and thus p*|D is monic. Consider Ker p* n (F/[F,R])f. Firstly f[F,R] € Ker p* =* fp = 1 => f e R, so that Ker p* n (F/[F,R])f c Ker p* n (Rr»Ff)/[F,R] = Ker p* n D = 1. Thus p* is an isoclinic homomorphism from F/[F,R] onto K (see B.H. Neumann [33]); and therefore all covering groups for G are isoclinic with F/[F,R]. For more on this, see Tappe [38] and Beyl [4]. 4. SYLOW THEORY OF THE MULTIPLIER. SOME BETTER BOUNDS We know now that M(G) is finite, abelian, and of order bounded in terms of G. With A standing for a Sylow p-subgroup of a group A, we shall see now that M(G) < M(G ) for every prime p, so that it will be advantageous to look at p-groups. First one of the most useful of the simple results: Lemma 4.1 (Schur [35], Jones and Wiegold [26]). Let H be any group, A a central subgroup of finite index* G = H/A. Then H 1 n A is an epimorphic image of M(G).

Wiegold: The Schur multiplier

144

Proof. Let H = F/R with F free, A = T/R so that G = F/T and thus M(G) = (F! rtT)/[F,T]. But H 1 n A = (F'RHT)/R = (Ff HT)R/R as R C T, so that Hf n A = (Ff ni)/(F ! n R) . But F1 n R D [F,T] as A is central, so that (F1 nT)/(F f n R) is an epimorphic image of M(G). In fact, Schur proved that H1 n A is a subgroup of M(G), using characters. Since M(G) is finite and abelian, this is the same result. Sylow theory is based on the following result which can be found in [11], or rather more simply in [26]: Theorem 4.2. Let G be a finite group and A a subgroup of index n. Then the group 3 (M(G)) of n-th powers of elements of M(G) is a subgroup of M(A). Proof. Let (H,L) be a maximal defining pair for G, so that L = M(G). Put A = B/L and let T be the transfer of H into B/Bf. Then T(X) = x n B' for all x in L, as x e Z(H); but T(X) = 1 as x e H 1 . Then x n e B1 and so Bn(L) < B f n L < M(A). Corollary 4.3. The exponent of M(G) divides the index of every cyclic subgroup. In particular, it divides |G|. See Huppert [19] for a proof for cyclic normal subgroups. Corollary 4.4. M(G) < M(G ) . Proof. Put A = G , so that n = |G : A| is prime to p. Corollary 4.5 (Schur [35]). The group $f (M(G)) of elements of order prime to n is contained in M(A). This follows since B'(M(G)) C 3 (M(G)); every element of order n n prime to n is an n-th power of itself. Clearly, $f(M(G)) c $ (M(G)) in many cases. While on this subject: it was a long-unsolved problem as to whether the exponent of M(G) divides the exponent of G. Macdonald and Wamsley see [3] settled this by constructing a group of exponent 4 (and order 22l) with multiplier of exponent 8. Schur proved that if G is a p-group of order p , then M(G) has exponent at most p . These methods can be used to prove this result,

Wiegold:

The Schur m u l t i p l i e r

145

and a l s o : Exercise, |G|

= p

If |G| = p m+

m

and G has exponent p m , then G a ZZm x 2Zm«

and G has exponent p m , then G i s one of:

TL

If

P P x 7L 1,

p p mm p p 2Z m x 2 x z , < a,b|a = b = [a,b,a] = [a,b,b] = [a,b] = 1 >. p p P So far we have only a poor idea of how big M(G) can be. To get better answers, we need only look at p-groups, and remember the strategy: take H/N s G with N C Hf n Z(H) and N a M(G). If we can make a good stab at |H1|, we'll be getting somewhere. p

Theorem 4.6 ([42]). Let H be a group sueh that H/Z(H) is a finite p-group of order p n . Then Hf is a p-group of order at most p 5 n ^ n " ^. Proof. Proceed by induction on n. For n < 1 all is well, so assume n > 2 and that the theorem is true for groups with smaller central factor-groups. Choose a e Z2 (H)\ Zi (H) and set N = < [a,h] : h e H >. Then N is central; since [a,hih2] = [a,hi][a,h2] in this situation, N consists of these commutators so we shall count them to see how big N is. Clearly [a,hi] = [a,h2] ~ h"1 aht = h^1 ah2 «-* C(a)hi = C(a)h2 , so |N| = |H : C(a) | . But C(a) > Z(H) so that |N| < p n . Now look at Z(H/N). It contains Z(H)/N, but it also contains aN as [a,H] C N. Thus Z(H/N) > Z(H)/N, so that |H/N : Z(H/N) | < p n . Thus by induction |H'/N| < piCn-IDCn-2) and go |H,| < pJn(n-l) as ^ < p n " 1 . What really makes this proof tick is the fact that a € [a,H] =» a = 1 in a nilpotent group H. It would be interesting to know precisely what groups have this property: it is easy to see that it is a criterion for nilpotency for finite groups. As an immediate corollary we have: Corollary 4.7 (Green [10]). If G is a finite p-group of order p11, then We shall prove that these bounds are attained, l a t e r . For now we observe that they have been strengthened in a number of ways, for example: Theorem 4.8 (Gaschutz, Neubiiser, Ti Yen [7]). then

If G is a finite -p-group^

Wiegold:

The Schur multiplier

146

| |G.|dCG/ZCG))-l See Vermani [39] for a homological version of t h i s . Theorem 4.9 (Jones [ 2 4 ] ) .

If G is a finite

p-group

of order p n with

d(G) = d, then

This result of Jones is one of the more elegant of these simply-achieved bounds. Let us return to a general group G with central factor-group (Xi

O'V

(Xy

of order q. Write q = pj p2 . . . p, , where the p. are primes, pi < P2 < . . . < p, and a l l the a. are non-zero. Then by all that has K

1

gone before

|G-nZ(G)| < |M(G/Z)| < p l ° ' ( « ' - i y * ( * ' 1 ) . . . P v 4 ^ " 1 ' 1

2

K

so, with Z = Z(G) , |G'|

= |(G/Z).| |G'OZ| < p » » « ( - * " p l « « i C - . * l ) . . 1

2

*°k(V" K

This can be attained only when G/Z is perfect and has elementary abelian Sylow-subgroups, for example when G/Z a PSL(2,5). In fact a l i t t l e bit of arithmetical adhoccery gives: K g (i) If k > 2, then |G'| < q {(logPlq-1) (ii) If k = 2, then |G'| < q l(logo-l) pi (iii) If k = 1, then |G'| < q

Theorem 4.10 ([42]).

pi

q-k+2)

Item (ii) requires a knowledge of solubility for small values of q; and (iii) is Theorem 4.6. Altogether one has |Gf| < q P , where p is the smallest prime divisor of q = |G/Z(G)|. We shall see later that this bound is attained if and only if q is a p-power (other than p, of course). 5.

MULTIPLIERS OF DIRECT PRODUCTS. ABELIAN GROUPS. DEFICIENCY PR0BLE14S

The main r e s u l t here i s : Theorem 5.1 (Schur [ 3 6 ] , Wiegold [44]). then M(AxB) a M(A) x M(B) x (A<8>B).

If A and B are any finite

groups*

Wiegold:

The Schur multiplier

147

The proof I give will seem quite natural, and I think easier than those in standard texts. Firstly recall that the tensor product A ® B is the group generated by formal pairs a ® b ("tensors") with defining relations a ® b i b 2 = (a®bi) (a<8>b2);

ai a2 ® b = (ai ®b) (a2 ® b ) .

This group is always abelian (in fact it is isomorphic with [A,B]/[A,B,F], where F = A * B: see MacHenry [31] or Wiegold [41]), and it is isomorphic with A/A1 B/B 1 . Right, we see what we can find out about M(A x B). Suppose that H/N = A x B where N C H 1 n Z(H), and let X,Y be the subgroups of H such that X/N = A, Y/N = B. They are normal subgroups of H and we have H = XY, [X,Y] C X n Y = N. But H 1 = X»Y![X,Y] and I claim that N = (Xf HN)(Y f ON) [X,Y]. For, let h e N. Then h e H 1 and so h = xyu with x e X ! , y 6 Y ' , u € [X,Y]. We have x = h(yu)" 1 € Y n X as h and (yu)"1 both lie in Y, so that x € N; and similarly y e N. We have proved that N C (Xf HN)(Y f nN)[X,Y], and the reverse inclusion is clear. But now Xf n N C M(A), Yf n N c M(B) by Lemma 4.1, and we need to know something about [X,Y]. It is a homomorphic image of A <8> B, as the following shows. For each a e B, b G B we choose x € X, y, € y a D such that x N = a, y, N = b. Then the map a & b i > [x ,y, ] extends to a D a D a homomorphism from A ® B to [X,Y]. The verification uses the facts that [X,Y] and N are central in H, and a bit of commutator calculation. Putting this all together shows that |M(A*B)| < |M(A) I |M(B) I |A<S>B|. To prove the isomorphism, the trick is to construct a defining pair for A x B out of covering groups U,V for A,B respectively, where the second element is isomorphic with M(A) x M(B) x (A<8>B). This is then bound to be a maximal defining pair, and the theorem is proved. We choose the second nilpotent product S (see Golovin [9]) of U and V. The salient features are that every element s of S has a unique expression s = uvw with u € U, v € V, w € [U,V], and that [U,V] is central and isomorphic with U ® V. We know further that U/L - A where L C U ' n Z(U), L a M(A); and V/M a B where M C V n Z(V) and M a M(B). In S we choose the subgroup T = < L,M,[U,V] >. An easy calculation shows that it is central; it is in Sf and S/T ^ u/L x V/M a A x B,

Wiegold:

The Schur multiplier

because of the normal form above.

148

Also because of it,

T a M(A) x M(B) x (U ® V) , and the last thing we need to observe is that U ® V a U/Uf

<8> V / V

a A / A ' <8> B / B ' .

This now gives the multipliers of abelian groups, since we know that cyclic groups have trivial multipliers and we can use the obvious generalisation of Theorem 5.1: M(Ai x... x A ) a M(Ai) x... xM(AJ n

n

x

TJ* l
(A. <8> A . ) . X

3

Thus the multiplier of an elementary abelian p-group of order p

has order p

best possible.

, which proves that the bound in Corollary 4.7 is So is the bound for |G'| in terms of |G/Z(G)| = q given

just after Theorem 4.10; the second nilpotent product of n cyclic groups of order p has central factor-group of order p n and commutator subgroup . Jn(n-l) c of order p Going back to direct products, it is clear that they tend to have non-trivial multipliers.

In fact Ai x... xA

has trivial multiplier

if and only if all A. do and A. ® A. = 1 for i ^ j. where Ai = A2 - .. . - A n = A; writing A

n

A particular case is

for the direct power of n copies

of A, we see that M(A n ) = 1 if and only if M(A) = 1 and A = A f .

Here is

an example of such a thing:

Theorem 5.2 (Kervaire [28], see also Iwahori-Matsumoto [20]). covering group for a perfect group G. Proof.

Let A be a

Then M(A) = 1 and A = A 1 .

I thank Rudolf Beyl for pointing out the Kervaire and Iwahori-

Matsumoto references:

I was unaware that this result was in print any-

where, but quite aware that it was well-known! That A = A' is clear. and M a M(G).

Here's an easy proof.

Suppose that A/M a G with M C Z(A) n A r

To prove that M(A) = 1, let (H,N) be a defining pair for A,

so that H/N a A, N C Z(H) n H f . K/N = M then K C Z2 (H).

Look at H.

Firstly H = H', and if

However Z2(H) = Z(H) as H is perfect, so that

K c Z(H) and H/K

a H / N / K / N a A/M

a

G;

as K C H' n Z(H) we have that (H,K) is a defining pair for G.

But then

|H| < I A.I by maximality, so that N = l and M(A) = 1, as required.

Wiegold:

The Schur multiplier

149

Here is a fantastically difficult problem:

If G is any finite group ± 1, does def(Gn) tend to oo as

Problem 5.3. n •*• oo ?

Since M(G n ) as M(G) n x (G ® G)^n'l\ when M(G) = 1 and G = G!.

the only problem arises

Absolutely nothing is known about this except

some results of Gruenberg [12] in the "relations abelianised" case, where it is shown that the "abelianised deficiency" does not tend to oo.

Even

for the smallest problem case, that of the binary icosahedral group B, one B itself has deficiency zero, whereas def(B2) is unknown.

is stumped.

The difficulties are clear:

unless one can hit on a fortuitous

presentation, there is little hope.

I am prepared to conjecture that B2

has non-zero deficiency, and a (not very) conceivable approach would be to show that As2

needs at least n + 3 relations in any n-generator

presentation (here As is the alternating group, not somebody's Sylow 5subgroup).

For if B2 had an n-generator n-relation presentation, killing

the centre needs two further relations to give Ag2 . better to try to show that B by showing that As19

19

Possibly it would be

has no 2-generator 2-relation presentation

has no 2-generator 21-relation presentation.

Why?

Well, because there is just one normal subgroup N of the free group F of rank 2 such that F/N = A ^ 9 , namely the intersection of the 19 normal subgroups with As as factor-group (see P. Hall [14]), so there is absolutely only one way of presenting As19

as a 2-generator group.

However, even this is not much help, and though I like Problem 5.3, I think it will be a long time before anything happens to it.

In this area,

problems are easy to invent! Finally in this section, the Swan examples and their generalisations.

Swan proved that the group

< ai ,a2 ,...,an,c|c3=l, a?=a2., [a^a.] = 1,

1 < i, j. < n >

has trivial multiplier and deficiency tending to oo with n.

It is an

(n+1)-generator group; we do something very similar with 3-generator groups, partly as a further illustration of the elementary approach adopted here.

Let T, = < a,b > for the free k-th nilpotent product of two

7-cycles, T k = < a,b|a7 =b 7 =1, [ti ,t2 ,... ,t k + 1 ] =1, t± e {a,b} >.

Wiegold:

The Schur multiplier

150

By a result of Haebich [13], M(T,) is an elementary 7-group isomorphic with Y k + 1 (F)/Y k + 2 ( F )> where F = ZZ? * Z 7 ; thus def(T.) -* oo as k •* oo since d(M(T, )) -*• oo as k ->• oo . Now let G, be the splitting extension of T, by a 3-cycle < c> induced by ac = a2,

b c = b2 .

k+1 The claim is that G, has trivial multiplier whenever 2 - 1 is prime to 7, that is, two thirds of the time.

Suppose that H/L = G, with

L C H 1 n Z(H), and let aL = a, 3L = b, yL = c. our aim is to prove that L = l.

Then H = < a,3,Y >, and

Firstly we have congruences modulo L:

a7 = 37 = y3 = 1, a Y = a 2 , 3 Y = 3 2 , where 8. £ {a,3K

[0i ,62 ,... , 8 ^ ] s 1,

Thus using centrality a lot, Y

L^l ,D2 , . . . , OK -, J ~ v ~1

L°l 9 °2 * * • * > ° V i 1 -I i\.^*J.

i

~

-

2

2

2

n

L°I >°2 > • • • ) ° V Ji\~A .1 J

whereas

k+1 As we are assuming that (7,2 -1) = 1, we can now deduce that < a,3 > is nilpotent of class k. Next, oJ = a2p and 3 Y = 32a for some p,a £ L, so that [a,y] = ap and [3,y] = 3a.

Moreover L is of exponent 7 since the Sylow

7-subgroup of G, has elementary multiplier, while the Sylow 3-subgroup is cyclic; thus [a,y]7 = a 7 , [ 3 , Y ] 7 = 37 . Concentrate on a.

We have [a,y] = ap so that [a,y>Y] = [a>Y]

as p is central; thus 1 = [«,Y3] = [a,YY2] = [a,Y2][a,Y][a,Y,Y2]

Thus a7 = 1, and similarly 37 = 1, so that < a,3 > - < a,b > since they are both nilpotent of class k generated by two elements of order 7, and < a,b > is the largest such. Finally, what about H f ?

It is the normal closure of the

elements [a,3], [OI,Y] = ap, [3,Y] = 3a. normal closure of ap, 3a. like

Since [ap,3a] = [a,3], H ! is the

It is even generated by them since relations

Wiegold:

(ap) hold.

The Schur multiplier

151

= a[a,6]p, (ap) Y = a2 p2

But (ap)7 = (3a)7 = 1 and < ap,$a > is of class k; since H f L = G'

= T, , we have finally to conclude that L = l, as required. By results of Swan [37], or using Reidemeister-Schreier, (def T , + 1) < 3(def G, + 1), the 3 being the index of T, in G,.

Thus

d e f G, + oo as k •*• oo . After the lecture where I had done this, Bernhard Neumann pointed out how to make 2-generator examples.

Just embed G, as a sub-

group of index 2 in the splitting extension of T, by the automorphism d of order 6 induced by a

= b, b

= a2 .

This group still has trivial

multiplier and deficiency tending to oo with k; however, calculations are a bit easier with G, than those involved in attacking the Neumann group head-on.

My thanks to him.

6.

SOME HARDER PROBLEMS CONCERNED WITH MULTIPLIERS Here I must confess that the elementary approach is severely

restricted, and there are some problems that remain unsolved, or have been solved only using homological methods.

Perhaps the nicest papers

in the second category in recent years are those of D.F. Holt [16],[17] on the local control of Schur multipliers, which I would love to understand. Perhaps the most persistent problem is:

Problem 6.1. (1)

When exactly is the multiplier of a p-group trivial? Firstly, a p-group with trivial multiplier must have at most

3 generators.

This is connected with the famous result of Golod and

Shafarevich stating that a finite p-group G with d(G) = d needs more than d2/4 relations to define it. lectures:

This is well beyond the scope of these

see Huppert [19], and also Gaschiitz and Newman [8] for better

results. For each odd p, there are infinitely many 3-generator p-groups with trivial multiplier (see Andozhskii [1]). Previously to that, one had for example the famous 3-generator 3-relator groups due to Mennicke [32]: T ^ u ib 1+t , c , 1+t a J. = < a,b,c|a = a , b = b , c = c

1+t ^ >.

For prime t, the maximal nilpotent factor-group of J. is a t-group with trivial multiplier. Two-generator groups with trivial multiplier are easy to

Wiegold:

The Schur multiplier

152

construct, for example the groups m G = < a,b | a P

n = bP

= [a,b] >.

See also I.D. Macdonald [30]. (2)

David Johnson [21] has shown that a non-cyclic p-group with

trivial multiplier is not generated by elements of order p. this goes for groups of prime exponent.

In particular

It is interesting that his proof

makes no reference to the generating number; it is homological, and I believe that it should be possible to prove the result with the methods used here and to give some lower bounds. For instance, let G be of exponent p (p odd), nil-2 and set dCG) = d.

By Theorem 4.9, p i d (d-l)

<

|GI||M(GJ|B

and so the only problem arises when |Gf| = p 2 ^ "

However |G!| < p i d (d-l) . But then G has very

large multiplier, namely as large as the third term of the lower central series of the third nilpotent product (Golovin [9]) of d p-cycles. (3)

A rather small contribution of my own is [43] , where it is

shown that a centrally decomposable p-group has non-trivial multiplier, so that in particular if G has trivial multiplier, then Z(G) C $(G). Wamsley has suggested (oral communication) that a p-group with trivial multiplier must be tricyclic.

This may be so, but we do not

know the answer to: Problem 6.2.

Is every p-group a subgroup of a p-group with trivial

multiplier? It is not hard to see that every group is a subgroup of a group with trivial multiplier. Problem 6.3.

Is every p-group a subgroup of a direct product of p-groups

with trivial multiplier? °f

a

One might have more luck with:

Is every p-group a subgroup of a direct power

p-gvoup with trivial multiplier? Kite-flying, I know:

but I guess that something more about

the elusive multiplier will be discovered if it is tackled. Problem 6.4.

Let G be a p-group of rank r (that isj every subgroup can

be generated by r elementss and some subgroup actually needs r generators). Is M(G) of rank bounded in terms of r? Jones [25] proved that there is a bound in terms of r and the nilpotency class of G, and he conjectures that something like a quadratic

Wiegold: The Schur multiplier

153

bound is correct. For the origin of this problem, see Bachurin [2]. REFERENCES 1. A.V. Andozhskii, On some classes of closed pro-p-groups, Izv. Akad. Nauk SSSR Ser. Mat. 3£ (1975), 707-738. 2. G.F. Bachurin, On the multipliers of torsion-free nilpotent groups, Mat. ZametH S_ (1969), 541-544. 3. A.J. Bayes, J. Kautsky § J. Wamsley, Computation in nilpotent groups (application), in Proc. Second Internat. Conf. Theory of Groups* Springer-Verlag, Berlin (1974), 82-89. 4. F.R. Beyl, Isoclinisms of group extensions and the Schur multiplicator, these Proceedings. 5. F.R. Beyl § J. Tappe, Group extensions* representations and the Schur multiplicator* Lecture notes in preparation. 6. C M . Campbell § E.F. Robertson, A deficiency zero presentation for SL(2,p), Bull. London Math. foe. Yl^ (1980), 17-20. 7. W. Gaschiitz, J. Neubiiser $ Ti Yen, Uber den Multiplikator von pGruppen, Math. Z. JLO£ (1967), 93-96. 8. W. Gaschiitz § M.F. Newman, On presentations of finite p-groups, J. reine angew. Math. 245_ (1970), 172-176. 9. O.N. Golovin, Nilpotent products of groups, Mat. Sb. 27_ (69) (1950), 427-454; Amer. Math. Soc. Transl. Ser. 2 2_ (1956), 89-115. 10. J.A. Green, On the number of automorphisms of a finite group, Proc. Roy. Soc. A(237) (1956), 574-581. 11. K.W. Gruenberg, Some cohomological topics in group theory* Queen Mary College Mathematics Notes (1967). 12. K.W. Gruenberg, The partial Euler characteristics of the direct powers of a finite group, Arch, der Math. 35_ (1980), 267-274. 13. W. Haebich, The multiplicator of a regular product of groups, Bull. Austral. Math. Soc. 7_ (1972), 279-296. 14. P. Hall, The Eulerian functions of a group, Quart. J. Math. 7_ (1936), 134-151. 15. P. Hall, The classification of prime-power groups, J. reine angew. Math. 182_ (1940), 130-141. 16. D.F. Holt, On the local control of Schur multipliers, Quart. J. Math. 2_(28) (1977), 495-508. 17. D.F. Holt, More on the local control of Schur multipliers, Quart. J. Math. 2J31) (1980), 191-208. 18. H. Hopf, Fundamentalgruppe und zweite Bettische Gruppe, Comm. Math. Helvetid 14 (1941/42), 257-309. 19. B. Huppert, EndlicKeGruppen I* Springer-Verlag, Berlin (1967). 20. N. Iwahori $ H. Matsumoto, Several remarks on projective representations, J. Fac. Sd. Univ. Tokyo Sect. I IO_ (1964), 129-146. 21. D.L. Johnson, A property of finite p-groups with trivial multiplicator, Amer. J. Math. 98_ (1976), 105-108. 22. D.L. Johnson § E.F. Robertson, Finite groups of deficiency zero, in Homological group theory9 edited by C.T.C. Wall, LMS Lecture Notes, Vol.36, Cambridge University Press (1979), 275-289. 23. M.R. Jones, Numerical results on multiplicators of finite groups* Ph.D. thesis, University of Wales (1973). 24. M.R. Jones, Multiplicators of p-groups, Math. Z. 127 (1972), 165-166. 25. M.R. Jones, Some inequalities for the multiplicator of a finite group, Proc. Amer. Math. Soc. 39 (1973), 450-456.

Wiegold:

The Schur multiplier

154

26. M.R. Jones § J. Wiegold, A subgroup theorem for multipliers, J. London Math. Soc. (2)(5 (1973), 738. 27. M.R. Jones § J. Wiegold, Isoclinisms and covering groups, Butt* Austral. Math. Soc. 11_ (1974), 71-76. 28. M.A. Kervaire, Multiplicateurs de Schur et K-theorie, in Essays on Topology and Related Topics, Memoirs dedies a Georges de Rham, edited by A. Haefliger and R. Narasimhan, Springer-Verlag, Berlin (1970), 212-225. 29. W. Ledermann § B.H. Neumann, On the order of the automorphism group of a finite group II, Proc. Roy. Soc. A 235_ (1956), 235-246. 30. I.D. Macdonald, On a class of finitely presented groups, Canad. J. Math. 1£ (1962), 602-613. 31. T.S. MacHenry, The tensor product and the second nilpotent product of groups, Math. Z. T5. (I960), 134-145. 32. J. Mennicke, Einige endliche Gruppen mit drei Erzeugenden und drei Relationen, Arch, der Math. liO (1959), 409-418. 33. B.H. Neumann, Groups with finite classes of conjugate subgroups, Math. Z. 63_ (1955), 76-96. 34. B.H. Neumann, Some groups with trivial multiplicator, Publ. Math. Debrecen £ (1955), 190-194. 35. I. Schur, Uber die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen, J. reine angew. Math. 127 (1904), 20-50. 36. I. Schur, Untersuchungen liber die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen, J. reine angew. Math. Ybl^ (1907), 85-137. 37. R.G. Swan, Minimal resolutions for finite groups, Topology 4_ (1965), 193-208. 38. J. Tappe, Isoklinismen endlicher Gruppen, Habilitationschrift, Rhein. - Westfalische Technische Hochschule, Aachen (1978). 39. L.R. Vermani, An exact sequence and a theorem of Gaschiitz, Neubuser and Yen on the multiplicator, J. London Math. Soc. (2)1_ (1969), 95-100. 40. J.W. Wamsley, Minimal presentations for finite groups, Bull. London Math. Soc. 5. (1973), 129-144. 41. J. Wiegold, Nilpotent products of groups with amalgamations, Publicationes Mathematicae Debrecen 6_ (1959), 131-168. 42. J. Wiegold, Multiplicators and groups with finite central factorgroups, Math. Z. 89 (1965), 345-347. 43. J. Wiegold, Some groups with non-trivial multiplicators, Math. Z. 120 (1971), 307-308. 44. J. Wiegold, The multiplicator of a direct product, Quart. J. Math. (2)2£ (1971), 103-105.

155 A PROCEDURE FOR OBTAINING SIMPLIFIED DEFINING RELATIONS FOR A SUBGROUP D.G. Arrell Leeds Polytechnic, Leeds, LSI 3HE, England S. Manrai Leeds Polytechnic, Leeds, LSI 3HE, England M.F. Worboys Leeds Polytechnic, Leeds, LSI 3HE, England

1. INTRODUCTION Given a finitely presented group G and a set of generators for a subgroup H of finite index in G, the Todd-Coxeter algorithm [10] gives a systematic method for determining the index of H.

This algorithm

has been the subject of much investigation over the last fifteen years. Various computer implementations have been devised to improve its computational efficiency (see, for example, [4], [5], [6] and [11]) and it has also been modified to allow the construction of a set of defining relations for H, ([1],[9]). The main disadvantage of this automatic approach to obtaining subgroup presentations is that, even when the index of H is small, the presentations often contain either a large number of generators, many of which are redundant, or many and long relations - or both.

(See, for example, [7].)

It is possible to improve these

presentations by performing a sequence of Tietze transformations ([8], Chapter IV) to simplify the relations and remove redundant generators, but by doing so we may lose control over the subgroup generators:

that

is, the resulting generating set in the simplified presentation of H may not be equal to the original set of generators.

Since, in some

investigations, we are looking for a set of defining relations on a specific set of subgroup generators this approach is not always appropriate. There are several methods given in the literature for obtaining the presentation in terms of the specified set of subgroup generators, for example [1] and [9]. We shall describe here an implementation of McLain's method and show how we can still apply Tietze transformations to simplify the presentation without modifying this specified set of subgroup generators.

Arrell et at.:

2.

Defining relations for a subgroup

156

THE MODIFIED ALGORITHM We describe briefly the modified algorithm given by McLain in

[9] and we use the notation of [9]. The algorithm is based on that of Benson and Mendelsohn [2]. We shall abbreviate it to the M-B-M algorithm and refer the reader to [9] for details. Suppose G has the presentation
|Ri=...R

=1>

and that H is a subgroup of finite index generated by {hi ,.. . ,h }. Following McLain, we shall assume that the h. are actually group generators; for if not, then we may add h. = w.(g) to the list of group generators and at the same time add a new relation h.w.(g)" 1 = 1.

When

the M-B-M algorithm terminates it has constructed a coset table T, the columns of which are indexed by the group generators g., i = l,...,n, and their inverses and the rows of which are indexed by the coset representatives x , X = 1,...,|G:H|, where xi = 1.

The (x.,g?)th entry

A

A

1

of T, e = ±1, will contain the ordered pair (w . (h) ,x ) , where w . (h) Ai y AI and x are such that y

McLain shows in [9] that if we apply each x.A to each R. to obtain, by 3 repeated applications of (1), a set of words S.A (h) such that

3

x^R. = S..(h)x. then H may be presented as A J JA A
|S m

(h) = 1, j = l,...,r, X = 1,...,|G:H| >.

j

A

The problem with implementing this method is that the words w..(h) may become very long, thus adding to the storage requirement of the basic Todd-Coxeter algorithm.

One way round this is to note that the

w..(h) are constructed as the algorithm progresses in the form Al

w . (h) = w ^ w ^ 2 , ei ,£2 = ±1, where wi and W2 are two previously defined entries of T.

It is therefore sensible to store not the w .(h) but a

pair of pointers pi (X,i) and p 2 (X,i), where the pa(X,i) point to the w , a = 1,2.

Hence, the (x.,g?)th entry of T would now correspond to A

Ot

Pi(X,i)

p2(X,i)

1

x^.

(2)

(We may represent w"1 by the use of a negative pointer.) p (X,i) may also be stored by means of pointers.

In turn the

This recursive pointer

presentation gives rise to two data structures in our implementation of the M-B-M algorithm:

the modified coset table T, as in [9], whose entries

Arrell et at.:

Defining relations for a subgroup

157

are of the form (2) and a corresponding binary tree B containing the information which allows us to construct the w .(h) from sequences of Al

pointers p ^ f y j ) , as in Fig. 1. This method of storing the w ^ ( h ) saves considerably on storage but has the disadvantage that when we come to construct the subgroup relations S. (h), a considerable amount of time-consuming tree J* searching has to be performed. Since many of the S..(h) turn out to be 3A redundant this method becomes very inefficient when |G:H| ceases to be small. Notice however that if we apply a coset representative x^ to a relation R. our method of data storage will first give a sequence of pointers s"jx = pf1 (Xi,ii)Pa61 (Xi ,ii) ... pi et (x t ,i t )P2 t (x t ,i t ). In fact we may regard the p (X,i) as Schreier generators for H and the S.

as subgroup relations in the hi ,...,h , p (X,i) , a = 1,2, e = ±1,

X = 1,...,|G:H|, i = l,...,n.

These relations will be considerably

shorter than the S. and can often be simplified by performing sequences JX of Tietze transformations. Indeed computational experience has shown that these transformations often eliminate all the extra generators p (X,i) leaving a presentation in terms of the hi ,...,h

only.

If they

do not, then only a limited amount of tree searching may be required to express those p (X,i) which cannot be eliminated by the Tietze transformations in terms of those that can.

Further sequences of Tietze

transformations will now yield a simplified presentation in terms of the

Fig. 1

w xl ao

/\ generators of H

Arrell et at.:

3.

Defining relations for a subgroup

158

PRACTICAL CONSIDERATIONS Computer programs that perform Tietze transformations are

available.

If we take as input to such a program the relations S.. which 3* are the output of the M-B-M algorithm then we can allow the Tietze transformation program to eliminate as many p (X,i) as possible.

If we

find that some p (y,j) cannot be eliminated then we may read from the tree B that p (y,j) = pil (y,j)p22 (y,j), say, and hence we may represent this as a new relation pf'(u,j)p|:2(p,j)Pa(u,j)'1 = 1.

(3)

It is the ability to deduce new relations of this form that distinguishes the M-B-M algorithm from the usual Reidemeister-Schreier algorithm.

By

repeating step (3) as often as necessary we will obtain a presentation for the subgroup in terms of the original subgroup generators only. By linking our implementation of the M-B-M algorithm to a program that performs Tietze transformations we are able to obtain a procedure that outputs simplified subgroup presentations in terms of specified generating sets.

Efficient programs to perform Tietze

transformations are available, an example of which is that written by Kenne and Richardson of the Institute of Advanced Studies at the Australian National University.

It is easy to modify this program to

accept output from our M-B-M program and automatically perform substitutions of the type (3) above.

Experience has shown that it is

important to retain control over the timing of the substitutions (3) since, depending upon when these are made, we may obtain presentatipns of widely differing total lengths.

For this reason we suggest that any

implementation of this modified algorithm be made interactive.

4.

AN APPLICATION Campbell and Robertson give in [3] the three generator

deficiency -1 presentation for SL(2,2 n ), 3 <^ n <_ 9, < x,y,z | x2 = y , x2 = (yz)2 , (xz)3 = z2 , xyxy" xy

= 1 >

where £ = 2n-l and k = n for n = 3,4,6,7,9, k = 12 for n = 5 and k = 13 for n = 8, and ask whether these groups have deficiency zero presentations. Since SL(2,2n) is in fact generated by xy and z one can modify this question to ask if we can obtain two generator deficiency zero presentations (as is the case when n = 3 ) . The modified M-B-M algorithm

Arrell et al. :

Defining relations for a subgroup

159

described here may be helpful in the search for such presentations since, by enumerating over < a,b > with a = xy and b = z, it is easy to use the M-B-M algorithm in conjunction with a Tietze transformation program to convert the three generator presentations of [3] to two generator presentations; this would not be possible with the usual ReidemeisterSchreier algorithm since no new information would be obtained.

Acknowledgement.

We would like to thank Dr. E.F. Robertson for drawing

our attention to the Tietze transformation program of Kenne and Richardson and for allowing us to experiment with his modified version of the program.

REFERENCES

1. M.J. Beetham § C M . Campbell, A note on the Todd-Coxeter coset enumeration algorithm, Proo. Edinburgh Math. Soc. 20^(1976), 73-79. 2. C.T. Benson § N.S. Mendelsohn, A calculus for a certain class of word problems in groups, J. Combinatorial Theory 1_(1966), 202-208. 3. C M . Campbell $ E.F. Robertson, The efficiency of simple groups of order < 10s , Comm. Alg. (to appear). 4. J.J. Cannon, A general purpose group theory program, Proo. Second International Conf. Theory of Groups, Canberra (1973), 204-217. 5. J.J. Cannon, L.A. Dimino, G. Havas § J.M. Watson, Implementation and analysis of the Todd-Coxeter algorithm, Math. Comp. 2_7_(1973), 463-490. 6. H. Felsch, Programmierung der Restklassenabzahlung einer Gruppe nach Untergruppen, Numer. Math. 3^(1961), 250-256. 7. G. Havas, A Reidemeister-Schreier program, Proc. Second International Conf. Theory of Groups* Canberra (1973), 347-356. 8. D.L. Johnson, Topics in the theory of group presentations., Cambridge University Press (1980). 9. D.H. McLain, An algorithm for determining defining relations of a subgroup, Glasgow Math. J. 18_(1977), 51-56. 10. J.A. Todd § H.S.M. Coxeter, A practical method for determining the cosets of a finite abstract group, Proc. Edinburgh Math. Soc. 5^(1936), 26-34. 11. H.F. Trotter, A machine program for coset enumeration, Canad. Math. Bull. 7(1964), 357-368.

160 GL n AND THE AUTOMORPHISM GROUPS OF FREE METABELIAN GROUPS AND POLYNOMIAL RINGS S. Bachmuth University of California, Santa Barbara, CA 93106, U.S.A. H.Y. Mochizuki University of California, Santa Barbara, CA 93106, U.S.A.

1.

INTRODUCTION For an integer n >_ 2, we consider the following three groups:

GL

..(R), the group of invertible (n - 1) x (n - 1) matrices with entries in R = ZZ [xi ,.. . ,x ] or K[xi ,. .. ,x

+ 1

] , m >_ 1, the polynomial

ring over the integers 2 or a field K. Aut(M(n)), the group of automorphisms of the free metabelian groups M(n) = F(n)/F"(n) of rank n where F(n) is the free group of rank n and F"(n) its second derived group. Aut K (P(n)), the group of K-automorphisms of the polynomial ring P(n) = K[xi ,...,x ] in n indeterminates over the field K.

A study of these groups has split into three distinct phases corresponding to values n = 2, n = 3 and n >_ 4.

Our theme is that the

three groups have many similar or analogous characteristics and should be considered as one and the same study. Consider for example an immediate question confronting one in a study of these groups.

What is a reasonable set of generators?

More

specifically, do any of these groups possess elements other than the obvious or tame ones?

(See §2 for definitions).

Apart from the case

n = 3, evidence is pointing to the fact that these groups possess only tame elements.

The exceptional case n = 3 is indeed very different from

the cases n ^ 3, as we shall see, but apparently similar for all three groups.

We attempt to justify this apparent similarity in our

explanations below. Before discussing the cases n = 2, n = 3 and n _>_ 4 for these groups in §5, §6 and §7, respectively, we need to introduce some background material in §2, §3 and §4.

Bachmuth § Mochizuki: 2.

GL n and automorphism groups

161

THE TAME ELEMENTS The subgroup of GL

.. (R) generated by the elementary matrices

and the invertible diagonal matrices is denoted by GE

(R). (An

elementary matrix is a matrix which differs from the identity matrix by at most one off-diagonal entry.)

We will call an element of GL _-,(R)

which belongs to GE _-.(R) a tame element and non-tome otherwise. The groups Aut(M(n)) and Aut^CPfn)) contain subgroups that are analogues of GE . in GL , and which in each case we call the subn-1 n-1 group of tame automorphisms. We first define the tame automorphisms of Aut(M(n)).

It is

well-known that the automorphism group Aut(F(n)) is generated by the elementary automorphisms, i.e., those which are induced by elementary Nielsen transformations [12, Chap.3].

Let yi,...,y

be a free set of

generators of F(n). Among the elementary automorphisms are the analogues of the elementary matrices and the diagonal matrices in GL (2); namely, the automorphisms defined by y. •> y. , i ^ j and y. •> y.yv or y, y. where 1

1

j

3 K

K j

j ^ k and e = ±1 and the automorphisms which fix some of the generators and map the other generators to their inverses. These elementary automorphisms suffice to generate Aut(F(n)). elements in Aut(F(n)) tame.

Thus, we agree to call all

Since F"(n) is a characteristic subgroup of

F(n), there is a natural homomorphism of Aut(F(n)) into Aut(M(n)).

The

images of the elements of Aut(M(n)) under this homomorphism are defined to be the tome automorphisms of Aut(M(n)). We next consider Autj,(P(n)).

The "triangular" subgroup of

AutK(P(n)) consisting of all automorphisms of the form xi -> ai xi + f2 (x2 ,.. . ,xn) x2 -> a2 x2 + f3 (x3 ,. . . ,xn) x T -> a ,x . + £ v(x ) n-1 n-1 n-1 n n^ x

n

•> a x

n n

+ b,

where the a. are nonzero elements of K, b is also in K, and the f. are in K[x.,x.+1,...,x ] , will be denoted by J(n). called Jonquieve automorphisms.

The affine subgroup A(n) of Autj,(P(n))

consists of all automorphisms of form xi *> ai i xi

+ ...

Automorphisms in J(n) are

+ alnxn

+

bi

Bachmuth § Mochizuki:

GL n and automorphism groups

162

where the b. are in K and (a..) is in GL (K). The elements of the subgroup of Aut^fPfn)) generated by J(n) and A(n) are called tome automorphisms. Thus, to summarize, in all three groups GL

1(R),

Aut(M(n))

and Aut^(P(n)), the tame elements are merely the obvious elements which these groups possess. GL

Whether, in fact, any of the groups in the families

-.(R), Aut(M(n)) and AutK(P(n)) possesses any non-tame elements is a

difficult problem.

As might be expected, most success to date has been

in showing that all elements in a given group are tame.

In fact, what

this paper aims to present is some evidence for the conjecture that in all three families there exist only tame elements with the one exception n = 3, in which case most elements are non-tame.

3. THE GROUPS G L ^ f R ) We recall that R = TL [xt ,. .. ,x ] or K[xi ,.. . ,x

.] where

m -I 1 and K is a field. GLi (R) is the multiplicative group of units in R.

Hence,

GLi (R) = GEi (R) . P.M. Cohn [8] showed that GL2 (R) contains non-tame elements. In fact, it becomes evident from studying the amalgamated free product structure of GL2 (R) that "most" of GL2 (R) is non-tame. D. Wright [17].)

(See, for example,

But, as far as we know, the difficult problem of giving

a set of generators of GL2(R) has not been solved. For n M

we have:

Theorem 1 (Suslin [15]). GL

For n >_ 4, GL

l(R)

= GE _ 1 ( R ) , and thus

^(R) is finitely generated if R = 7L [xi ,.. . ,x ] or R = K[xi ,.. . ,xm]

and K is a finite field. Hence, GL

_(R) is reasonably well-understood, and we will

indicate in §5, §6 and §7 how GL

1 (R)

can possibly serve as a model for

Aut(M(n)) and Aut r (P(n)). It is very interesting to compare GL _^(S) where S = 2 [xi ,xr! ,...,x m ,x^] or S = K[xi ,Xi"! ,. .. >\+1>^+1]

with G L ^ C R ) .

GLi (S) is, of course, the multiplicative group of units of S.

The

authors [4] have shown that GL2 (S) contains non-tame elements if m >_ 2, but the following question remains unanswered.

Question.

Does GL2 {7L [xi ,xf* ]) (respectively* GL2 (K[xi ,xi-1 ,x2 ,xZl ] ) , K

Bachmuth a field)

contain

$ Mochizuki:

any non-tame It is very

non-tame has

elements

that

intriguing GL

and automorphism

groups

GL2 (K[xi ,xil ,x2])

does

that

and Mochizuki

[4]).

REPRESENTATIONS

of both Aut(M(n)) between

and Aut^(P(n))

may be applicable

and why there

these groups on the one hand and GL

we formulate the following propositions. explain

the results

in §6 and

appears

The first one is also needed

and F(n)/F' (n) = gp < xi ,...,x

rank

the

isomorphic

is

Using proposition. Proposition direct by

y. -> x.

a theorem

of Magnus,

There

exists

Aut (F(n)/Ff

one can prove

details

an embedding

f

product

of A u t ( F ( n ) / F ( n ) )

a' -> [a, (a..)]

where

(i)

a e Aut(F(n)/F'(n))

(n))

is

of Aut(F(n))

= GL

(Z)

and

into

the

(Z

[xi ,xi

-1

by a 1

is i n d u c e d 1

GLn(2Z [ x i , x i " , . . . , x n , x ^ ] )

following

references.)

of Aut(M(n))

and G L

1

(ii)

generators

surjective.

(See [5] for further 2.

to

> the free abelian group of

(i = l,...,n).

to GL (2Z), and the natural homomorphism

Aut(F(n)/Ff(n))

other,

§7.

yi ,y2 ,. ..,y

mapping

to

to be a

(R) on the

Let F(n) be the free group on the free set of n under

[15]

.(S) = GE .(S). n-1 n - 1

TWO

connection

contain

For n >_4, Suslin

In order to see why the same techniques the study

163

elements?

(Bachmuth

also proved 4.

GLn

into ,...,x

the

semi-

!

given

,x" ])

G

Aut(M(n)),

satisfies

1 -

1 -

, and

(a. .) 1 (iii)

multiplication

in the semi-direct a..)

Moreover,

[a, (a..)]

product

is given

by

..) represents

an element

o/Aut(M(n))

if and only

if

(ii)

holds. Levin representation see Bergman formulate

[11] generalized

of groups and Dicks

a result

the basic

to a representation

[6]).

(similar

theorem

of associative

This generalization to Proposition

of Magnus rings

enables one

2 above)

about

from a

an

(also

to automorphism

Bachmuth § Mochizuki:

164

GL n and automorphism groups

group of a ring which is closely related to P(n) = K[xi ,...,x ] . Let K < ui ,...,u > be the free algebra over the field K in non-commuting indeterminates ui,...,u . Let A be the augmentation ideal of K < m ,...,u > generated by ui,...,u , and let C = [A,A] be the commutator ideal of A. Thus K < m ,. . . ,u >/A = K and K < ui ,.. . ,u >/C = K[xi,...,x ] = P(n). Aut°(K < ui,...,u >/AC) will denote the subgroup n i\ n of Aut^(K < ui,...,u >/AC) which leaves A/AC (and hence C/AC) invariant, R n and finally Aut°(P(n)) will denote the subgroup of Autv(P(n)) which leaves invariant the augmentation ideal of P(n), the ideal generated by xi ,...,x .

Proposition 3. There exists

an embedding of Aut°(K < u i , . . . , u >/AC) K n into the semi-direct product of Aut° (K[xi , . . . ,x ]) and GL (K[xi , . . . ,x ]) given by a1 -*• [a, ( a . . ) ] where (i) a e Aut°(K[xi , . . . , x n]) is induced by a' e Aut°(K < ui , . . . , u n >/AC), K is. (ii)

(& ) e GLn(K[xi , . . . , x n ] )

satisfies

Xi

(a. .)

, and

(iii) multiplication in the semi-direct product is given by

Moreover^

[ a , ( a . . ) ] represents 13

an element

of Aut°(K < ui , . . . , u R

n

>/AC) if

and only if (ii) holds, 5.

CASE n = 2 For Aut(M(n)) and Aut»(P(n)) complete information is known

only in the case n = 2. For some time it has been known that Autj,(K[xi ,X2 ]) consists entirely of tame automorphisms [9]. A more precise result is known ([16], [13], [10] and [17]).

See §2 for the

definition of "triangular" subgroup J(2) and the affine subgroup A(2) of AutK(K[xi ,x 2 ]). Thereom 4. Aut1.(K[xi ,x2]) is the free product of the subgroups J(2) and A(2) with their intersection amalgamated.

Bachmuth $ Mochizuki:

GL n and automorphism groups

165

It is also true that Aut(M(2)) contains only tame automorphisms, and in fact, an equally strong structure theorem for Aut(M(2)) has been proved.

Theorem 5 ([1]).

Let Inn(M(2)) denote the normal subgroup of inner

automorphisms of Aut(M(2)).

Then, Aut(M(2))/Inn(M(2)) a GL2 (2)

From this theorem it follows that all automorphisms of M(2) are tame.

We mention that Theorem 5 has been generalized to auto-

morphism groups of a wide class of 2-generator groups in [2].

6.

CASE n = 3 We recall (see §3) that whereas GLi (R) , R = 7L [xi , ..,,x ] or

K[xi ,...,x

-] with m _>_ 1, is just the multiplicative group of units of

R, GL2 (R) is an interesting group in that it consists "mostly" of nontame elements.

Indeed, any set of generators of GL2 (R) must contain

infinitely many non-tame elements. The study of Aut(M(3)) closely parallels that of GL2 (R).

We

have

Theorem 6 ([5]).

Any set of generators of Aut(M(3)) must contain in-

finitely many non-tame elements. Thus, as in the case of GL2 (R), "most" elements of Aut(M(3)) are non-tame.

The analogy with GL2 (R) is even stronger yet, as we

explain now. Chein [7] was the first to show that Aut(M(3)) contains a non-tame automorphism.

In [3], the authors, using a different method,

reproved and extended Chein's theorem.

The method was to find a

representation of the kernel £(3) of the natural homomorphism Aut(M(3)) •*- Aut(M(3)/M(3) ') into GU (7L [xi ,xf1 ,x2 ,xi"1 ]) in such a way that the tame automorphisms in £(3) map into GE2 (Z [xi ,xi-1 ,x2 jX^1 ]) , and then to show that the image of K(3) in GL2 (2 [xi ,xf! ,x2 ,x2"1]) contains elements not in GE2 (ZZ [xi jXf1 ,x2 ,X2l ]).

In this way we have much more

than just an analogy between tame and non-tame elements of Aut(M(3)) and GL2 (R).

This method was not enough to prove Theorem 4 but showed

instead that K(3) is infinitely generated - an important first step. Since Aut(M(3))/K (3) = Aut(M(3)/M(3) •) a Aut(F(3)/F(3) •) a GL3(2), acts in a natural way o n £ ( 3 ) .

GL3(Z)

Thus, to show that Aut(M(3)) is

infinitely generated, we essentially had to show that K(3) is infinitely generated as a GL3 (Z>operator group.

Bachmuth § Mochizuki:

GL n and automorphism groups

166

Because most elements of Aut(M(3)) and GL2 (R) are non-tame and because of the similar formalism between Aut(M(3)) and Aut.,(P(3)), P(3) = K[xi,X2,X3], (or rather an automorphism group closely related to Aut K (P(3)); see §4, Propositions 1 and 2 ) , one is led at this stage to conjecture that most elements of Autv(P(3)) are non-tame as well. is.

How-

ever, not even one element of AutK(P(3)) has been shown to be non-tame, although such elements have been conjectured to exist.

In fact, Nagata

[14] wrote down a candidate for a non-tame element of Aut K (P(3)). One of the difficulties with AutK(P(3)) compared to Aut(M(3)) is the lack of a suitable subgroup of Aut.,(P(3)) to play the role of X(3) to help prove the existence of non-tame automorphisms.

Instead one

appears to be immediately plunged into all the technical difficulties inherent in Aut(M(3)) in contrast with the technically easier Z(3). However, it is only recently that methods were developed in [3] to deal with Z(3) and subsequently extended to study GL2 (Z [xi ,xi-1 ,...,x , x - 1 ] ) , GL2 (K[xi ^ f 1 ,...,x m + 1 ,x m 1 + 1 ]), and Aut(M(3)) ([4] and [5]). It is our hope that the methods are now in place and only technical problems remain, and a theorem for Aut (P(3)) similar to Theorem 6 will eve eventually te However, the determination of the structure of GL3 (R),

be achieved.

Aut(M(3)) an and AutK(P(3)) will require much more work and most probably new methods.

7.

CASES n >_ 4 For n > 4, there is evidence that the situation becomes

better.

If the analogy between Aut(M(n)), Aut..(P(n)), and GL

.. (R) is

n-1

j\

indeed valid, then the evidence is rather convincing. First we recall Suslin's theorem (Theorem 1, §3) that if n > 4, then GL (R) = GE (R). — n— x n~1 Thus, in higher dimensions all elements are tame and GL generated. same way.

-(R) is finitely

There is strong evidence that Aut(M(n)), n ^_ 4, behaves the Namely, if one embeds Aut(M(3)) in Aut(M(4)) in the obvious

way (by first embedding M(3) in M(4) in the obvious way), then we have examples of non-tame elements of Aut(M(3)) which become tame as elements of Aut(M(4)).

Could these be isolated examples?

Proposition 2 in §4 once again.

Possibly, but consider

This result associates Aut(M(n)) with a

proper subgroup of GL n (2 [xi jXi""1 ,...,x jx""1]).

As a consequence of

Proposition 2, one can associate Aut(M(n)) with a subgroup of GL where S is a certain localization of 7L [xi ,xil ,...,x ,x~* ].

-(S)

If Suslin's

theorem is a guide, Aut(M(n)), n >_4, is associated with a subgroup of GL, , k ^ 3, where all elements are tame, while Aut(M(3)) is associated

Bachmuth § Mochizuki:

GL n and automorphism groups

167

with a subgroup of GL2 . Thus, our examples may not be accidents, but, in fact, Aut(M(n)), n >_ 4, may be finitely generated, although a proof of this fact will not be easy and is still some distance away.

But, at least a

conceptual method for such a proof is suggested. Everything that has been said for Aut(M(n)) can be carried over to Aut K (P(n)).

If Aut (P(n)), n >_ 4, contains only tame elements,

then these groups in principle should be less complicated and easier to study than Aut^(P(3)).

Perhaps, skillful matrix calculations will be

enough to prove the analogues of Suslin's theorem for Aut(M(n)) and Aut K (P(n)), n ^ 4.

One can hope that the methods in the proof of Suslin's

theorem will offer valuable guidance.

Acknowledgement.

Research support from the NSF is gratefully acknowledged

by both authors.

REFERENCES

1. S. Bachmuth, Automorphisms of free metabelian groups, Trans. Amer. Math. Soc. U8_ (1965), 93-104. 2. S. Bachmuth, E. Formanek § H.Y. Mochizuki, IA-automorphisms of twogenerator torsion-free groups, J. Algebra 40^ (1976), 19-30. 3. S. Bachmuth § H.Y. Mochizuki, IA-automorphisms of free metabelian groups of rank 3, J. Algebra 55_ (1978), 106-115. 4. S. Bachmuth § H.Y. Mochizuki, E2 t SL2 for most Laurent polynomial rings, submitted for publication. 5. S. Bachmuth § H.Y. Mochizuki, The automorphism group of the free metabelian group of rank 3 is not finitely generated, in preparation. 6. G. Bergman § W. Dicks, On universal derivations, J. Algebra 36_ (1975), 435-462. 7. 0. Chein, IA-automorphisms of free and free metabelian groups, Comm. Pure Appl. Math. 21. (1968), 605-629. 8. P.M. Cohn, On the structure of the GL2 of a ring, Inst. Hautes Etudes Sci. Publ. Math. 3£ (1966), 365-413. 9. H.W.E. Jung, Uber ganze birationale Transformationen der Ebene, J. reine angew. Math. 18£ (1942), 161-174. 10. T. Kambayashi, On the absence of nontrivial separable forms of the affine plane, J. Algebra 35_ (1975), 449-456. 11. J. Lewin, A matrix representation for associative algebras I, Trans. Amer. Math. Soc. 188^ (1974), 293-308. 12. W. Magnus, A. Karrass § D. Solitar, Combinatorial group theory, Wiley, New York (1966). 13. M. Nagata, On automorphism group of k[x,y], Lectures in Math. No.5, Kyoto Univ., Kinokuniya Tokyo (1972). 14. M. Nagata, Polynomial rings and affine spaces. Regional conference series in mathematics No.37, Amer. Math. S o c , Providence, R.I. (1978).

Bachmuth § Mochizuki: GLn and automorphism groups

168

15. A.A. Suslin, On the structure of the special linear group over polynomial rings, Isv. Akad. Nauk. 1^ (1977), 221-238. 16. W. Van der Kulk, On polynomial rings in two variables, Niew Archief voor Wiskunde I_ (1953), 33-41. 17. D. Wright, The amalgamated free product structure of GL2 (k[xt ,...,x ]) and the weak Jacobian theorem for two variables, J. Pure ana Applied Algebra 12_ (1978), 235-251.

169 ISOCLINISMS OF GROUP EXTENSIONS AND THE SCHUR MULTIPLICATOR

F.R. Beyl University of Heidelberg, Heidelberg, West Germany

In this talk we discuss how various properties of the middle group G of the central group extension e : 0

> A -^-> G — — >

Q

> 0

(0.1)

of A by Q depend on the extension class [e] e H 2 (Q,A).

Most of the

properties involved (e.g. isoclinism) can be formulated with the aid of the commutator function of e or relate to the verbal subgroup VG (with respect to a variety defined by commutator laws).

The recurring theme

is that many interesting properties of G are already determined by a certain subgroup U(e) of the Schur multiplicator M(Q) of the (finite or infinite) group Q.

In particular (Theorems B and C ) , if U(ei) = U(e2)

for central extensions ei and ei by the same group Q, then ei and ei are isoclinic and TTIZ(GI) = TT2Z(G2). The subgroup U(e) is defined as the kernel of the so-called "homology transgression" 0*(e) : M(Q)

> A of the extension e.

Though

the cohomology theory of groups served as an inspiration, language and reasoning are group-theoretic throughout.

For example, M(Q) is defined

by the familiar Sahur-Hopf formula M(Q) = S n [F,F]/[S,F] and also 6*(e) has a very simple description in terms of a free presentation S <J F — > > Q of Q. In §1 the generalized representing groups e of Q ("hinreichend erganzte Gruppen" in the sense of Schur [18, p.23]) are characterized by U(e) = 0 .

In §3 we discuss the central subgroup Z*(Q) of Q which measures

how much Q deviates from being capable, i.e. isomorphic to the central factor group of some group. Here we give a survey of results due to many authors, as the list of references indicates.

Our principal contributions were announced

in [3]. We include proofs to the extent that they are not or not readily available in the literature.

It is hoped to cover most of the material

given here, as well as the representation-theoretic aspects, in [4]. We

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170

suggest the fundamental paper [13] by Leedham-Green and McKay as "further reading"; these authors treat isologism as a generalization of isoclinism in relation to varietal (co)homology.

Let us finally mention that [11]

includes a large bibliography on isoclinism. 1.

GENERALIZED REPRESENTING GROUPS (THEOREM A) Every complex vector space V determines an extension a y : <E* > — - — > GL(V)

nat

»

PGL(V) ,

(1.1)

where (E* = (C \{0} and 6 (c) for c G (C* is the dilation v • image of 6 is precisely the center of GL(V).

> c-v.

The

An extension e as in (0.1)

is called a generalized representing group of Q if it is central* i.e. K(A) C Z(G) the center, and every complex projective representation y :Q

> PGL(V) can be lifted to a linear representation 3 : G

with nat ° $ = y ° IT.

> GL(V)

(Instead of (C, any algebraically closed field of

characteristic 0 yields the same concept.)

A generalized representing

group is called a representing group of Q ("Darstellungsgruppe" in the sense of Schur [18]), if it is moreover stem* i.e. K A C [G,G]. Let the Sohur multiplioator of Q be the abelian group M

N ) = S?|Fr^

(Schur-Hopf Formula),

(1.2)

defined in terms of a chosen free presentation eQ : S C of Q.

>

»

F

Q

(1.3)

It is due to Hopf [9], in a special case to Schur [19], that any

pair of free presentations of Q determines a unique isomorphism in (1.2), see also [25]. Likewise, if a homomorphism y ' Qi

> Q2 is given, any

lift of y to the free groups induces a homomorphism M(7) : M(Qi) — > M(Q 2 ), depending only on y.

In this manner, M is a functor; it can be shown to

be naturally equivalent to H 2 (-,Z ) , the second integral homology group. Every group extension e :0

> N ——>

G ——>

Q

> 0,

(1.4)

with N arbitrary, determines the 5-term exact sequence M(Q)

^> rxT

rA

> G ,

originally due to Hochschild and Serre. 1

[N,G] for K" [ K N , G ] .

> Q ,

> 0,

(1.5)

(By misuse of language, we write

If e is central, then the range of 6*(e) is

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Isoclinisms of group extensions

considered N rather than N/0.)

171

Here we rely on the elementary treatment

which (1.5) receives e.g. in [5], cf. also [4]. The 5-term exact sequence is natural with respect to morphisms of group extensions, i.e. translations of (1.4). as follows.

Let e

be evaluated.

The homomorphism 6*(e) in (1.5) can be described

in (1.3) be the free presentation at which M(Q) shall

Choose any morphism (a,$,lQ) : e

> e of extensions.

Then 6*(e) is the composite (1.6) which is independent of the choices involved (a

Theorem A.

induced by a ) .

A central extension e as in (0.1) is a generalized

representing group of Q if, and only if, U(e) : = Ker 8*(e) = 0. Consequently, e as in (0.1) is a representing group of Q precisely when 6*(e) is an isomorphism.

In case Q is finite, Yamazaki

[26, §1.4] has given an equivalent description in terms of the transgression H1 (A,(C*)

> H2 (Q,(C*); however, the identification of M(Q) with

2

H (Q,(C*) is not appropriate for arbitrary Q. In the sequel, we need the Universal Coefficient Theorem and the calculus of induced extensions again and again. describe these.

We are going to

Let Cext(Q,A) be the group of congruence classes of

central extensions of A by Q, with the Baer sum; it is isomorphic to H2 (Q,A).

Congruent extensions ei and e2 are related by a morphism

(1,3,1) : ei

> e2 with fixed ends, we write ei = e2 and let [ei ]

denote the congruence class. and homomorphisms a : A

Given a central extension e as in (0.1)

> Ai and y : Qi

> Q, there are induced

extensions ae of Ai by Q and ey of A by Qi , together with morphisms (a,a#,l) : ae

> e and (1,7',7) : e7

> e.

MacLane's "conceptual

treatment" [14, p.70] of induced module extensions carries over almost literally - we now use multiplicative notation and replace d by the direct product X.

Details can be found in [20, II §4] and [4]. For

example, e7 is constructed by appeal to the pullback = fibre-product of TT and y in the category of groups.

A major part of the "calculus of

induced extensions" is subsumed under el = e and le = e and the following property.

(1.7)

Let ei and e2 be central extensions.

A morphism (a,.,7) : ei — > e2

of extensions with prescribed a and y exists if, and only if,

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172

[aei ] = [e2y] e Cext(Qi,A 2 ). Recall that Ext(Q , ,A) denotes the group of congruence classes of abelian extensions of A by Q , .

Theorem 1.8 (Universal Coefficient Theorem) . Let Q be a group and A an abelian group. 0

Then there is a natural short-exact sequence

> Ext(Q ab ,A) — - 4 — > Cext(Q,A) — ^ — >

which is split.

Here 9* = [e] i

Hom(M(Q) ,A)

> 0

> 0*(e) as in (1.6) and ip[e] = [(e)ab]

is the pull-back of e along the abelianization ab : Q

> Q , .

The description of the map 6* appears in [5, Thm. 2.2]. A related formulation of Theorem 1.8, good for finite groups, was given by Yamazaki [26]. Actually, Theorem 1.8 is a special case of the Universal Coefficient Theorem of algebraic topology [14, Thm. III.4.1].

The above

formulation allows a proof by the calculus of induced extensions, see [4]. As the Universal Coefficient Theorem describes all extensions e by Q whose 0*(e) is a given homomorphism M(Q)

> A, it is useful to

characterize properties of the extension group G in terms of 0*(e).

Proof of Theorem A.

a) Note that every homomorphism A

> (E* can be

extended to any abelian group B 3 A, since (C* is divisible as an abelian group.

Thus Ext(Q b,
> Hom(M(Q) ,(C*) is an

isomorphism. b) Assume that 0*(e) is monomorphic, let y : Q jective representation.

> PGL(V) be a pro-

Since (C* is divisible and 0*(e) : M(Q)

monomorphic, there is a homomorphism a : A

> A

> (C* with a0*(e) = 8^(a^y).

Then 0*[ae] = a0*(e) = 0*[av7] and thus [ae] = [o^y] by (a). There exists (a,3,7) : e

> a^ by (1.7) and 3 is a representation lifting 7-

c) Conversely, let e be a generalized representing group and 0 ± x e M(Q); we shall show 0*(e) x ^ 1 e (C*.

Invoke from (1.3) the free presentation

e

> S : = S/[S,F] is monomorphic by (1.6)

of Q.

Since 0*(e ) : M(Q)

and C* is divisible, there is a homomorphism f : S f0*(e ) x ^ 1.

> (E* with

Regard f as a 1-dimensional representation of S and form

the induced representation t of F := F/[S,F] with representation space V = (E(F) 8 ~(C, where (C(F) denotes the complex group ring.

Since S c Z(F),

the restriction t S decomposes into 1-dimensional representations equivalent to f, thus is a dilation.

We obtain a commutative diagram

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Isoclinisms of group extensions

173

» Q

oy

: (C* > — - — > GL(V)

which induces y : Q

»

PGL(V)

> PGL(V) with o^y = fc(e ) by (1.7).

Since e is

a generalized representing group, there is a homomorphism $ : G lifting 7.

Let a = 01 A,(C*.

Then ae E a r = fc(e ) by (1.7).

> GL(V) Using (1.6).

we find e*(c(eQ)) = e*(eQ) and a6*(e) = 6*(ae) = fe*(c(eQ)) = £0*(e Q ), thus a0*(e) x £ 1.

Final remark:

if Q is finite, then V is finite-

dimensional; using a minimal invariant subspace of V instead of V, we may construct y as an irreducible protective representation.

2.

ISOCLINISMS OF EXTENSIONS (THEOREM B) We generalize P. Hall's concept of isoclinism from groups to

central extensions. c e

Any central e as in (0.1) determines a set function

= {ff(gi) x 7T(g2) i

function.

> [gi,g2]> : Q x Q

> [G,G], the commutator

Now central extensions K.

e. : A. > 1

TT

— > G.

are c a l l e d i,8ocliivic9 and E, : [ d ,Gi ]

= — » Q. for i: = 1,2 x

l

1

(2.1) v

'

i

ei ~ e2 , i f t h e r e are isomorphisms n : Qi

J

> Q2

> [G2 ,G2 ] such t h a t

Qi x Qi

-

> [Gi ,Gi] (2.2)

Q2

x

Q2

E3

> [&,(

is commutative, with c. denoting the appropriate commutator functions. By associating with a group G the extension eG

: Z(G) C

>

G

- ^ - »

G/Z(G),

we regain Hall's [8] notion of group isoclinism:

(2.3) G ~ H <=E*> e^ ~ e H .

ei and e2 as in (2.1) are isoclinic extensions, then d

If

and G2 are iso-

clinic as groups; the point is that the commutator functions detect TTIZ(GI) respectively TT2Z(G2).

The justification for the generalization

is that the isoclinism classification splits into two fairly separate problems:

a well-rounded theory of isoclinism of extensions (Theorem B)

and the study of the precise center of a central extension group (Theorem C).

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Isoclinisms of group extensions

Lemma 2.4 (cf. [8, p. 134]). with [G,G] n Ker f = 0.

Let f : G

174

> H be an epimorphism of groups

Then Ker f c Z(G) and f determines an isoclinism

(£ = f|[G,G],[H,H]; n = f j of groups. The salient point is f -1 Z(H) = Z(G). These examples and the inclusions Gi

c

— > G with Gi-Z(G) = G are the only "easy" situations

giving rise to group isoclinism [8]. An epimorphism f : G an isoolinio epimorphism of groups . extensions.

> H with [G,G] n Ker f = 0 is called Let ei and e2 as in (2.1) be central

If there exists an epimorphism 8 : Gi

> G2 with

[Gi ,Gi ] n Ker 8 = 0 and 8"1 (Ker7r2 ) = Ker TTI , then the uniquely induced morphism (a,8,7) : ei

> e2 is called an isoclinic epimorphism of

extensions; in particular, 7 is bijective and (£;7) with £ = 31[Gi ,Gi ] , [G2,G2] is an isoclinism of extensions.

Remark 2.5.

The isoclinic epimorphisms of groups with common range H can

easily be determined.

An isoclinic epimorphism f : G K f

central extension e : A >

> G

»

with Im 8*(e) = Ker (A -»• G ->- G , ) = 0.

> H affords a

H with K ( A ) n [G,G] = 0, hence By the same token, f in the

central extension e is an isoclinic epimorphism if 0*(e) = 0.

The

Universal Coefficient Theorem characterizes those e as backward-induced along H

>> H , from an abelian extension A >

congruence).

> E

>> H , (up to

It is automatic to prolong the above construction to iso-

clinic epimorphisms of extensions with range Ai >

> H

>> Qi , here

n = 7 can be prescribed as an arbitrary isomorphism Q * Qi . Theorem B.

Let central extensions ei and e2 be as in (2.1).

Then the

following are equivalent: (i) (ii)

ei and e2 are isoclinic extensions; there is a central extension e : A > isodinio

(iii)

epimorphisms

(a.,$.*n.) : e

there exists an isomorphism n : Qi

> G

»

Q together with

> e., i : = 1,2; > Q2 such that M(n) U(ei) =

U(e 2 ) in M(Q 2 ). If Gi and G2 are finite3 G in (ii) may also be chosen finite. This theorem shows that the isoclinism classes of extensions with factor group (isomorphic to) Q are in bijective correspondence with the Aut(Q)-orbits of the subgroups of the Schur multiplicator M(Q).

A

special case of (i) *=• (iii), in a cohomological formulation, was proved by Tappe [21]. Closely related results, mainly in the framework of

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175

Isoclinisms of group extensions

varietal cohomology, have been obtained by Leedham-Green and McKay [13]. Actually Theorem B, in the treatment given here, generalizes to isologism without difficulty. In Gruenberg's category (—-) of central extensions by Q [7, §9.9], the appropriate concept is that of special isoclinism3 defined as an isoclinism (£;n) with Q = Qi = Q2 and n = 1. isomorphism n : Q

> Q

f

We are free to pull an

into the projection IT of e, i.e. to replace e

as in (0.1) by

>> Q f

en"1 = e 1

In this manner, isoclinism is reduced to special isoclinism and the following theorem implies Theorem B. Theorem 2.6.

Let central extensions e. : A. >

given for i : = 1,2. (i) (ii)

>> Q be

Then the following are equivalent:

ei and e2 are special isoclinic extensions; there is a central extension e : A >

> G — »

isoclinic epimorphisms (a.,3-,1) : e (iii)

> G.

Q together with

> e^, i : = 1,2;

U(et) = U(e 2 ) C M(Q).

If d and G2 are finite3 G in (ii) may also be chosen finite. In particular, the totality of special isoclinism classes in (—-) is bijectively related to the set of subgroups of M(Q).

For the

proof, we invoke the generalization of an elementary pullback construction of Weichsel [24], who in turn acknowledges a suggestion by G. Higman. Lemma 2.7. For ei and e2 as in Theorem 2.6, let A : Q diagonal homomorphism and < ei ,e2 > = (ei x e 2 )A.

> Q x Q be the

Then < ei ,e2 > is

central and fits in commutative diagrams ,e2 > : N >

-» Q

-> G

(2.8)

°i K. 1

1

-» Q

for i := 1,2 with the following properties: (a) N = Ni x N2 and a. is the projection onto the i-th factor; (b)

G = Gi x

(c)

Ker K

(<

G2 and 3- is induced by the projection onto G.; Ker 0*(e ,e2 >) = Ker K e j ) n K 0 ( 2 ). )

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Proof.

Isoclinisms of group extensions

176

Assertions (a) and (b) are more or less the definitions of

< ei ,e2 >, a. and 3..

The naturality of 6*, when applied to (2.8), gives

a. o 6*(< ei ,e2 >) .

Since (Ker ai)

n

(Ker a 2 ) = 0, we conclude

(c).

Proof of Theorem 2.6. a) Claim:

(I thank Ralph Strebel for a helpful discussion.)

(i) <==> (ii) .

Clearly (ii) implies (i) .

and e2 , we invoke e = < ei ,e2 > and (ou,3.,l) : e

Given isoclinic ei > e. from Lemma 2.7.

Since the a.'s are onto and 7 = 1 , the &.fs are epimorphisms with 371 (Ker TT.) = Ker TT.

The formula [G,G] n Ker 3. = 0 essentially carries

over from [10, II p.73]. b) Claim:

(ii) *=* (iii).

For a central extension e as in (0.1), by (1.5)

there is a natural exact sequence 0

> Ker e,(e)

> M(Q) -§-> [G,G]

where 0 denotes the restriction of K6*(e) : M(Q) any map (a.,3-,1) : e

> e. of central extensions, we obtain a commutative

diagram

U(e)

c

> M(Q)

>

[G.G]

> M(Q)

> [Gi,Gi]

»

[Q,Q]

»

[Q,Q]

where a. is an inclusion and 3-* the restriction of 3-•

Now assume (iii).

With e = < ei ,e2 > we have U(e) = U(ei) = U(e 2 ) by Lemma 2.7(c).

HenGe

for both i = 1 and i = 2, a. is the identity map, 3.* an isomorphism, and [G,G] n Ker 3- = 0. extensions.

Thus (a.,3-,1) are isoclinic epimorphisms of

Conversely, assume (ii). Then 3 t * and 3 2 * in (2.9) are iso-

morphisms, and hence U(ei ) = U(e) = U(e 2 ).

By the initial remarks, Theorem B implies a similar characterization of isoclinism classes of groups.

Of course, (i) ^^ (ii) for group

isoclinism is due to Jones and Wiegold [17] and was used in the proof of Theorem 2.6.

Note that now Q must be capable and only certain subgroups

U(e) C M(Q) are allowed - those for which K A is the precise center of G. This problem is treated in §3, see Remark 3.6 for the extremal case U(e G ) = 0.

Proposition 2.10 (King [11, Thm. 5.7]).

The following properties of a

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Isoclinisms of group extensions

177

group G are equivalent: (i)

G is isoolinio to a finite group;

(ii)

G/Z(G) is finite;

(iii)

G is isoolinio to a finite subquotient of itself.

Proof.

Our approach allows a short direct proof of Kingfs result. The implications (iii) =* (i) =* (ii) are trivial.

Q = G/Z(G) is finite.

Assume

Then there exists a finitely generated subgroup Gi

of G with G = Z(G)-Gi . Thus Z(Gi) = Z(G) n Gi and G is isoclinic to the subgroup Gi [8, p.134]. ei : Z(Gi)

Look at

c

> Gi

»

Q.

As a central subgroup of finite index in a finitely generated group, Z(Gi) is finitely generated abelian.

Decompose Z(Gi) = T x A where T is the

(finite) torsion subgroup and A is torsionfree.

As M(Q) is finite with

Q by [18, p.26] or [7, p.212], Z(Gi) n [& ,Gi ] = Im 0*(ei) is also finite, hence lies in the torsion part. clinic epimorphism.

It follows that Gi

>> Gi/A is an iso-

Thus G is isoclinic to its subquotient Gi /A and Gi /A

is finite as an extension of Z(Gi)/A « T by Q. 3.

THE PRECISE CENTER OF A CENTRAL EXTENSION GROUP (THEOREM C) If e as in (0.1) is central, then sequence (1.5) can be

enlarged to the longer exact sequence G

ab

8 A

X(e)

where the Ganea map x( e ) extensions.

—>

is

M

(G)

M(iT)

— > M(Q) ...

natural with respect to morphisms of central

According to [5], the following formula describes x(e) in

terms of a free presentation R < F — - — » and a : S

: G , a Z(G)

a)

(3.2)

> M(G).

If e as in (0.1) is a central extension, then TTZ(G) = {x G Z(Q) | Vq e Q^

: xQ(q«x) e U(e) C M(Q)}.

(3.3)

Let X denote the right-hand side of (3.3).

Claim:

map e G

Let S = Ker(Trp) < F

* a(s)) = [f,s]-[R,F] e M(G).

The Ganea map of G is x G = x ( O

Proof.

G of G.

> A be the restriction of p; then X(e)(p(f)[G,G]

Theorem C.

(3.1)

TTZ(G) C X.

The naturality of x* when applied to the obvious

> e Q , and (1.5) give

Beyl: 6*(e)

where TT1 : Z(G)

Isoclinisms of group extensions xQ(Trab 0 *•)

= 0*(e)

M(TT)XG = 0,

> Z(Q) r e s t r i c t s IT. Thus x Q O

= U(e) for a l l z € Z(G) and x € G , .

178

Since TT ,

b00

a fr(z)) e Ker0*(e)

: G,

> Q , i s onto,

the claim follows. b)

Claim:

X C TTZ(G).

We invoke the notation of ( 3 . 2 ) .

by c e n t r a l i t y , also U(e) = Im M(TT) = (R n [F,F])/[S,F] . y G G with iry = x and w e F with pw = y.

Then [S,F] C R For x e X choose

Since XQC^P (f) [Q>Q] « x) =

[ f , w ] « [ S , F ] , we conclude [f,w] e R n [F,F] C R for a l l f e F.

Thus

w

[g*y] = [g»P ] = 0 for a l l g e G and y i s central. In p a r t i c u l a r , Theorem C t e l l s us whether e as in (0.1) i s strictly

centrals

i . e . KA = Z(G), and draws attention to the characteristic

central subgroup Z*(Q) = {x € Z(Q) | Vq € Q ^ : x Q (q Q x) = 0 } .

(3.4)

Thus, U(e) must be "small" for strictly central extensions e. If Z*(Q) ^ 0 , no extension by Q is strictly central. Proposition 3.5 ([2]). Alternative descriptions of Z*(Q): a) Z*(Q) is the kernel of x Q : Z(Q) > Hom(Q ab ,M(Q)), where xQ denotes the adjoint homomorphism of x0J b) Z*(Q) is the image of the center of any generalized representing group* e.g. a centralized free presentation of Q; c) Z*(Q) = n TTZ(G), where e ranges over all central extensions by Q, as in (0.1); d) Z*(Q) is the smallest normal subgroup N of Q such that Q/N is capable; e) Z*(Q) is the largest central subgroup A of Q such that M(Q) > M(Q/A) is monomorphic. A group Q is called unicentral [6] if TTZ(G) = Z(Q) for every central extension TT : G » Q. By Proposition 3.5(c), Q is unicentral precisely when Z*(Q) = Z(Q). On the other hand, by Proposition 3.5(d), Q is capable precisely when Z*(Q) = 0. And Proposition 3.5(a) implies: if Q is capable and if Q , and M(Q) have finite exponent, then the GCD of these exponents annihilates Z(Q). Remark 3.6. For any group Q, all groups G underlying the representing groups IT : G » Q of Q lie in one isoclinism class of groups, viz. the class consisting of precisely the generalized representing groups of the

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Isoclinisms of group extensions

capable group Q/Z*(Q).

179

This is relevant to the first problem of Jones

and Wiegold [10]. The first half of the assertion is essentially [7, Thm.7(i) p.217],

We now have complete information.

Indeed, if e as in

(0.1) is any representing group of Q, then Q/Z*(Q) s G/Z(G) by Proposition 3.5(b) and a : G

» Q

together with (1.5).

» Q/Z*(Q) satisfies M(a) = 0 by Theorem A

Hence Z(G) c

> G - ^ — » Q/Z*(Q) is a generalized

representing group of Q/Z*(Q) and it is isomorphic to e^. The converse now follows from Theorem B, case U(e) = 0. In the case of finite groups Q, Z*(Q) has already been considered in [17]. For Q arbitrary again, the methods of [2] give aids to compute Z*(G) and sometimes allow one to predict whether a group construction leads to capable or unicentral groups. Examples 3.7. a) Let G be a nilpotent periodic group, i.e. each element has finite order.

(Note that G is the restricted direct product of its

Sylow subgroups G .) Then G is unicentral (capable) if, and only if, each G b)

is unicentral (capable) [2, Prop. 6.2],

An extra-special p-group of order at least p5 (possibly infinite) is

unicentral [2, §8]. c)

A non-trivial torsion-free abelian group is capable precisely when

its rational rank exceeds one [2, Prop. 7.5]. d)

A capable finite metacyclic group has a faithful irreducible complex

projective representation [2, Cor. 9.4]. (The converse always holds.) 4. APPLICATIONS TO VARIETIES OF EXPONENT ZERO (THEOREM D) Our point of view in group varieties, including the terminology, is the one of Stammbach [20, Ch. I] except that we use the notation 91,91 ,<3i for the varieties of abelian groups, nil-c groups, solvable groups of length at most 1, respectively.

If V C F^ is a set of laws and 33 the

variety defined by it, the 93-verbal subgroup of a group G is denoted by VG; thus G e33 *** VG = 0. laws.

If V = VF^, then V is called a closed set of

If 93 contains 21, it is said to have exponent 0.

Remarks 4.1. a) LetSBbe any variety, defined by the closed set of laws W.

Then the groups G with G/Z(G) ^3B again form a variety, the variety

of center-by1^groups 5ft+- = center-by-$R

with the laws [ F ^ W ] .

For example,^! = % and

for c > 1. There is also the variety of abelian-by- 2K

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Isoclinisms of group extensions

groups G, given by the extensions A >

> G

Q €$B; it is defined by the laws [W,W].

180

» Q with A abelian and

For example, Si = 91 and ®

=

abelian-by-G, for & > 1. b) We say that an extension e as in (1.4) lies in 93 if G e33. and ei e 93, then certainly e ^93.

If e = ei

For Q ^93 and A e 9Jn93, let

Cextqj[Q,A) C Cext(Q,A) denote the classes of central extensions of A by Q in93.

By direct constructions or [20, Thm. III. 3.3], we find that

Cext~(Q,A) is a subgroup of Cext(Q,A) and a functor 93°P x (93^21) Definition 4.2.

Let 93 be a variety and Q € 93.

is a free presentation of Q, then S/VF

c

>?l.

Whenever e as in (1.3)

> F/VF

>> Q -£s called a

SB-free presentation of Q. Define KJJCQ) = Im{M(T) : M(F/VF)

> M(Q)}.

It is easily seen that K (Q) is independent from the choice of free presentation and is a functor of Q.

Let us mention that

Cext~(Q,A) and M(Q)/K (Q) are special instances of the varietal (co)homology groups of Leedham-Green [12]. Theorem D.

Let 93 be a variety of exponent 0 defined by the laws V and

let Q ^93.

If e as in (0.1) is any central extension^ then VG is the

image of K^(Q) tinder K ° 8*(e).

Thus e e93 precisely when 6*(e) vanishes

on K (Q) C M(Q). This theorem allows one to recover K (Q) as (isomorphic to) the 93-verbal subgroup of any generalized representing group of Q.

The

last assertion of Theorem D can be restated as: e £ $ < = * K^(Q) C U(e). Combining this with Theorem A, we find that the generalized representing groups of a given group generate the same variety of exponent zero. This improves [12, Thm. 4.5]. Proof of Theorem D.

We start with a free presentation p : F

put a = irp and S = Ker a. (4.3) of Fig. 1. (93) = 0.

»

G and

Thus we obtain a commutative diagram, diagram

Note that VF C S from 0 e 93 and VF C [F,F] from exponent

Evaluating M(x) and 6*(e) at (4.3) according to (1.2) and (1.6),

we eventually get Im{< o e*(e) o M(x)} = p(VF) = VG.

Beyl:

Isoclinisms of group extensions

181

Now we are going to discuss situations in which K^(Q) can be described internally, thus eliminating 23-free presentations.

The resulting

formulas for K (Q) give rise to applications of Theorem D in other areas. Proposition 4.4. Let$Bbe a variety (laws W, letQ be the variety of center-by-%B groups and Q €33. Then K (Q) = Ker{M(nat) : M(Q) f

where e Proof.

: WQ C

>Q

> M(Q/WQ)} = ImX(e')

» Q/WQ.

A free presentation as in (1.3) yields the commutative diagram

(4.5) in Fig. 1. With these free presentations of F/VF, Q, and Q/WQ, we have Im M m

- (VFn[F,F])-[S,F]

_ Sn[S-WF,F]

Under the present assumptions, e1 is central and VF = [F,WF] C S. Thus Ker M(nat) = Imx(e') by (3.1) and (VF n [ F , F ] ) • [ S , F ] = VF • [ S , F ] C S H [ S - W F , F ] , S n [S«WF,F] C [S«WF,F] C [ S , F ] • [WF,F] = VF • [ S , F ] . C o r o l l a r y 4 . 6 (Evens [ 6 ] ) . Let Q be nilpotent K = Ker{M(nat) : M(Q) r

_G = K 8st(e)K.

precisely

If e as in (0.1) is central*

Thus G is nilpotent

Let 3B be a variety

of abelian-by-Wigroups

Fig. 1 Diagram (4.3)

e

:s o i

C

P

then

of class n rather than (n + 1)

defined by the laws W, let 93 be the

and let Q ^93. Then

K^tQ) = Im{(incl) : M(WQ)

VF C

n and

when U(e) 2. K-

Proposition 4.7. variety

> M(Q/r Q)}.

of class

Diagram (4.5)

> p

» F/VF

>

>> Q

F i

> M(Q)}.

IM

I

e:A >—-—> G — - — » Q

VF C

e :S C

^> p

1

> p

o

» F/VF F/V

1-1

>> n

*

I B S-WF C

|nat >p

» Q/WQ

Beyl: Proof.

Isoclinisms of group extensions

18 2

Again we construct a commutative diagram >

p

» F/VF

» Q

t

incl

t

_> S • WF

»

WQ

with F free and hence S • WF i s also free. I m M(T) =

[WF,WF].rS>P] <-

As VF = [WF,WF] C S,

Im(incl)

=

(Sn[ S .WPS.WF]).[S,F] _

The reverse inclusion follows from [S • WF,S • WF] C [WF,WF] • [S,F] C S. Related problems for finite groups and the varieties GL have been treated by Yamazaki [26, p.164] with somewhat involved cohomological arguments. Let Q be a nilpotent group, say of class n. A central extension ei :Q/2Z >

> M

» Q is called induced-central if there is a

central extension e as in (0.1) and a homomorphism f : A KA = r

> Q/2Z with

.G and fe = ei . This concept was introduced by Passi [16] in his

study of the Dimension Conjecture.

We regard "induced-central" as a

property of the congruence class [ei ] £ Cext(Q,Q/Z ) rather than of the extension ei . Proposition 4.8.

Let Q be a nilpotent group of class n and

K = Ker{M(nat) : M(Q)

> M(Q/r Q)}. Then ei e Cext(Q,Q/Z) is induced-

central precisely when K + U(ei) = M(Q). This proposition has also been obtained and generalized by Vermani [23]. Proof.

If e as in (0.1) is any central extension then, by Corollary 4.6,

KA = r

G precisely when 6*(e)K = A; this implies K + U(e) = M(Q). Now

assume that ei is induced-central and let e,f be as required.

Then

ei E fe implies e*(eO = fe J e ) ; thus U(e t ) 3 U ( e ) and K + U(e t ) = M(Q) . Conversely, given ei with K + U(ei) = M(Q), let A = Im 8*(ei) C Q / 2 and

M(Q) — S — » A C—I—> Q/2Z

Beyl:

Isoclinisms of group extensions

be the image decomposition of 6*(ei).

183

By the Universal Coefficient

Theorem 1.8, choose any e e Cext(Q,A) with 8*(e) = g.

Then Ker g =

Ker 8*(ei) and g is onto, hence 6*(e)K = A and KA = r + ,G. 6*(fe) = f o

g

= e*(ei) and 0* : Cext(Q,Q/ZZ)

isomorphism, since Q/Z is divisible.

Moreover,

> Hom(M(Q) ,Q/Z ) is an

We conclude fe = ei or that ei is

induced-central.

Example 4.9.

In general the subset of (congruence classes of) induced-

central extensions in Cext(Q,Q/Z) is not a subgroup. apparently open question.

This answers an

Let n = 2, D be the dihedral group of order 8,

Q = D x 2Z/2 and let TT : Q

»

Q , » (2 /2) 3 denote the abelianization.

According to Schur [19], M(D) * 7L /2 and M(Q) « M (Q a b ) * ( Z / 2 ) 3 ; the latter result is known as the "Kunneth Formula", see also [25]. Then K = KerMfir) « 2 / 2 by (1.5). fi,f2

: M(Q)

> ZZ/2

> Q/Zwith fi + f2 + 0, Ker(fi+f2) => K and

K n Kerfi = K n Kerf2 = 0. i := 1,2.

One easily finds homomorphisms

Choose e i € Cext(Q,Q/ZZ) with eje..) = f± for

Then ei and e2 are induced-central by Proposition 4.8 but

ei + e2 is not.

Theorem 4.10. Let^&be any variety and 23='5Bv9I and Q ^28. Then K (Q) = K (Q). The following are equivalent: (i) (ii)

Q is absolutely-^ i.e. all central extensions by Q lie in SB; some generalized representing group of Q lies in 93;

(iii) iyQ) = o. Details for Theorem 4.10 and the following examples will be given in [4].

Examples 4.11.

a) An abelian group Q is absolutely-abelian precisely

when M(Q) = Ker{M(Q)

> M(0)} = 0.

These groups have been classified

by Moskalenko [15] and again in [1]; the interesting ones are infinite! b) A group Q is absolutely-^ precisely when [Q,Q] C Z*(Q).

This

condition expresses the fact that every representing group G of Q has G/Z(G) « Q/Z*(Q) abelian.

A nilpotent periodic group Q is absolutely-SRj

if, and only if, so are all its Sylow p-subgroups.

For Q, [Q,Q] and Z*(Q)

are restricted direct products for their respective Sylow subgroups [2, Prop. 6.2].

Beyl:

Isoclinisms of group extensions

184

REFERENCES 1. F.R. Beyl, Abelian groups with a vanishing homology group, J. Pure Appl. Algebra 1_ (1976), 175-193. 2. F.R. Beyl, U. Feigner § P. Schmid, On groups occurring as center factor groups, J. Algebra 61_ (1979), 161-177. 3. F.R. Beyl, Commutator properties of extension groups, C.R. Math. Rep. Aoad. Sci. Canada 2_ (1980), 27-30. Zbl 428.20021. 4. F.R. Beyl $ J. Tappe, Group extensions9 representations* and the Sohur multiplicator* Lecture Notes in preparation. 5. B. Eckmann, P. Hilton § U. Stammbach, On the homology theory of central group extensions: I - The commutator map and stem extensions, Comment. Math. Helv. 47 (1972), 102-122. 6. L. Evens, Terminal p-groups, Illinois J. Math. \2_ (1968), 682-699. 7. K.W. Gruenberg, Cohomological topics in group theory * Lecture Notes in Mathematics Vol. 143, Springer-Verlag, Berlin, Heidelberg, New York (1970). 8. P. Hall, The classification of prime-power groups, J. reine angew. Math. 182^ (1940), 130-141. 9. H. Hopf, Fundamentalgruppe und zweite Bettische Gruppe, Comment. Math. Helv. 1£ (1942), 257-309. 10. M.R. Jones $ J. Wiegold, Isoclinisms and covering groups, Bull. Austral. Math. Soo. U_ (1974), 71-76. 11. S.C. King, Quotient and subgroup reduction for isoclinism of groups * Dissertation (Ph.D.), Yale University, New Haven CN (1978). Available from: University Microfilms International, Order No. 79-16616. 12. C.R. Leedham-Green, Homology in varieties of groups I, Trans. Amer. Math. Soc. 162_ (1971), 1-14. 13. C.R. Leedham-Green § S. McKay, Baer-invariants, isologism, varietal laws and homology, Acta Math. 137_ (1976), 99-150. 14. S. MacLane, Homologyj Grundlehren der math. Wissenschaften vol. 114, Springer-Verlag, Berlin, Gottingen, Heidelberg (1963). 15. A.I. Moskalenko, On central extensions of an abelian group by using an abelian group, Siberian Math. J. 9_ (1968), 78-86. MR 37 # 312. 16. I.B.S. Passi, Induced central extensions, J. Algebra 16^ (1970), 27-39. 17. E.W. Read, On the centre of a representation group, J. London Math. Soc. (2) 16_ (1977), 43-50. 18. I. Schur, Uber die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen, J. reine angew. Math. 127 (1904), 20-50. 19. I. Schur, Untersuchungen iiber die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen, J. reine angew. Math. \S2_ (1907), 85-137. 20. U. Stammbach, Homology in group theory3 Lecture Notes in Mathematics Vol. 359, Springer, Berlin, Heidelberg, New York (1973). 21. J. Tappe, On isoclinic groups, Math. Z. 14j[ (1976), 147-153. 22. J. Tappe, Isoklinismen endlicher Gruppen* Habilitationsschrift, Rheinisch-Westfalische Technische Hochschule Aachen, Aachen (1978). Zbl 433.20020. 23. L.R. Vermani, A note on induced central extensions, Bull. Austral. Math. Soc. 2£ (1979), 411-420. 24. P.M. Weichsel, On isoclinism, J. London Math. Soc. 38^ (1963), 63-65. 25. J. Wiegold, The Schur multiplier: an elementary approach, these Proceedings.

Beyl:

Isoclinisms of group extensions

26. K. Yamazaki, On projective representations and ring extensions of finite groups, J. Fao. Sci. Univ. Tokyo Sect. I Kl_ (1964), 147-195.

185

186 THE MAXIMAL SUBGROUPS OF THE CHEVALLEY GROUP G2 (4) G. Butler University of Sydney, Sydney 2006, N.S.W., Australia

1.

INTRODUCTION The aim of this paper is to prove the following theorem.

Theorem 1.

The simple Chevalley group G2 (4) of order 2 1 2 3 3 5 2 7«13 has

eight conjugacy classes of maximal subgroups. their representatives

The isomorphism types of

are

2 2 -2 8 -(3xAs)

2 4 -2 6 • (3xA5 ^>

3-L3(4)-2

HJ

U 3 (4)-2

Us (3)-2

L2 (13)

A5XA5

where p n denotes an elementary abelian group of that order3 A

is the

alternating group of degree n, L (q) is the protective special linear group of dimension n over a field of q elements3 U (q) is the protective special unitary group of dimension n over a field of q2 elements> and HJ is the sporadic simple group of Rally Janko^ and Wales.

The group A-B has

a normal subgroup isomorphic to A such that the factor group is isomorphic to B. The groups G2 (q) were originally discovered by Dickson [6], [7].

The group G2 (4) is important because it occurs in the chain

Z4 <_ U3 (3) <^ HJ <_ G2 (4) <_ Suz, as noted by Suzuki [14] in his discovery of the sporadic simple group Suz.

Moreover, it is the subgroup of least

index in Suz, so it seemed natural to begin our investigation of Suzuki's group by studying G2 (4).

The subgroup structure of HJ has already been

determined by Finkelstein and Rudvalis [9]. The conjugacy classes of elements and the character table of G2 (4) were determined by Wright [16]. We list the classes in Table 1. The structure of the centralizers is taken from [15] and [16]. For a finite simple group G, every maximal subgroup is the normalizer in G of a characteristically simple group.

A characteristically

Butler:

Maximal subgroups of Gj (4)

187

simple group is a direct product of isomorphic simple groups.

Hence, an

abelian characteristically simple group is an elementary abelian p-group for some prime p.

The classes of elementary abelian subgroups and their

normalizers are determined either from the centralizers in G of elements of prime order or from the Sylow normalizers. The local analysis, counting arguments, and the techniques of Finkelstein, Rudvalis, and Young [8], [9], [10], [17] determine the isomorphism types of non-abelian simple subgroups of G2 (4), the number of classes of each type, and the centralizer and normalizer of a representative of each class. Properties of the simple groups (other than G2(4)) mentioned in this paper can be found in Dickson [5], Fischer and McKay [11], and Table 1

Classes of G2 C4)

class

centralizer

1A

212.33.52.7-13

2A

212-3-5

2B

8

2 -3-5 6

3

powers

22-28-As

representative

J

6

2 -A5

Ji

3A

2 -3 -5-7

3-L3(4)

T

3B

22-32-5

3xA s

Ti

4A

29-3

2A

R

4B

28-3

2A

Ri

4C

29

5AB

22-3-52

5xA 5

5CD

22-3-52

5xA s

6A

26-3 2

2A

R2 IT TTi

3A,2A

TJ

3B,2B

TiJi

6B

2 -3

7A 8A

3-7 2s

4A

K

8B

25

4C

Ki

2

a

10AB

2 -5

5AB,2B

TTJI

1OCD

22-5

5CD,2A

TTI

4

J

12A

2 -3

6A,4A

TR

12BC

24-3

6A,4B

TRi

13AB

13

15AB

3-5

5AB,3A

ITT

15CD

3-5

5CD,3B

TTiTi

21AB

3-7

7A,3A

aT

b,b -1

Butler:

Maximal subgroups of G* (4)

188

Young [17]. The character tables of these groups appear in McKay [13], whose notation we use for their conjugacy classes.

For example, the

"first" class of elements of order three is denoted 3A, and 13AB denotes the union of two classes of elements of order thirteen whose elements generate the same class of cyclic subgroups. The notation of Gorenstein [12] is used.

If H is a subgroup

of G 2 (4) then C(H) and N(H) denote respectively the centralizer and normalizer of H in G2 (4), while if Z is a conjugacy class of elements of G2 (4) then C(K) and N(X) denote the centralizer and normalizer of the subgroup generated by an appropriate representative of K.

Acknowledgement.

This work was partially supported by the National

Research Council of Canada.

2.

TECHNIQUES OF FINKELSTEIN, RUDVALISj AND YOUNG Let G be a group and let H be a subgroup of G.

C3 be conjugacy classes of elements of G.

Let C\ , C2 and

H is Ci-^pure if H\l C d

and

H is of type (C\3Ci*C*) if there exists x G C\ ,y G C2 such that xy G C3 and H = < x,y >.

Let t be a fixed element of C3 and define C2jC3)G

= {(x,y) G Ci x C2 I xy = t } , and

The subscript is dropped when it is clear which group is meant.

The

structure constant #(CijC2jCz) can be calculated from the character table of G (Burnside [1]). If Ki , K2 and K& are conjugacy classes of elements of H contained in C\ , C2 and C$ respectively then # (Ki jK2 jK.3 )„ <_ #(C\ 3C2 3C$ ). This result is used to determine the inclusion of classes of a subgroup in the classes of G, as well as being used in non-existence proofs of subgroups of various isomorphism types. The next two results concern the action (by conjugation) of C G (t) on L(Ci,C2,Cz).

Theorem 2 (Finkelstein and Rudvalis [9, Lemma 2.1]).

If C (t) acts

transitively on the set A(C\,C2*C3) then the set T = {< x g ,y g > | (x,y) G h(CijC2jC3 )j g G G } is a conjugacy class of subgroups of G. morej if (x,y) G h(CijC2jC3) in the action of C G (t) on

Further-

then C G (< x,y >) is the stabilizer of (x,y)

Butler:

Corollary 3.

Maximal subgroups of G2(4)

189

If C G (t) has m orbits on k(C\ >C2>C3) then V is the union

of at most m conjugacy classes of subgroups of G.

Furthermorej if (x,y)

is an element of the i-th orbit A. then C G (< x,y >) is the stabilizer of (x,y) in the action of CG(t) on A^. A knowledge of the orbits of CG(t) on A (Ci., C2., C3 J can be obtained by considering which elements of C G (t) commute with elements of Ci and C2 . The following theorem is useful in G2 (4) when p=5.

Theorem 4 (Finkelstein and Rudvalis [10, Theorem 6.1]). Let p >_ 5 be a prime and assume the elements of C\ > C2 and C3 have order 2, 3, and p respectively.

Let t e c3 be fixed and assume that t is conjugate to t m

in G if and only if m is a square modulo p. Let Ao be the subset of A (Ci j C2 3 C3 ) consisting of the pairs (x,y) such that < x,y > is isomorphic to L2 (p).

Then the number of

conjugacy classes of subgroups of G isomorphic to L2 (p) and of type (Ci;C2jtCi) is the number of orbits of CG(t) acting on Ao by conjugation.

3. (3.1)

LOCAL ANALYSIS

p=13.

A Sylow 13-normalizer is isomorphic to F13 . We will see

that such groups are contained in subgroups isomorphic to L2 (13). (3.2)

p=7.

Let P be a Sylow 7-subgroup.

elements of order three lying in 3A.

Then C(P) = 3x7 with the

Hence N(P) <_ N(3A).

Actually,

N(P) = < x,y z I z7 = x3 = y6 = [z,x] = 1, z y = z3 , x y = x"1 >. (3.3)

p=5.

As a Sylow 5-subgroup P of HJ is a Sylow 5-subgroup of G2 (4),

we make use of the results of [9]. As elements of 5A fix precisely one point in the degree 416 representation, N(5A) = N HT (5A) = F5xAs , and P fixes precisely one point. Hence N(P) = N UT (P) s 52 (2xE 3 ).

Let w G 5C.

Then C(w) = 5xAs , where the

2

As is of type (2A,3B,5A), and N(< w >) = F5 xAs , as G2 (4) has no subgroups isomorphic to E5 . We will see that N(5C) is contained in a subgroup isomorphic to A5 xA5 . (3.4)

p=3.

subgroup.

Again, we draw on the results of [9]. Let P be a Sylow 3-

Then P is extraspecial of order 27 and exponent 3,

Z(P)-1 C 3A, and P-Z(P) C 3B.

Hence, C(3A) = 3-L3 (4) and C(3B) s 3xAs

gives:

Lemma 5. 9.

G2 (4) has unique class of elementary abelian subgroup of order

A representative contains 2 elements of 3A and 6 elements of 3B.

It

Butler:

Maximal subgroups of G2 (4)

190

is self-centralizing and its normalizer is contained in N(3A).

Further-

more, N(3A) = 3*L3(4)*2 and N(3B) = I3 *A5 . By Sylow's theorem and the fact that Aut(P) = 3 2 «SL(2,3), N(P)/P is isomorphic to a Sylow 2-subgroup of SL(2,3). We will see that N(3B) is contained in a subgroup isomorphic to As xAs . From the analysis of L2 (7) subgroups we see that of the 3 classes of L2 (7) in L3 (4),

two are interchanged by the automorphism,

and the other is normalized.

Thus the automorphism is the transpose-

inverse automorphism x foio]. [100J (3.5)

p=2.

Our analysis in this section will rely heavily on the work

of Thomas [15].

Following his notation, the elements involved in the 2-

local analysis are x r (a)

a € GF(4) = {0,1,w,u)2 },

r ^ {a,b,a+b,2a+b,3a+b,3a+2b},

k = h(a+b,w), where for all a, $, r

w = w a , j = h(b,co) ,

t = wfe,

= x r (a+3), [xa(a),xb(3)] =

[x

[

b(a)'X3a+b(3)]

= X

Vb(a)'X2a+b(3)]

3a+2b(a3)'

=

X

3a + 2b ( a 3 ) >

and all other commutators are trivial; k3 = 1 and acts by conjugation thus x a (a) - xa(a32a),

W

0 0

+ Vb(a)Sa)'

x 2 a + b (a:) ^ X 2 a + b ( u a ) '

X

3a + 2b ( a ) -

X

3a + 2b ( o ) ;

w2 = 1, k = k"1 , and w acts by conjugation thus x

bCa) ~ X3a+b(a)' V b C a ) " x2a+bCa)'

X

3a + 2b ( a ) "

X

3a + 2b ( a ) ;

a n d < w , x a ( o t ) > = A s v i a w -^ (J J ) a n d x f l ( a ) -> ( ^ J } . Furthermore, j 3 = 1, [k,j] = 1 and j acts by conjugation

Butler:

Maximal subgroups of G2 (4)

191

x a (a) (a)2a )

X

•

3a + b ( a ) * X 3a + b Ca:)

t2 = 1, (tx^fl)) 3 = 1, j

+ X3a+2b(wo);

= j " 1 , and [k,t] = 1, and t acts by conjugation a)

>

X

2a + b ( a ) " X2a+b(ot)>

X

3a + b ( a ) * X 3a + 2b (a) -

We define the subgroups S r = < xr(l),x (ID) >

r e {a,b,a+b,2a+b,3a+b,3a+2b},

S =

S S S a b a + b S 2a+b S 3a + b S 3a+2b' n - C c c c c U

Y

" Va+b b 2a+b b 3a+b 3a+2b* = S

3a+b S 3a+2b J

Z = S

3a+2b* C = Ci = < S,k,w > = EX S ,k,w >, a M = S

a S a+b S 2a+b S 3a+b S 3a+2b-

Then S i s a Sylow 2-subgroup o f G, Z = Z(S) = Z(C), C = D = 0 2 ( C ) . and C / D ^ . and C2 / 0 2 (C 2 ) = As . M = C S (Y).

C2 = C ^ x ^ C D ) .

0 2 (C 2 ) =

Y i s an elementary a b e l i a n group o f order 2 4 and

NG(M) = < S , k , j , t > = M ( < k > x < j , t >) = M(Z3 xAs ) . The involutions of S have one of the following forms &

x

a(o:)VbC6)x2a+b(a6)x3a+bMx3a+2b(6)'

(ii)

x b (a)x a + b (B)x 2 a + b ( Y )x 3 a + b ( a -'e Y )x 3 a + 2 b (6), a ^ 0,

(iii) V b ( a ) x 3 a + b ( e ) x 3 a + 2 b ( Y ) ' X

2a + b ( o ) x 3a + b ( 6 : ) x 3a + 2b ( Y ) '

X

3a+b(a)x3a+2b(e)'

X

3a+2b(a)'

a

a

a

a

* °'

* °> a

* °>

* °>

^ °-

They are conjugate in S (with centralizer order) to x (a) (2 6 ); a x, (a)x2 vfY+a"^ 2 ) (28) (12 classes depending on values of a and Y +

a - V ) ! x a + b (a) (2 8 ); x 2 a + b (a) (2 8 ); In GiC4), #(2A,2A,2B) = 0.

Hence the set (En 2A) U T is a

subgroup, for any elementary abelian group E.

Therefore we need only

Butler:

Maximal subgroups of G2 (4)

192

consider the pure elementary abelian subgroups. The involutions of S in 2A are ( i i ) with y = a2 $ 2 , (v) and (vi). From Table 2 we see that C has 4 classes of 2A-pure elementary abelian subgroups of order 4, with representatives Z, and V(ci) = < x 3 a + 2 b ( 1 ) > X 3a+b (o0 As N(Z) = C< j > we have: Proposition 6. containing

>

a = 1 a) a)2

'

' '

'

I n G2

v

W>

(^)tj =

V

W-

The 2k-pure elementary abelian subgroups of order 8

Z are conjugate

in N(Z) to Z < x, , (1) >.

An elementary abelian subgroup containing V(a) has no involution of type ( i i ) as the product of an involution of type ( i i ) and one of type (v) has the form *b ( a ) xa+b ( 6 ) X2a+b M

X

3a + b ( a ~' e Y + c " ) X 3a + 2b ( 5 )

a 4- 0 t ai , which i s never an involution. Propostion 7. contain

Hence we have:

The 2A-pure elementary abelian subgroups of order 8 which

V(a) are contained in Y. The Ik-pure elementary abelian sub-

groups of order 16 are conjugate elementary

abelian

to Y, and there are no 2k-pure

subgroups of order 32.

We further investigate the elementary abelian subgroups of order 8 containing V(ot).

To generate a subgroup < V(a) ,y > of Y not

containing Z, we require an element y £ Y\Z<x«

u(°0 ^ - There are 8

choices of y, forming 2 cosets of V(a) with representatives x, X

3a+b(3)x3a+2b(a23)j

6

Table 2 representative X3a+2b(1)

* a'

In N

cCV(a))

= M
[class]

|centralizer|

1

2 1 2 .3-5 12

centralizer

2 -3.5

Ci

2 1 2 .3-5

Ci

x3a+b(l)

20

2 1 0 -3

M

X

20

2 1 0 -3

3a+b^

10

class

Ci

1

3a+2b(tt>

and

Conjugacy classes of involutions of Ci

1

X

+ b (3)

> x b ( a 2 ) >, these two

4(v) + 16(ii) 4(v) + 16(ii)

20

2 -3

M

240

28

S

960

26

S Y

S

b a+b a

4(v) + 16(ii) Y

Ui Kii)+48(iii)+48(iv

Butler:

Maximal subgroups of G2 (4)

elements are conjugate.

193

Thus the elementary abelian subgroups of order 8

containing V(a) are conjugate in Nr(V(a)) to S 7 conjugated by t to Z<x<$

+b(

1

,< x_

o,

(1) >.

This

>

) *

Any 2A-pure elementary abelian subgroup of G is conjugate to one of <

X

3 a + 2 b ^ >•

Z

= S 3a + 2b>

< * 3 a + 2 b ( D . x 3 a + 2 b («) >, Z< x 3 a + b (l)>, Y As M is normalized by k, the normalizers of V(l), V(u)), and Z< x-

, (1) > are contained in N(M). Also, C < N(Z) = C < j >, and

JE+D

N(Y) = N(M). Hence, the normalizer of a 2A-pure elementary abelian subgroup is conjugate to a subgroup of either N(Z) = D(< j >x< k,w >) = D(Z 3 xA 5 ), or N(M) = M(< k >x< t,j >) a M(Z 3 xA 5 ). These are not conjugate, as their 02-subgroups are D and M respectively, Z(D) = Z, Z(M) = Y, and so their O2-subgroups are not conjugate. We will now consider the 2B-pure subgroups.

A Sylow 2-sub-

group of C2 is C s (x 2 a + b (l)) = S b S 2 a + b S 3 a + b S 3 a + 2 b whose involutions in 2B are (ii)

x b ( a ) x 2 a + b ( Y ) x 3 a + 2 b ( 6 ) , a * 0 ^ y,

fiv)

x

2a+b(a)x3a+b(6)x3a+2b(Y)'

These are conjugate in C g (x 2 (ii)

+b (l))

a

* °'

to

xb(a)x2a+b(Y), |x2a+b(a)x3a+b(6)'

B

* °

In C2 these are conjugate (with centralizer order) to x^ +u(°0 (28*3*5) ; x 2 a + b (a)x 3 a + b (l) (2 8 ); and x b (l)x 2 a + b (a) (2 6 -5).

Hence C2 has 3 classes

of 2B-pure elementary abelian subgroups of order 4, with representatives S

2a + b' <

Let E be any of these 3 subgroups. E contains an element of the form - x^

+b (°0

"•

Then, for each a = l,a),aj2, Therefore, a 2B-pure

elementary abelian subgroup of order 8 containing E contains no further element of type (ii) or (iv). Hence G has no 2B-pure elementary abelian subgroup of order 8.

Butler:

Maximal subgroups of G2 (4)

194

Furthermore, E is normalized by k, so N(E) = C(E)< k > which is a subgroup of N(M).

Therefore, the maximal 2-local subgroups are N(Z)

and N(M) . Summary.

The maximal local subgroups are 6

N(13A) = Fi3 N(5A)

2

s Fs

N(3A) = 3-L3 (4)2 As

N(3B) = Z3XA5

2

N(5C)

a F5" x As

N(Z)

ss 2 2 -2 2 (3xA s )

N(5 2 )

= 52 (2xZ3;

N(M)

= 24 -26 (3xAs )

HJ contains conjugates of N(5A) and N(5 2 ); L2 (13) contains N(13A); the normalizers of subgroups isomorphic to As contain N(5C) and N(3B).

The remaining subgroups are maximal. 4.

NON-ABELIAN SIMPLE GROUPS We begin by limiting the order of the subgroups of G2 (4).

Theorem 8. Proof.

G2 (4) has no proper subgroups of index less than 416.

The degrees of the rational characters which are less than 416

are 1, 65, 78, 350, 364, and 378. Hence, if i\> is a permutation character of degree less than 416 then either ty = 1 + x> X irreducible, or ip = 1 + X2 + X10 of degree 1 + 65 + 78 = 144. first case, i|/ takes negative values. contrary to 8B2 = 4C.

For all such x in the

In the second case K 8 B ) > ^(4C)

Hence, there is no permutation character of de-gree

less than 416. Corollary 9.

If H is a simple subgroup of G2 (4) then |H| < 106 . As these groups have been classified [4], the possible simple

subgroups in G2 (4) are, by order consideration, A n , n = 5,6,7,8; L2 (q), q = 7,8,13,25,27,64; L3 (q), q = 3,4; U3 (q), q = 3,4; Sz(8) and HJ.

It is

well known that G2 (4) has subgroups isomorphic to U3 (4) and HJ; and hence subgroups isomorphic to As, L2 (7) , and U3 (3) . Young [17] exhibits a subgroup isomorphic to L2 (13).

We will show that G2 (4) has no subgroups

of the remaining isomorphism types. G2 (4) has precisely one class of elementary abelian subgroups 2

of order 3 .

They contain 2 elements of 3A and 6 elements of 3B.

A

Sylow 3-subgroup of Ae is elementary abelian and contains 4 elements with cycle type I 3 3 and 4 elements with cycle type 32 . Hence G2 (4) has no

Butler:

Maximal subgroups of G2 (4)

19 5

subgroup isomorphic to A6 , nor isomorphic therefore to A7 , A* , or L3 (4) . Lemma 10. G2 (4) has no subgroups isomorph-ic to L2 (25). Proof.

The classes of L2 (25) are given in Table 3. As 6A is a square,

6A fuses to 6A, and hence 3A, 2A fuses to 3A and 2A. (2,3,13)-group and #(2A,3A,13) G .,. = 0 .

L2 (25) is a

Hence the result.

As L2 (q) contains an elementary abelian group of order q and a cyclic group of order e(q-l), where e = gcd(q-l,2), G2 (4) has no subgroups isomorphic to L2 (27) or L2 (64).

The Sylow 13-normalizer of Sz(8)

is isomorphic to F13. Therefore, G2 (4) has no subgroups isomorphic to Sz(8). Lemma 11. G2 (4) has no subgroup isomorphio to L3 (3). Proof.

Using the notation of [13] for the classes of L3 (3), the fusion

of L3 (3), if it exists, must be L3 (3)

: 1A 2A 3A 3B 4A 6A 8AB 13ABCD

G2 (4)

: 1A 2A 3A 3B /4A J4A \4C

6A J8A /8 13AB \8 18B

The restriction of X2 is not a character of L3 (3) and therefore G2 (4) has no subgroup isomorphic to L3 (3). Hence, the isomorphism types of the simple subgroups of G2 (4) are known.

We will now consider each isomorphism type in turn.

Table 3 class

|centralizer| 3

1A

2 -3-5 -13

2A

2 3 -3

3A

22-3

4A

22-3

5A

52

5B

52

6A

22-3 2

12AB

2 -3

13A-F

13

powers

2

2A

3A,2A 6A,4A

Butler:

(4.1)

Maximal subgroups of Ga (4)

196

HJ.

Lemma 12.

The fusion of HJ is HJ

:1A 2A 2B 3A 3B 4A 5AB 5CD 6A 6B 7A 8A 10AB 10CD 12A 15AB

G2 (4):1A 2A 2B 3A 3B 4A 5AB 5CD 6A 6B 7A 8A 10AB 10CD 12A 15AB Proof.

Since 2A is a square, 2A fuses to 2A.

Hence 6A fuses to 6A, 3A

fuses to 3A, 10CD fuses to 10CD, and 5CD fuses to 5CD.

As HJ contains a

Sylow p-subgroup of & (4) for p = 3,5, it follows that 3B fuses to 3B and 5AB fuses to 5AB.

Theorem 13. Proof.

The rest follows from the power maps.

G2 (4) has one class of subgroups isomorphic to HJ.

By restricting the permutation character it follows the HJ has

three orbits on 416 points.

The indices of the maximal subgroups of HJ

force the orbit lengths to be 1,100, and 315.

Hence every subgroup

isomorphic to HJ is conjugate to the point stabilizer (in the degree 416 representation).

(4.2)

U 3 (4).

Lemma 14.

Proof.

The fusion of Ife (4) is U3 (4) :

1A

2A

3A

4A

5A-D

5EF

10A-D

13A-D

15A-D

G2 (4) :

1A

2A

3B

4C

5CD

5AB

10CD

13A

15CD

As 2A is a square, 2A fuses to 2A.

Hence 10A-D fuses to 10CD,

5A-D fuses to 5CD, 15A-D fuses to 15CD, and 3A fuses to 3B. contains a Sylow 5-subgroup of G2 (4), 5EF fuses to 5AB.

As U3 (4)

The restriction

of X2 is irreducible, therefore 4A fuses to 4C.

U3 (4) is a (2,3,13)-group and #(2A,3B,13) = 13. has a unique subgroup isomorphic to U3 (4).

Hence Gj (4)

As there is a unique subgroup

isomorphic to U3 (4) containing a given Sylow 13-subgroup, the Sylow 13normalizer implies that C(U3 (4)) = 1 and N(U3 (4)) = U3(4)-2. (4.3)

U 3 (3).

Lemma 15.

The fusion of Ife (3) is U3 (3) :

1A

2A

3A

3B

4AB

4C

6A

7AB

8AB

12AB

G2 (4) :

1A

2A

3A

3B

4A

4A

6A

7AB

8A

12A

Butler:

Proof.

Maximal subgroups of G2 (4)

As 2A is a square, 2A fuses to 2A.

fuses to 3A.

Hence 6A fuses to 6A, and 3A

If 12AB fuses to 12BC then 4AB fuses to 4B and the fusion

of 8A is impossible. 8A fuses to 8A. to 3B.

197

Hence 12AB fuses to 12A and 4AB fuses to 4A, and

As U3 (3) contains a Sylow 3-subgroup of G2(4), 3B fuses

Determining the restriction of x*

as a sum

of irreducibles of

U3(3) gives that 4C fuses to 4A.

Theorem 16. Proof.

G2 (4) has one class of subgroups isomorphic to U3 (3).

U3 (3) is a (2,6,7)-group and #(2A,6A,7) = 21.

A two-point

stabilizer is isomorphic to U3 (3) and has trivial centralizer.

Hence

C(7A) acts transitively on A(2A,6A,7).

For the two-point stabilizer, N(U3 (3)) = U3(3)«2.

(4.4)

L 2 (13).

Lemma 17.

Proof.

The fusion of L2 (13) is U (13) :

1A

2A

3A

6A

7ABC

13AB

G2 (4)

1A

2B

3B

6B

7A

13AB

:

L2 (13) is a (2,3,13)-group.

If 6A fuses to 6A then 3A fuses to

3A and 2A fuses to 2A, contradicting #(2A,3A,13) = 0.

Hence we have the

above fusion.

Theorem 18.

G2 (4) has one class of subgroups isomorphic to U (13).

They

are self-normalizing. Proof.

Let P be a Sylow 13-subgroup of G 2 (4).

order 6 normalizing P.

containing P; and hence P<s> whose product is s.

Let s be an element of

Suppose Y is a subgroup isomorphic to U (13) In Y there are 6 pairs of involutions

For any pair (xi , x 2 ) , Y = < P,s,xi >.

In G2 (4)

there are precisely 6 pairs of involutions in 2B whose product is s. Hence G2 (4) has a unique class of subgroups isomorphic to L2 (13).

As

N(P) = N y (P), the rest follows.

(4.5)

L 2 (7).

Theorem 19. L2 (7).

G2 (4) has at most three classes of subgroups isomorphic to

Every subgroup isomorphic to L2 (7) has a cyclic centralizer

generated by an element of 3A.

Butler:

Proof.

Maximal subgroups of G2 (4)

The involutions of U (7) are squares.

198

Hence they lie in 2A.

As

#(2A,3A,7) = 0 and #(2A,3B,7) = 21, G2 (4) has at most three classes. From the structure of C(3A), it is clear they are centralized by an element of 3A.

We found representatives of two distinct classes and computed their normalizers using the algorithm of [2], [3]. They were isomorphic to Z3 xL2 (7) and (Z3 xL2 (7))-2.

This completely determines how an element

normalizing 3A acts on the three classes of L2 (7) in C(3A).

Thus G2 (4)

has precisely two classes of subgroups isomorphic to L2 (7). (4.6)

A5. The non-zero structure constants of type (2X,3Y,5Z) are #(2B,3A,5C) = 5,

#(2B,3B,5A) = 300,

#(2B,3B,5C) = 300,

#(2A,3B,5A) = 80.

Proposition 20.

G2 (4) has a unique class of subgroups isomorphic to As

of type (2B,3A,5C).

If H is a representative then C(H) is isomorphic to

A5 of type (2A,3B,5A). Proof.

As C(5C) = 5*A5 , with the A5 of type (2A,3B,5A), the results

follow immediately from #(2B,3A,5C) = 5.

Corollary 21.

C(3B) = 3xAs with As of type (2A,3B,5A) .

Proposition 22.

G2 (4) has a unique class of subgroups isomorphic to As

of type (2B,3B,5C).

They are self-normalizing^ and contained in a sub-

group isomorphic to HJ. Proof.

C(5C) = 5xAs with As of type (2A,3B,5A).

As no element of 2A or

5A commutes with an element of 3B, the lengths of the C(5C)-orbits on A(2B,3B,5C) are divisible by 2 2 *5 2 . As no element of 3B commutes with an As of (2B,3B,5C), C(5C) is transitive on A(2B,3B,5C).

Hence there is a

unique class, they have trivial centralizer, and are clearly selfnormalizing.

The fusion of HJ in G2 (4) and [9] give the last statement.

Proposition 23. type (2B,3B,5A). Proof.

G2 (4) has four classes of subgroups isomorphic to As of Each has a conjugate of S 2

+b

as its centralizer.

C(5A) = 5xAs with As of type (2B,3A,5C).

No element of 3A or 5C

commutes with an element of 2B, so 75 divides the C(5A)-orbit lengths on A(2B,3B,5A).

Therefore the centralizer of an As of this type is conjugate

Butler:

to a subgroup of S^

Maximal subgroups of G2 (4)

,.

As C(S~

199

, ) contains an As of this type, and

since #(2B,3B,5A) = 300 = |C(5A)|, it follows that each centralizer is non-trivial.

As C2 = C(S 2

.) the orbit lengths are [754] and the result

follows from Theorem 4.

Proposition 24.

The C(5A)-orbits on A(2A,3B,5A) are [5,253] or [5,75].

The subgroups of G2 (4) of type (2A,3B,5A) have centralizer isomorphic to As and A» or to As and Z2xZ2 in the respective oases. groups of each oentralizer is conjugate to S^ Proof.

Let C(5A) = < f >xA.

The Sylow 2-sub-

, .

There is a unique class of subgroups of

type (2A,3B,5A) with centralizer isomorphic to A5 because C(A) = A5 of type (2A,3B,5A).

By Theorem 4, C(5A) has precisely one orbit of length 5

on A(2A,3B,5A). The subgroups of A have index 5,6,10,12,15,30 and 60.

There-

fore the C(5A)-orbits of length not equal to 5 have length 25,30,50,60,75. , ) , if 2 divides the order of the centralizer then 2 2 does.

As C2 = C(S 9

Hence there is no orbit of length 50.

The result follows.

By the above results we have:

Corollary 25.

G2 (4) has a unique class of subgroups isomorphic to A5 xAs .

They are self-normalizing. The normalizers of subgroups isomorphic to As are contained in conjugates of HJ, As xAs or N(S 2

5.

,) .

CONCLUSION As the inclusion of the normalizers of the characteristically

simple subgroups has been discussed, we have proved

Theorem 26.

G2 (4) has eight conjugacy classes of maximal subgroups.

isomorphism typess orders^ and indices are 2 2 •2 8 -(3xAs)

212325

1365

2 4 •2 6 .(3xA 5 )

212325

1365

3- L3 (4)-2

27335 7

2080

7

3

2

HJ

2 3 5 7

u3 (4).2 u3 (3).2

2 7 3 5 2 13 26337 2

L2 (13)

2 3 7 13

As xAs

243252

416

2016 20800 230400 69888

The

Butler:

Maximal subgroups of G2 (4)

200

BIBLIOGRAPHY

2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

15. 16.

17.

W. Burnside, The theory of groups of finite order* 2nd Edition, Dover, New York (1955). G. Butler, Computing normalizers in permutation groups, preprint. J .J. Cannon, Software tools for group theory, to appear in "Proceedings of Santa Cruz conference. C. Cato, The orders of the known simple groups as far as one trillion, Math. Comp. 31. (1977), 574-577. L .E. Dickson, Linear groups with an exposition of the Galois field theory, Dover, New York (1958). L .E. Dickson, Theory of linear groups in an arbitrary field, Trans. Amer. Math. Soc. 2_ (1901), 363-394. L .E. Dickson, A new system of simple groups, Math. Ann. 60^ (1905), 137-150. L. Finkelstein, The maximal subgroups of Conway's group C3 and McLaughlin's group, J. Algebra 25_ (1973), 58-89. L. Finkelstein § A. Rudvalis, Maximal subgroups of the Hall-JankoWales group, J. Algebra 24 (1973), 486-493. L . Finkelstein § A. Rudvalis, The maximal subgroups of Janko's simple group of order 50,232,960, J. Algebra 30^ (1974), 122-143. J . Fischer § J. McKay, The nonabelian simple groups G, |G| < 106 maximal subgroups, Math. Comp. 32^ (1978), 1293-1302. D. Gorenstein, Finite groups* Harper and Row, New York (1968). J . McKay, The simple groups G, |G| < 106 - character tables, Comm. Alg. 7_ (1979), 1407-1445. M. Suzuki, A simple group of order 448,345,497,600, in Theory of finite groups* edited by R. Brauer $ C-H. Sah, Benjamin, New York (1969), 113-119. G. Thomas, A characterization of the groups G2 (2 n ), J. Algebra \Z_ (1969), 87-118. D. Wright, The irreducible characters of the simple group of M. Suzuki of order 448,345,497,600, J. Algebra 29_ (1974), 303323. K-C. Young, Computing with finite groups* Ph.D. thesis, McGill University (1975).

201 GENERATORS AND RELATIONS FOR THE COHOMOLOGY RING OF JANKOfS FIRST GROUP IN THE FIRST TWENTY ONE DIMENSIONS G.R. Chapman University of Guelph, Ontario NIG 2W1, Canada

1. INTRODUCTION If G is a finite group, and A a G-ring, then generators and relations for H*(G,A), the cohomology of G with coefficients in A, may be obtained via the Hochschild-Serre spectral sequence [5]. Examples include [2],[3],[7],[10],[13].

Prerequisite to such a calculation is the

existence of a proper normal subgroup of G, so that for simple groups this method is not available.

Perhaps, in such cases, a more appropriate

method is to extract the p-part of the cohomology C[H*(G,A)] ) from the cohomology ring of the Sylow subgroup for each prime p dividing |G|, the order of G ([1, p.259]).

In general, this requires a knowledge of the

intersection of the Sylow p-subgroups, but has the following two simplifications.

Let G

of automorphisms of G

be the Sylow p-subgroup of G, and $

the group

induced by inner automorphisms of G.

Lemma 1 ([12, Lemma 1]). If G

is abelian, then [H*(G,A)]

consists of

those elements of H*(G ,A) which are fixed under the action of *

on

H*(G ,A).

Lemma 2 ([12, Theorem 2]). If p is odd and G

is cyclic, then the p-

iperiod of G is twice the order of $ . In [12], Lemma 1 is stated only with integer coefficients, but the proof applies to coefficients in an arbitrary G-ring. Janko!s first group [6], which we denote by J, has order 2 .3.5.7.11.19 and Sylow 2-subgroup elementary abelian.

If p G ir(J), the

set of primes dividing | j | , let Z denote the integers, and Z/p the cyclic J-module of order p, both with trivial J-action.

In this paper we use the

above two lemmas to calculate generators and relations for H*(J,Z/p) for each prime p e w(J). Since J

is cyclic or elementary abelian, it is

well known that pH (J ,Z) = 0 (n>0). from

Hence the homology sequence arising

Chapman:

0

> Z

The cohomology ring of Janko's group

> Z -i-> Z/p

202

> 0

may be decomposed into short exact sequences

0

> [H n (J,Z)] p - J — > Hn(J,Z/p) - ^ — >

[H n + 1 (J,Z)] p

> 0 ( n > 1).

The Bockstein homomorphism A n : Hn(J,Z/p) is defined to be j n +

> H n+1 (J,Z/p)

© &n 9 so it follows that [H*(J,Z)]

may be obtained

either as ker(A*), or im(A*). For the odd primes in TT(J), the Sylow subgroups are cyclic and Lemma 2 applies.

In these cases the results may be read off from Janko's

original paper [6]. For p=2, we use the description of Nj(J2) (the normalizer of J2 in J) given in [4]. We calculate Hn(J,Z/2) for ascending values of n, and while it is known that a minimal set of cup product generators will be finite [14], we know of no results which give a bound for the highest degree in which a cup product generator may occur.

Thus we cannot determine when a complete set of generators has

been achieved.

The expense of the calculation in degree n depends upon

n4, so, while the method is applicable for any n, we calculate only up to n=21,

and offer no results concerning possible generators and relations

which may occur in higher degrees. follows.

Our reason for this choice is as

One may regard the calculation as a problem in invariant theory,

since H*(J,Z/2) is the subring of R invariant under the action of N_(J2), where R is the ring of polynomials in three variables over Z/2.

We seek

an integrity basis for the ring of invariants, and a complete set of syzygies.

The proof that the ring of invariants is finitely generated

offers no bound on the degrees of the generators when the field in question is of prime characteristic [9]. However, were the field the complex numbers, then [8] would show that |N T (J2)| is an upper bound, and in our case, this number is 21.

The reader is referred to [11] for

details.

2.

METHOD For p an odd prime in TT(J), J

is cyclic so that by [1],

H*(J ,Z/p) has ring generators x in degree 1, £ in degree 2, subject only to the relation x 2 =0, p x = p £ = 0 .

It follows from [12, Lemma 3] that

The cohomology ring of Janko!s group

Chapman:

H*(J,Z/p)

i s g e n e r a t e d by ( x ^ ^ "

1

,

£%)>,

where in

* |N_(J ) / C _ ( J ) f

p

and CT(J ) is the centralizer of J in J. J p p is generated by %m^

and so [H*(J,Z)]

203

«J

p

u

p

It is well known that A(x) = £,

alone.

If g € J

(g 7* e), then we

may write m = |N (J ) | •h(g)/|j|, where h(g) is the class number of g in J P P J. The values of m may now read off from Janko's original paper [6], and are displayed in §3. For p=2, J2 is elementary abelian of order 8.

Since

C T (J 2 ) = J2 [6], N T (J 2 ) may be regarded as a subgroup of GL(3,2). J

By [4]

J

p.32, there is a choice of generators ti,t2 ,t3 for J2 for which NT(J2) is J generated by elements 1 0 0

0 0 1

1 1

0

g= 1

1

0

0 0 1

1 0 1

Now H1(J2 ,Z/2) - Hom(J2 ,Z/2) and if we

of order 3, 7 respectively.

define x,y,z € H 1 (J 2 ,Z/2) to correspond under this isomorphism to the homomorphisms tl

tl

+ 1

+ 0

t2 ->• 0

t 2 -»- 1

t3 + 0

t3 + 0

t3

-•

respectively, it is well known that H*(J2 ,Z/2) is the polynomial ring on x,y,z with mod 2 coefficients. A(s) = 22 .

Furthermore, A (a:) = x2 , A (2/) = y2 ,

The dimension of Hn(J2 ,Z/2) as a vector space is

we denote by d(n).

n 2

, which

If Fi ,Gi are the automorphisms of H1 (J2 ,Z/2) induced

by f, g then

Fi (x) = x

Gi (x) = y

Fi (y) = z

Gi (y) = z

Fi (z) = x + y + ;

Gi (z) = x + z.

We proceed inductively on the degree of the cohomology group. Let & be a minimal generating set of H*(J,Z/2) (as a ring), and suppose we have determined those elements of & which have degree < n-1. this set &(n-l).

Call

Suppose further that we have a set of those relations

in H*(iJ,Z/2) which occur in degree < n-1, called (R(n-l) . We may take

(a)

Select a basis of Hn(J2 ,Z/2) and write, as d(n) x d(n)

matrices, the automorphisms induced on H (J2 ,Z/2) by f and g. these by F , G

respectively.

Denote

If I is the d(n) x d(n) identity matrix, let

Chapman:

204

The cohomology ring of Janko's group F n

- I

be the 2d(n) x d(n) matrix with upper d(n) rows those of F - I , and lower d(n) rows those of G - I . According to Lemma 1, H n (J,Z/2) is the nullspace of M . n (b)

Row reduce M

(by Gauss-Jordan reduction), and find a basis

for the nullspace of M . Denote this basis by {vi ,v2 ,... »varn") }• (c)

Calculate the set of polynomials of degree n which may be

obtained by multiplying elements in &(n-l).

Identify these polynomials

in accordance with the relations in (R(n-l).

Denote the set of

polynomials so obtained by {ai ,a2 ,... ,\rn^ }• (d)

Form the b(n) x a(n) matrix Q(n) whose coefficients are

defined by a(n)

a, =

I Q, ,(n) v,

Reduce this matrix.

(1< i
Each row of zeros in the reduced matrix yields a

relation, and these are appended to &(n-l) to yield
Put

Then &(n) is obtained as the union of &(n-l)

tne with c(n) elements of {vi ,v2 ,...> v a r n ) choice of which elements being clear from the form of the reduced matrix.

(e) The results for integer coefficients may now be obtained via the Bockstein homomorphism. 3. (a)

EESULTS If p is an odd prime in TT(J), we have the following table p m(p)

(b)

3

5

7

11

19

2

2

6

10

6

p=2. In the first 21 degrees, H*(J,Z/2) is generated by

>v in degrees 3,4,5,6,7, where a = x3 + y3 + z3 + x2 z + xy2 +y2 z + xyz, $ = x4 +y* + z4 +x2y2

+x2z2

+y2z2

+xyz(x+y+z),

Y = xs +ys + zs + x4 z + xy4 +y4 z + xyz (xy+xz+yz) , y = x6 +y6 + z6 +x4y2

+x2z4

+y2z4

+xyz(x3+y3

+z3),

Chapman:

205

The cohomology ring of Janko's group

v = xyz (x3y + x3 z + xy3 + xz3 + y3 z + yz3) . The relations are av+$y+y2 = 0

(degree 10), a4+a2 y+B3 +yv+y2 = 0

(degree 12),

and the following table gives a n = dim . Hn(J,Z/2) for 1 < n < 21.

n

1

2

3

4

5

6

7

8

9

10

a(n)

0

0

1

1

1

2

2

2

3

3

n

11

12

13

14

15

16

17

18

19

20

21

a(n)

4

5

5

6

7

7

8

10

10

11

13

If this set of generators were complete, then a good polynomial basis for H*(J,Z/2) (in the sense of [11, p.99]) would consist of free invariants {a,3,v} and transient invariants {y,y,yy}.

Moreover the Poincare series

would be

The Bockstein homomorphism gives A(a) = 6, A(B) = 0, A(y)

a 2 , A(y) = v, A(v) = 0,

so that [H*(J,Z)]2 has generators S,T,U,V,W,X,Y in degrees 4,6,7,9,10,12, 15, where S = $, T = a2 , U = v, V = 3y+a3 , W = y2 , X = y2 , Y = a3y+y3 • Relations in the first twenty one degrees are S4+SX+ST2+TW+UV = 0

(degree 16),

S2W+T3+V2

(degree 18),

= 0

SY+T2U+WV = 0

(degree 19),

S2X+TU2+W2

(degree 20),

= 0

SWU = (T2+S3+X)V + TY

(degree 21).

Further, the following relations follow from relations in H*(J,Z/2). UY = STX+W(T2+S3+X)

(degree 22),

WU2 = T 2 X+X 2 +S 6 +T 4

(degree 24),

VY = SW2 +f +T2 S3 +T2 X

(degree 24),

Y

2

= fx+W

3

(degree 30).

REFERENCES

1. H. Cartan $ S. Eilenberg, Homologioal algebra, Princeton University Press, Princeton, N.J. (1956).

Chapman:

The cohomology ring of Janko's group

206

2. G.R. Chapman, The cohomology ring of a finite abelian group, Proo. London Math. Soc, to appear. 3. L. Evens, On the Chern classes of representations of finite groups, Trans. Amer. Math. Soc. US_ (1965), 180-193. 4. M. Hall, Jr., Computers in group theory, in Topics in group theory and computation, edited by M.P.J. Curran, Academic Press (1977). 5. G.P. Hochschild § J.P. Serre, Cohomology of group extensions, Trans. Amer. Math. Soc. 74 (1953), 110-135. 6. Z. Janko, A new finite simple group with abelian Sylow 2-subgroup and its characterization, J. Algebra _3 (1966), 147-186. 7. G. Lewis, The integral cohomology ring of groups of order p 3 , Trans. Amer. Math. Soc. U2_ (1968), 501-529. 8. E. Noether, Der Endlichkeitssatz der Invarianten endlicher Gruppen, Math. Ann. 1J_ (1916), 89-92. 9. E. Noether, Der Endlichkeitssatz der Invariantentheorie endlicher linearer Gruppen der Charakteristik p, Nachr. Ges. D. Wiss. Gottingen (1926), 28-35. 10. D. Quillen, The mod 2 cohomology rings of extra-special 2-groups and the spinor groups, Math. Ann. 1_1_ (1972), 197-212. 11. N.J.A. Sloane, Error-correcting codes and invariant theory: new applications of a nineteenth-century technique, Amer. Math. Monthly 84- (1977), 82-107. 12. R.G. Swan, The p-period of a finite group, III. J. Math. £ (1960), 341-346. 13. C.B. Thomas, The integral cohomology ring of S ^ Mathematika 2\_ (1974), 228-232. 14. B.B. Venkov, Cohomology algebras for some classifying spaces, Dokl. Akad. Nauk SSSR 127 (1959), 943-944. M.R.21#7500.

207

THE BURNSIDE GROUP OF EXPONENT 5 WITH TWO GENERATORS

M. Hall Jr. California Institute of Technology, Pasadena, CA 91125, U.S.A. C.C. Sims Rutgers University, New Brunswick, NJ 08903, U.S.A.

1. INTRODUCTION It has been known for some time [1, Ch.18] that finitely generated groups of exponent 2, 3, 4 or 6 are finite. much less known on groups of exponent 5.

So far there is

It was shown by Kostrikin [4]

in 1955 that the largest finite group with two generators has order at most 5 3 4 .

In 1956 Graham Higman [3] used a combinatorial argument to

show that for any finite number of generators there is a largest finite group of exponent 5.

In 1974 Havas, Wall and Warns ley [2] showed that the

largest finite two generator group of exponent 5 has order exactly 5 3 4 and they found a detailed table of commutator relations which describe this group exactly. In this paper in an attempt to prove finite B(5,2) the Burnside group of exponent 5 with two generators, it is shown that B(5,2) has a normal subgroup Ki of index 51 ° . explicit elements.

Ki is the normal closure of 24

Ki should be an Abelian group of order 5 2 4 . This

reduces the proof of the finiteness of B(5,2) to a proof that Ki is Abelian.

2.

THE RESTRICTED BURNSIDE PROBLEM A weakened form of the Burnside conjecture is the following:

Conjecture.

B

: For a positive integer r, there is an integer b

n, r n, r such that every finite group of exponent n that can be generated by r elements has order at most b n,r This is known as the restricted Burnside problem.

If

G = B(5,2) is the Burnside group of exponent 5 with two generators, it was shown by Kostrikin [4] in 1955, using methods from Lie rings, that if G. is the ith term of the lower central series for G, then G13 = G14 = ... so that if G is finite (being a 5-group) some G showed for G, the restricted group, that |G| < 5

= 1 and so G n 34

.

= 1.

He

Hall § Sims:

Burnside group of exponent 5

208

In 1974, Havas, Wall and Wamsley [2] showed that |"G| = 5 3 4 . They used a commutator calculus and gave a table of commutator relations which describe (T exactly.

The table given here is a later version

developed by George Havas. The generators of G are 1 and 2 and further commutators are 3,...,34 defined recursively in the following way.

Table 2 . 1 Class 2

3 = (2,1)

Class 3

4 = (3,1) 5 = (3,2)

Class 4

6 = (4,1) 7 = (4,2)

11 = (9,1) 12 = (9,2)

Class 6

Class 9

23 = (19,1) 24 = (19,2)

13 = (10,1)

25 = (20,1)

14 = (10,2)

26 = (20,2)

15 = (11,2)

27 = (21,2)

8 = (5,2)

16 = (12,1)

28 = (22,2)

9 = (6,2)

17 = (12,2)

10 = (7,2)

18 = (13,2)

Class 5

Class 7

Class 10

30 = (24,2)

19 = (15,1) 20 = (15,2)

Class 8

29 = (23,2) 31 = (27,2)

Class 11

32 = (29,2)

Class 12

34 = (32,2)

21 = (16,2)

33 = (30,2)

22 = (18,2)

In the following table all commutators (i,j) i > j not listed are understood to be the identity.

Table 2.2

Thus (12, 11) is the identity.

Non-identity commutators

= 102 12 3 13' 14 4 15 2 16 2 17' 18'

(2,1)

= 3

(3,1) (3,2)

= 4

19' 21 2 24 2 26' 29 4 31 2 32 4 3 3 '

= 5

34 2

(4,1) (4,2)

= 6

(4,3)

(5 . 3 )

(5 . 4 )

22 4 24' 25 4 26 2 27 2 28 3 30 3 31*

= 7 4

2

4

4

4

= 9'll'l2 15 18 19 20 21

2

33 2 34'

224 25' 272 284 294 30' 31 4 33 3 34

(5,1)

(6 ,D

= 7'9'l02ll'l22133144153 3

3

3

2

16 20 2l'24 25'26 27 28 2

3

3

3

4

29 30 31 32 33 34 = 8

= 11' 15' 162 19 2 2 0 4 2 1 2 2 3 ' 24' 25 2 26 3 27' 29 4 30 3 31 4 32 3 33 4

3

3

f5.2)

= 122 13' IS 4 16 3 1 7 3 1 8 4 2 0 3 21 3

2

34 3 2

(6 . 2 )

= 9

(6 , 3 )

= 11' 16 4 20 3 2 1 ' 2 6 4 2 7 2 3 1 2 33 4 34 2

Hall § Sims: Burnside group of exponent 5 (6.4)

= 19' 23' 242 25 2 29" 304 32 2 33 2 34

(6,5)

1

2

4

4

4

(10.2)

= 14

(10.3)

= 17 2 18 3 20 3 21 1 22 3 24 1 25 2 26 3 27 4 28 4 29 3 30 4 31 3 32 3 34 3

2

= IS 16 19' 2 0 2 1 2 4 ' 25 26' (10.4)

= 20 4 21 3 24 3 26 1 27 4 32 1 33 3 34 4

= 9 II 12 13'l5'l6 17*20'

(10.5)

= 22 4 26 2 28 1 30 2 31 4 32 4 33 3 34 2

21 3 22 1 23* 244 25 4 26 2 27' 28 4

(10.6)

= 244 254 291 301 321 332 342

29 2 31 1 32 3 33 2 342

(10.7)

= 262 274 304 31 2 341

27*29" 32 4 33X 34 2 (7.1)

209

2

2

2

3

(7.2)

= 10

(10.8)

= 28 1 31 4 33 3 34 1

(7,3)

= 12 3 13 1 16 3 17' 18' 20 3 21 3 22 2

(10.9)

= 32 3 34 2

24 2 25' 26 1 27' 29 3 304 31 3 32 4

(11.1)

= 191 242 253 291 301 323 333 342

33 3 34 3

(11.2)

= 15

(11.3)

= 191 25 4 30 2 33 2 34 3

(11.4)

= 23 1 29 2 32 3 34 1

(11.5)

= 24 2 25 4 29 4 30 4 32 1 34 1

(11.7)

= 29 ! 32 4 34 2

(11.8)

= 30 2 32 3 33 2 34 4

(7,4)

= I S 4 16 3 19 3 20 3 21 3 24 4 25' 27 2 30 3 31 4 33" 34 2

(7,5)

1

3

2

2

1

4

3

4

2

= 17 18 20 21 22 24 25 27

2

28 2 29 2 32 2 34 4 2

3

1

1

(7,6)

= 19 24 25 29 30' 32 33

(8,1)

= 10 3 12 2 13 2 14 3 15 2 16 1 17 3 19' 4

2

3

3

2

3

3

2

1

1

(12.1)

= 16

= 14 17 22 25 26 27* 29' 30'

(12.2)

= 17

(12.3)

= 20 4 21 3 24 4 25 3 26 3 27 4 29 1 30 4

24 27 28 29 30 32 33' 34

(8,2)

3

31 2 33 3 34 4

(8,3)

2

27" 28" 30 31 32 4

29 32 33' 34

(8,6)

4

= 17 3 1 8 4 2 0 ' 22 1 24 4 25 3 26 3 27 2 2

(8,5)

31 3 32 4

= 14 2 17 3 18' 20 J 21 2 22 4 24' 26 2 2

(8.4)

(11.10) = 323

2

3

= 22 3 26' 27 1 28 1 30 3 31 4 33 3 34 4 3

4

4

2

3

3

4

= 20 21 24 25 26 27 29 31

3

32 2 33 3 3

1

4

(8,7)

= 22" 27 3 28' 30 3 1 ' 32 33' 34

(9,1) (9,2)

= 11 = 12

(9,3)

= 15 4 16 1 19" 20 1 21 2 24 2 25 2 26 2

(12.4)

= 25 1 29 4 30 2 32 2 34 3

(12.5)

= 26 3 27 2 30 2 31 3 32 4 34 4

(12.6)

= 29*323

(12.7)

= 30 2 32 2 33 4 34 2

(12.8)

= 31 1 34 1

(12.9)

= 32 3 34 4

(12.10) = 33 4 34 3 (13.1)

= 15 2 l6 2 20 2 21 2 23 1 26 3 29 2 31 4 321 334 344

(13.2)

= 18

27' 29 3 30 3 31 2 33' 34 3

(13.3)

= 21 2 24 3 25 2 29 1 31 4 32 1

(9,4)

= 19 2 24 2 25 2 29 1 32' 34 1

(13.4)

= 253 291 302 323 334 341

(9,5)

= 2 1 3 25 J 27' 28 4 30 3 32 1 33 3

(13.5)

= 26 4 27 1 30 1 31 4 32 2 34 4

(9,6)

= 23 1 29 1 32 4 34 2

(13.6)

= 29 1 32 4 34 2

(9,7)

= 24 2 25 4 29 1 30 3 32' 33' 34 2

(13.7)

= 30 4 33 1

(13.8)

= 31 4 33 1 34 3

(13.9)

= 323

2

1

3

2

4

3

(9,8)

= 26 27 30 31 32 33 34

(10.11

= 13

2

Hall § Sims:

Burnside group of exponent 5

(13,10) = 33 1 34 1

(14.1)

2

4

4

2

3

(14.3)

4

4

4

= 17 18 20 21 22 24 25 27 1

29 31 33 34 (14.2)

3

1

1

= 31 4 33 4 1

1

1

r

4

4

4

3

1

1

3

= 21 26 27 28 30 31 32 33

3

344 2

(14.4)

= 26 27 30 31 33

(14.5)

= 28 1 31 1 33 4 34 3 4

1

2

(18.2)

= 22

(18.3)

= 26 l 271 302 31 4 323 333 344

(18.4)

= 30 1 32 2 33 3 34 3

(18.5)

= 31 2 34 4

(18.6)

= 32334X

(18.7)

= 33 3 34 4

(18,9)

= 343

(19.1)

= 23

(19.2)

= 24

(14.6)

= 30 32 33

(14.7)

= 31 4 33 4

(19.3)

= 29 4 32 3

(14.9)

= 33 4 34 J

(19,5)

= 32 2 34 3

(15.1)

= 19

(20.1)

= 25

= 20

(20.2)

= 26

(20.3)

= 30 4 32 4 33 3 34 3

(15.2)

4

1

1

1

1

2

4

210

(15.3)

= 24 25 29 30 32 33 34

(15,5)

= 32434*

(20.4)

= 32 2 34 4

(15.7)

= 32334X

(20.5)

= 33 3 34 1

(15.8)

= 33 2 34 3

(21.1)

= 24 3 25 1 29 1 30 1 32 4 33 2 34 4

(21.2)

= 27

(21.3)

= 30 1 32 1 33 3

(21.4)

= 341

(21.5)

= 33 1 34 1

(22.1)

= 26 3 30 3 31 1 321 343

(15.10) = 343 (16.1)

4

2

1

3

1

3

2

= 19 23 24 25 29 30 32 33

3

2

34 (16.2)

= 21 4

1

3

4

1

1

1

(16.3)

= 24 25 29 30 32 33 34

(16.4)

= 29 3 32 2 34 2

(22.2)

= 28

(16.5)

= 30 3 32 4 34 4

(22.3)

= 31 4 33 2 34 4

(16.7)

= 32334*

(22.4)

= 33 3 34 3

(23,2)

= 29

(24.1)

= 343

(24.2)

= 30

(24.3)

= 32 2 34 3

(24.5)

= 342

(16.8)

2

= 33 34

3

3

(16,10) = 34 (17,1)

4

4

3

4

4

4

1

= 20 21 24 25 26 29 30 31

3

3

(17.2)

= 34

(17.3)

= 26 1 27 1 30 2 32 3 33 2 34 3

(25.1)

= 29 4 32 1 34 1

(17.4)

= 30 32 34

4

(25.2)

= 32 2 34 3

(17.5)

= 31 2 33 3 34 4

3

4

(25.3)

= 32*344

(17.6)

= 32 34

1

(25,5)

= 341

(17.7)

= 33 3 34 4

(26,1)

= 30 4 32 2 33 4 34 3

(26.3)

= 33 1 34 3

(26.4)

= 342

(27,1)

= 30 4 32 1 33 4

3

3

(17,9)

= 34

(18,1)

= 20 2 21 4 24 4 25 1 26 3 27 3 30 2 31 3 3

33 34

2

Hall $ Sims:

(27,2)

= 31 4

3

2

3

(28.1)

= 31 33 34

(29.2)

= 32

(30,1)

Burnside group of exponent 5

= 32 34 3.

4

PLAN OF ATTACK:

FINITENESS

211

(30,2)

= 33

(31,1)

= 33 4 341

(32,2)

= 34

(33,1)

= 342

OF H

We s h a l l w r i t e G for B(5,2) t h e Burnside group with two g e n e r a t o r s a, b so G = < a,b > .

The r e s t r i c t e d group G = G/Gi3 , where

G13 i s t h e 13th term i n t h e lower c e n t r a l s e r i e s .

Here G i s t h e l a r g e s t

J

f i n i t e image of G. The elements a b a b g e n e r a t e G1 s i n c e i f a 3 OL 8, CL 82 r r W = a b 1 a b 2 ... a b e G1 then 04 + . . . + a = 0(5) and g _ a _$ 8 ot +a2 82 a 3 t + . . . + 3 = 0 ( 5 ) . If a, ± 0 then W = a 1 b 1 a 'b ' . b ' a 1 b a

r

3

...

a

, \ -a, - 3 , a b = a b a b W* where W* i s a s h o r t e r word i n G ' . I f at = 0 f 3 OL - 3 , - O L ) [ a 2 8, -OL - 8 . I " 1 J 2 W = b ! a 2 b a 2 W * = a 2 b ! a b Mw* where W* i s a s h o r t e r w o r d . Hence

r

by induction

We d e n o t e

these

elements

Table =

by

a"1 a . a

a" * b "

2

as Oi

Oi O2

06 O2

Ol

O3

Oi

O4

2

a b"

O4

a-'b

a b" 1

2 = a" 2 b " ' a b 2 2 2 2 = a~ b " a b 2

On

a" b

a8

= a" 2 b

a9

2

2

b"|o.b

u2 a D

O3

a6

(i,j = 1,...,4)

a b

a" ' b 2

as

a~1b"*Ja1b-J

ai,...,ai6-

3. 1

a"• ' b "

C7i O2

t h e 16 e l e m e n t s

a b"

2

a 2 b"'

a2 b " 1 a- 2 b

OSO41 O9 Oi

Os 06

Oi 0 02

Os On

Ol 1 O3

Os

Ol 2 04

Os

Os

Ol 3 Ol

09

Ol 0

Oio

=

a2 b " 2 a" 2 b 2

01 4 O2

O9

Oi 1

Ol 1

=

a2 b 2 a ,- 2 b" 2

ai 5 a3

09

Oi 2

012

=

a2 b ar'b-'

-1 ai 6 a4

-1 09

ai3

=

a b"1

014

=

a b"2

O\ 5

=

a b 2 a L~

b~

a b aL

b

Q

=

a"'b a"b 2

01

Oi3 O14

-1 02 -1 a3

-1 oi 3 at 5 -1

Ol 3 Ol 6

Ol3

generate

G'.

Hall § Sims:

Burnside group Qf exponent 5

212

Table 3.2 Automorphisms of G A : a -> a 2 , b -* b a

"*" ^

i

a

(01>0s, 0 i 3 , 0 9 ) (02 , 0 6 , 0 i 4 ,01 o ) (03 , 0 7 , 0 i s , 0 i i ) (04 , 0 8 , 0 i 6 , 0 1 2 )

i ^

y : a -> a , b -> b 2 0^

T a

i

•*•

(CJ^)P

(01

,02

,04

,03 ) (05

,06

,06

,07 ) (09 ,01 0 ,012

,01 1 ) (013

,014

,016

,01 5 )

: a -> b , b -*• a ~* ( - ° i ^ T

(.01 t O i

1

) ^ ! ,osl)(o3 ,o

(01 1 , 0 1 1 )

(012

,<JI s )

l 9

Cai 6

)(o4

t

oi3)(.o

6

,o6

) ( a 7 , o i l ) ( a 8 , 0i~4 )

)

The plan to prove that G is finite consists in finding a normal subgroup K which is Abelian, such that G/K = H is finite.

For if

K is of finite index, it is finitely generated and if Abelian is then finite. group.

From Table 2.2 commutators 11, 12, ..., 34 generate an Abelian We wish to find short words which lie in this group.

The

f

following 3 s have this property

Table 3.3 31 = 01020403

= 143173181223242262284302322334342

32

= 05 06 08 07

= 14217118320321122426i271282294323334343

3s = 09 01O012 0 H

= 14218220421322l242283294303312321331343

34 = 0 i 3 0 i 4 0 i 6 0 i 5

= 143171184202214222263273281293303314323333344

3s = 0105 013 09

= ll4162192204214234244253264273293303321334341

36

=02060i40io

= II1 19320121423224426227330l313321331344

37

= 0 3 07 0 i s 0 n

= 1111621932132332422642712913O1311322333341

3s = 0 4 0 8 0 1 6 0 1 2

= ll4161192202232251263274303323333341

We d e f i n e K = < 3i , 3 2 , 3 3 , 3 4 , 3 s , 3 6 , 3 7 , 3 s > G , H = G/K.

(3.4)

We easily find from Table 2.2 that K", the image of K in (T, is generated by commutators 11, 14, 15, 16, ..., 34 so that with H" = G/K", |H"| = 5 1 2

Theorem 3.1. Proof.

If H is finite* then |H| = 5 1 2 .

In K the following 22 3's and conjugates are independent:

3i ,3 2 ,3s ,34 ,3s ,3 6 ,3 7 , $ 8 , $f, $?, $f, 6?, 33a,34a,34b,35a,353,36a,36b,37a,3l)2 ,35a' . This was determined by computer, using Table 2.2.

Hall § Sims:

Burnside group of exponent 5

213

If H is finite, then K is finitely generated and so G/K' is finite.

From (3.5) |K/K'| > 5 2 2 .

Since |G/K'| = |H| . |K/K'| and G" is

the largest finite quotient of G, we have |H| < 5 1 2 . of H and so

12

> 5

|H|

But H is a quotient

.

Every fifth power can be expressed in terms of the 0's. The powers (a1b*J)s and conjugates are given in Table 3.6.

Table 3.6 (ab)S i

= 1

a

016 012 01107 -

1

-

0602

1

-

1

01 6 01 5 01 1 01 0 06

01 = 1

1

1

-

-.

05 01

1

-

1

040150140100905

1 a3 1

-1 - 1 08 04 03 014 013 09

.a4 1

-i -i -i 012 08 07 03 02 013

, = 1

012 04

05 = 1

(a2b)5

= 1

a

1

03 015 014 06

(3) ,A^ (4)

.a2 D a3 6

"i -i - 1 -i 01 2 01 1 03 02 01 4 01 3 05 - l i i = 04 0 1 6 0 1 5 0 7 06 0 9 1

=

,

= 1

r . (5)

(6)

1

(7) T =

•*•

,

o . (8)

(9)

—l —l —l 08 01 1 01 0 02 01 01 3 " 1

(a~2b)5

,

-, = 1

6

6

n

\.£)

=1

~i ~i ~i 016 08 07 010 09 01

a

(1) f

(10) (11)

11 a

08 016 015 03 02 010 09 = 1 -1 -1 -1 012 07 06 014 013 01 = 1

11

016 04

(13)

a2

03 01 1 01 0 05

= 1

(12)

3

11 a

08

11

04 012 01106

a4 s

(a " ' b ) = 16 a 16

a2 3

16 a

4

16

(15)

= 1

(16)

= 1

(17)

012 016 015 02

= 1

(18)

02 06

03 07

0105 05 09

= 1

06 010 09 013

(19)

= 1

(20)

01 S 01 1 09 05 -1 -1

08 04 -1

02 = 1 -1

(21)

2 01 5 01 3 09

01 2 08

06 02

0301301601201006

a4 2

(14)

= 1

a3 21 21

= 1

= 1

0801201101501401

04

21 a a

0109

05 013

04 08 07 01 1 01 0 014 01 3 -1 -1 -1

016 03

a

(ab2)5

21

07 015 014 02

07 03

2

5

0 1 0 1 6 014 010

(a b ) = a

01 1 07 01103

0 1 0 1 3 016 08

26a

015 07

05 012 010 02

05 01

01109 01

04 014

08 010

(24)

= 1 06

=

(25) 1

= 1 = 1

= 1

(22) (23)

= 1

04 016 014 06

03 015 013 05

= 1

= 1

(26) (27) (28) (29)

Hall § Sims:

Burnside group of exponent 5

26

a 7 a9 2

2

(a" b ) a

S

= 1

01204

a.2

- i - i

31

0 1 5 03

31

07

03 01109

(a

b

)

5

=

1

03 07

3

36 36

(31) (32)

02 010

06 014

05 09

= 1

=

=

(33) 1

(34)

1

012 016 014

(35) =

1

(36)

070110901301602 = 1 0 1 1 0 1 5 013 04 0206 = 1 -i -i -i 015 01 04 08 06 010 = 1 03 0 1 0 5 0 8 0 1 2 0 1 0 0 1 4 = 1

(37) (38)

-1

~*

(39) (40)

H1 = < 01 ,02 ,03 ,0s ,0« ,09 >.

Theorem 3.2. Proof.

08

~l

36 36

0 1 2 06

05 013 016 04

a a

(30)

-1

0109

31 2

= 1

07 015 013 01 04 012 010 = 1 -1 -1 -1 01105 08 016 014 02 = 1

31

- 1

02 014

214

Since H' = < 01,02,...,016 > it remains to express the remaining

a's in terms of 01,02,03,05,06,09. 016

= 0 9 05

06 02

From (19) in Table 3.6

03,

£(016 ) =

5.

Here £(x) is the length of x in terms of < 01 ,02 ,03 ,0s ,06 ,09 > as a free product of cyclic groups. From $ i = l and from 3s = 1 we have, respectively, -1 -1 -1 04 = 02 01 03 ,

5,(04) = 3;

-1 -1 -1 013 = 05 01 09 ,

£(013) = 3.

For the rest of this proof the left hand numbers indicate the relevant relation in Table 3.6. 15

012011 = 04J0i 3 0s * 06 , l(ot\aii) = 8

13

-1 010 011

1

0

8

0 1 4 = 01 3 a 5

&6 =

l

7

=

=

06 02

a

_i -1 05 0 1 6 04 03 a

l

a

1 6

a

a

l2alla3

1 2

a

i 0 = a i"4 C f 6" l a 2" 1

8

=

a

=

a

°2

=

l l

=

a

6

a

2

a

i 3a5

010905

= a.0^0^0,-1

0 7 0 t -;

1

01 2 = Cfl 1 ^7 ^6 a 2

4 °l 3 a 5

a

a

l 3a 5

^6r

A

10

^ i ^ l )

6 a 3 °2> & (°l 4 )

2"la3a6"la5ai"la4a5ai"3a6"la2"1 >

9as"la4lai3as"la6a3"la2a3la4>

37

a

06^2 ^3^4

=

=

K 0) =

a'1 aB 04-1 03 a^1 03 (o~\ o\ 2 )os = 1 so

From 4 and 8 combined a

-1 -1 -1 -1 -1 0 5 0 9 0S 06 02 03 04 0 3 , & ( 0 1 0 0 1 1 )

=

a

^(a8)

=

1 6

= 0 2 0 3 " 1 0 2 0 6 " 1 0 5 0 9 " 1 C J 1 3 09-1 , * ( 0 7 0 ; ; )

= 10

l a 16

= 090r3a9a5~la6a2"la3a2"la6a2"laia9as"la6a2"la3 > ^(^12)

=

1 8

14 1 6

=

16

Hall § Sims:

33

aj 5

= a 6 Qj 2 cr9 c^

Burnside group of exponent 5

215

a3 o^Oi1

U

6 U2

U

3 W2

U

6U2

u

l U9US

U

6U2

U

3 U4

u

o3 , J l ( a 1 5 )

l 3 U5

= 20

^5 > ~ C^i 1 3 "" 2 6

1

Hence H = < a1 ,a 2 ,a 3 ,a 5 ,a 6 ,a 9 >. Theorem 3 . 3 . 4

order 5 Proof.

Jrc H, < ot ,o2 ,o3 ,o4 > = < ot ,o2 ,o3 > is finite

or 5 . We begin with 10 relations

Bi = 1 aia 2 a 4 a 3 = 1 l

b

l

1

2

1 O2

4

(1)

o; a2o; o3a-l a4

b

7 b O

1

-

= 1

i Q3 O 2

C73

b

and is of

s

i O4 O2

i

_ - 2 _ -1 (J 4 CT3 CJ2 CT3 CJj -1

2

=

u C J = 3 4 2

i

l*

1

[ j j

Z

-, 1

ry|N

-1

/-rs

-1

Q O u o 4 1 4

(2)

b

*b

=

(7j C72

Bf = 1 o~l a^1 a4l o3l l

1

= 1

(1*)

l

o~2 a, a; o\o'4 a, = 1 -i O,

O 2 O^

i

-i O2 Oj

2 O-

-1

-1

-1

=

(2*)

-L

v."^

J

^4J

.

(D) v ^

A

T-

b

OA O< 4 1

Oil 0 * 4 3

.,

o>i On 4 2

OA 4

=

i.

/ T * >

1o *•

1 •'

Solving for a4 in the first 8 of these we have i i.

- l - i - i Q« Q4

=

=

Q<,

Q|

2

i

,*

Q-

C74

Z

O4

•— Oo

Oi

Oo

i

Q j O3

Q j CJ2

-

A

- i - i- i

1

1

-

**

C7|

1

2

From 1 and 3* we have cr2! a^1 a^1 = o2 o^1 o\o3l o2 or -1

°3

-2 0

2a3

=

a

2

a

-2

-1

i °2 °1 *

/x's

i.°J

From 1 and 1* we have o2* o^1 o3* = o3l o^1 o2* which gives —1 0" Q Q Q

2 0 0*o .

/* "7"\ I /J

Combining (6) and (7) gives

Here (6) and (8) show that < o\ ,02 > is normal and of index 5 in

From 1 and 2 o2l a^ 1 o3* = o\ o3l ox o2l ox o r — \ _, -1 CTo o . Q« Q | O-y

=

_~2 —1 - 1 Q< CJo Q | .

/-QN L^J

Hall $ Sims:

Burnside group of exponent 5

216

From 3 and 3* a2 a^1 a2 aj"1 a2 = a2aj"! aja^1 a2 giving

Multiplying (10) on the right by (9) gives

But from (7) this gives oxo2 - aj"1 a2 aj"2 a^1 aj"1 or

Now let us take the relation 1 = (baj)5 = (b" 4 a 1 b 4 )(b" 3 a 1 b 3 )(b" 2 a 1 b 2 )(b" 1 a 1 b)a 1

= a; 1 • (a; 1 a4 ) (a,"1 a 3 ) (aj"1 o2)ol .

(13)

Using 1 (13) becomes

From (7) o3olo2o3l

= o2ol . Using this and (6) gives

a^a^aj-'^aXXK1^! a

2aia2ai"la2arla2ai

=1*

or

(I 6 )

= 1-

From (12) a 2 aja 2 = aj"1a2aj"1 and substituting in (16) o^olof-o^alo^o^

=1

(17)

which becomes ajajoja^aaa" 1

= 1.

(18)

It i s convenient to write relations on < ai ,02 > in terms of i t s derived group, generated by yt , . . . , y 1 6 . Equations (19) Yi = ^la^lata3

y9

Y2 = Oi1*?*^

Y10 = ^ a ; 2 a j " 2 a 2 2

Y3 = o^ala.o;2

ytl = o\a\a;2 a2"2

Y4 = a" 1 (7,0, a ; 1

Y

Vs

= a

i" 2 a 2" l a i a 2

Y6 = a ^ a ^ a ' a

2

Y 18

Y14

=

o2la2to^2o2

= a 2 a a j " 2 a'x

Hall § Sims:

Burnside group of exponent 5

217

The mapping a •+ ai , b -* a2 is an endomorphism of G = < a,b > onto < oi ,o2 > and maps a^ •* y^, i = 1,...,16.

In particular replacing a's by

y's in the relations of Table 3.6 gives valid relations.

Also (12) and

!

Thus from (18) a~ o\o\ = a"1 at a"2 .

(18) give relations on the y's.

Multiplying on the left by af * and on the right by a~2 gives aj"2a2a2a22 = a"1 a^1 al a2 or y7 = yt . Y7 = Yt , Y9 = Y8> Y M

In the same way we readily find (20)

= Y 4 , Y 13 = Y6 , Yt s = Y2 , Yj« = Yx 0 •

Conjugating y7 = Yi > Yi i = Y4 and yt 5 = y2 by a2 gives T

Y

Y

7

Y

l '

ii

Y

15

Y

Y

S

4 '

Y

2 '

Y

T

8 Y

9

Y

l Y

i2

13Y16

"

Y

2 '

Y

6

Y

2

Y

i

'

Y

io

Y

2

Yt»

l

Y

3 '

Y

Y

14

2

Y

3 '

Y

Y

7

Y

M

Y

Y

4 '

Y

5

3

Y3

9

l 5Y l 3

Y

Y

4 »

Y

8

Y

6

J

4

Y2 > Y 1 2 Y 1 0 Y

3

'

Y

l 6Y l 4

=

Y4

Y3 »

Y

Y

4

^ ^

l *

From these we find Y

s

Y6 Y7 Y8

=

Y Y

I 3~1Y4

Y

Y3 Y2

9 = Y4Y3-1Ya

Yj 0

= Yi

Yl

Y2

1

Y i 2 = Y 4 Y 3 " Y 2 Yi"

Y3 Y 2 Y4

1

(22) 1

1

Y , « = Y 2 Y 3 " Yi" Y 3

T3

>16

>14

T

4

T i

" Y i Y 2 * Y t Y 2 Y3 Y 4 Y j Y 2 - Y4 Y3 * Y 3 Y t Y3 Y 2 Y4 Y 2 - Y4 Y 2 •

In a d d i t i o n the r e l a t i o n s yl Y2Y3"! = ylly2> Conjugating

Y4 Y 2

Y i s = Y2 1

= Y3' Y2Y4

Y 3 Yj

Yj 4

Y n = Y4

14

4

Y 1 3 = Y2Y," 1

3

= y6 and y9 = y8 give

Y4Y3"1Y2 = Y3"1Y2Y4-

(23)

(23) by powers of a2 shows t h a t < y t , y 2 , y 3 ,y 4 > i s Abelian

and so < a, ,a 2 > ' i s Abelian by (22) and we have Ys

= Y1Y3"1Y4

Y9 = Y 2 Y 3 " 1 Y 4

Yt 3

= YjYa" 1 ~ Y 2 Y4

Y6 "" Y 2 Y 3

Yi o ~ Yi Y 2

Yi 4

Y7

Y

Y i s = Y2

= Yi

M

= Y4

(24)

We find using the definitions of yx ,y2,y3 ,y4 in (19) and relations (6),(7),(8), (10) that

—1 U

3

—1 Y

2 3

Y

lY 4

W

3

—1 '4 U 3

—1 T

3'4

/*otr"\ v

J

Hall £ Sims:

Burnside group of exponent 5

218

A further relation i s (ba^ba)5

= ^lo3a2l°ia4a31°2ai1

=

!•

C26)

= 1 , or

(27)

If we take a4 = o2l o^1 o^1 t h i s becomes o a a l o 2 o 3 a ~ l o x o l l a " 1 o ^ 1 a,""1 o 2 o ~ l

Now (6) and (8) can be put in the form o~lola3

= a1Y3"1Y, *

cr"1 a2 a3 = o2y~\ = a^j" 1 •

Now (28) becomes a^1 Qj a^1 aj"1 ag"1 (a 2 Y2 * Yi"1 Y3 ^2 ) a 3 (29)

1

1

1

1

1

l

c?; (a 1 a 2 " a 1 " a 2 )Y" Yr Y 4 Y 4 Y 2 Y 4 " a 2 Y 2 " 1

1

1

This becomes a^ yj" Y2 Y3" Y4 a 2 y^ y^Yj1

1

1

=

1

anci

u s i n g (25) and

!

= 1 or a2" y1 3 Y^ 1 YI"1 Y2 Y4 ^2 Y2"1 = 1

= 1 or o~ yl2o2y~l

= 1 or

= 1, or

(30)

Y2"2Y3Y4"1 = 1. C o n j u g a t i n g by a 3

(29)

(31)

gives

Y1"2Y2Y3"1Y42 = 1.

(32)

Here (31) and (32) g i v e y 3 = y 2 y 2 " 2 , y4 = y 2 y 2 . Now (24) t a k e s t h e s i m p l e r Y3 Y4

=

Y2Y2"2 2

= Y Y2

Y6

Yi

Y7

= Y,

Y2

form Y8

Y 1 3 = Yi" 2 Y 2 " 2

= Y2' 1

Y 1 4 = Yj'2

Y9

= Yj"

Yio

=

Yx Y 2

Y i s = Y2

Yin i Y

9

Yi Y 2

T16

Yi

Y2

Yia =

This now shows that the derived group of < at ,o2 > i s of order 2

at most 5

so that < at ,02 > i s of order at most 54 and < 01 9Oi ,03 > i s

of order at most 5 s , and our thereom is proved. Defining relations for the group < ai,a 2 > as given here are a* = 1, a* = 1, a\ o~2 o\ a2 at a2 = 1,

Hall § Sims:

Burnside group of exponent 5

219

The last three relations are (12), (18) and y 2 y 9 = 1 of (30). On the 125 cosets of < ox > in < at ,o2 > we have by coset enumeration the following permutations: at = (1) (2,4,10,13,5) (3,7,18,21,8)(6,16,28,11,17)(9,24,23,35,25) (12,22,48,59,30)(14,34,68,54,26)(15,36,44,19,37)(20,46,88,89,47) (27,55,98,99,56)(29,57,39,75,58)(31,62,73,51,63)(32,64,106,107,65) (33,66,80,81,42)(38,67,52,61,74) (40,77,45,87,78)(41,72,92,49,79) (43,82,114,105,83) (50,84,94,53,93)(60,101,111,70,102)(69,95,108,85,97) (71,112,76,90,113) (86,109,122,91,117)(96,110,103,118,124) (100,116,121,119,125)(104)(115)(120)(123) o2 = (1,2,6,9,3)(4,11,29,31,12)(5,14,35,38,15)(7,19,45,28,20) (8,22,49,50,23)(10,26,46,21,27) (13,32,58,67,33)(16,39,76,55,40) (17,24,51,80,41)(18,42,77,84,43)(25,52,56,95,53)(30,60,65,103,61) (34,47,90,59,69) (36,70,100,57,71)(37,72,107,114,73)(44,85,93,118,86) (48,78,116,82,91)(54,96,74,115,97)(62,104,102,64,75) (63,98,101,125,105) (66,108,121,89,109)(68,110,112,81,111) (79,106,124,122,99)(83,117,123,92,94)(87,119,88,113,120). In the restricted group H, < 01 ,02 > is of order 53 and 1

Y1Y2"

~ 1-

^ u tw e haven't been able to prove this. By coset enumeration H = < a,b|(a1b*))s = $1 = 3 2 = .. = 8s = 1 >

has 3125 = 5s cosets on the subgroup < 01 ,02 ,03 ,05 >. Hence to prove the finiteness of H all that is now needed is to prove that [< CFI ,o2 ,03 ,0s > : < Oi ,o2 ,03 >] is finite.

In FT this index is 53 . Thus In H this is 56 .

we must prove [H' : < o\ ,a2 ,a3 >] finite.

Unfortunately

efforts in this direction have not been successful. Let us adjoin to K further elements all in class 6 or higher, to form a larger group Ki <Si

= Qi O4 O2 O3 Os Og 06 On

69

= 02 O3 04 Oi 06 CT7

62

~

610

=

OsG%O$OnO\3O\6O\4O\s

63

= Ci 3 O\ 6 O\ 4 O"i s O9 O\ 2 Ol 0 O\ 1

64

= O9 O\ 2 O\ 0 CJi 1 O\ O4 O2 O3

6s

= CJl CJl 3 CJ5 CJ9 CJ2 CJl 4 06 CJl 0

06 OTOSOS

On

OB OS

Ol s Ol6

O13

6 1 1 = 0 1 4 O\ 5 CJl 6 CTl 3 O\ 0 CJi 1 O\ 2 O9 61 2 61 3

= CJi 0 Oi 1 O\ 2 CJ9 CJ2 CJ3 CJ4 CJi =

CJS 09 01 3 01 06 01 0 01 4 02

66

= 02 01 4 06 CJl 0 04 01 6 08 01 2

6 1 4 = 06 01 0 01 4 02 08 01 2 01 6 CJ4

67

= 04 01 6 08 01 2 03 01 5 07 01 1

61 S

= 08 01 2 01 6 04 07 01 1 01 S 03

68

= 03 01 5 07 01 1 01 01 3 05 CJ9

61 6

= 07 01 1 CJl 5 03 05 09 01 3 01 •

Hall § Sims:

Bumside group of exponent 5

220

Let us define a new subgroup Ki in G by = <

Ki

3i y • • • i 38 , <Si , . . . , 6i 6 > •

Then G/Ki i s a f i n i t e group Hi . Let Hi = < a , b | ( a V ) 5

Theorem 3.4.

6i = . . . = 6i6 = 1 >. Proof.

= (a r b S a t b U ) 5 = 1, 3i = . .. = 3s = 1,

Then Hi is finite

and of order 5 1 0 .

By c o s e t enumeration i t i s found t h a t

[H^ : < a% ,a 2 ,a 3 > ] = 6 2 5 .

Now Hi i s a homomorphic image o f H so t h a t Theorem 3 . 3 a p p l i e s and t h e o r d e r o f < at , a 2 ,03 > i s 5 4 o r 5 s . 2

4

5 -5 -5

s

= 5

11

.

Hence t h e order o f Hi i s at most

Thus Hi i s a f a c t o r group o f G and [G : Ki ] = |Hi | .

Ki c o n t a i n s a l l commutators 1 1 , . . . , 3 4 and i s A b e l i a n . group Kj /Kj i s o f o r d e r a t l e a s t 5 2 4 |G/K;| i s f i n i t e and |G/K^ | < 5 3 4 . K

K

l i/ i I

>

524

i z

f o l l o w s

t h e commutator t a b l e

Acknowledgement.

t h a t

H

=

Hence t h e

and as Ki i s f i n i t e l y

generated

Thus |G/Kj || Kt /K; | < 5 3 4 G K

has

i / i order a t most 5 |Hi | > 5 1 0 and s o |Hi | = 511( 0

But

factor

10

and as .

But from

The work of the first author was supported in part by

NSF grant No. MCS 7821599 and that of the second author by NSF grant No. MCS 7802640.

REFERENCES

1. M. Hall Jr., Theory of groups, 2nd ed., Chelsea, New York (1976). 2. G. Havas, G.E. Wall § J.W. Wamsley, The two generator restricted Bumside group of exponent five, Bull. Austral. Math. Soc. 10_ (1974) , 459-470. 3. G. Higman, On finite groups of exponent five, Proa. Camb. Vhil. Soc. S2_ (1956) , 381-390. 4. A.I. Kostrikin, Solution of a weakened problem of Bumside for exponent 5, IzV. Akad. Nauk. SSSR Ser. Mat. 19_ (1955), 233-234.

221 THE ORIENTABILITY OF SUBGROUPS OF PLANE GROUPS A.H.M. Hoare University of Birmingham, Birmingham, B15 2TT, England D. Singerman University of Southampton, Southampton, S09 5NH, England

0. INTRODUCTION Let X be either the hyperbolic plane, the Euclidean plane or the sphere with its associated metric and let r be a group of isometries of X. r is properly discontinuous if for each compact set K C x, {y € r | y K n K / H is finite. It then follows that X/r is a surface, possibly with boundary, [1, p.52]. For example, if X is the Euclidean plane then the properly discontinuous groups r for which X/r is compact are the 17 crystallographic groups and X/r is either a torus, a sphere, a disc, an annulus, a Klein bottle, a Mobius band or a projective plane, [4]. If r is the hyperbolic plane we obtain infinitely many examples including the non-Euclidean crystallographic groups described in [9]. In the study of these groups the topological characteristics of the quotient surface X/r and the nature of the singularities of the natural projection p : X > X/r are often invariants of the group r. If A is a subgroup of r and if we know either the permutation representation of r on the A-cosets, or equivalently a Schreier coset graph for A in r, then these characteristics are computable for A, [7]. In particular the orientability of X/A is given by a canonical presentation of A which can be obtained from a canonical presentation of r and the action of r on the A-cosets. In this note we use a combination of geometrical and algebraic methods to determine the orientability of X/A. (A purely combinatorial proof of criterion (iv) in Theorem 2, for orientability was given in [8], using coinitial graphs.) In many cases the results here could be applied to higher dimensional spaces. However, it is not always the case that the quotient spaces will be manifolds and so it is not obvious what interpretation to give to the results. If, in the higher dimensional cases, the quotient spaces are manifolds the results will still be valid.

Hoare $ Singerman:

Subgroups of plane groups

222

1. GEOMETRIC PRELIMINARIES Let X and r be as above. By a polygon in X we mean a closed connected set with non-empty interior whose boundary is a union of geodesic segments. These geodesic segments are called sides, A fundamental polygon F for r is a polygon, possibly with infinitely many sides, such that:

F2.

F n Y F = 0 for all 1 t y € r where F denotes the interior of F.

F3. If b is a side of F there is a unique side b of F, possibly equal to b, and a unique element $ £ r such that 3b = b and $F is a polygon adjacent to F along b. We say that 3 pairs b with b. F4. F is locally finite; i.e. a compact set K c x intersects only finitely many images of F. A fundamental polygon exists for any properly discontinuous group r. An example is supplied by the Dirichlet region. This is defined as {x € X | d(x,p) < d(yx,p), for all y G r} where d is the metric on X and p is not fixed by any element of r except the identity. (See the articles by A.F. Beardon and L. Greenberg in [6]. There, only groups containing orientation preserving elements acting on the hyperbolic plane are considered but the proofs extend to the case where r contains orientation reversing elements as well, [10], and apply equally when X is the Euclidean plane or the sphere.) Remark. There are two ways in which we can have b = b in F3. Either 3 is a rotation about the mid-point of b through 180 degrees (and is orientation-preserving), or 3 is a reflection fixing the side b pointwise (and is orientation-reversing). In either case we will regard 3 as pairing b with itself. In the latter case only we will say that 3 fixes b. Proposition 1. The elements 3 which pair the sides of F generate r. Proof. Proposition 1.2.2 in Chapter 7 of [6]. Thus associated with a fundamental polygon F for r we have a generating set $ = (3i,32,...} consisting of those elements which pair the sides of F. Note that if 3 G $ then 3"1 e $, where 3 may equal 3" 1 ,

Hoare § Singerman: Subgroups of plane groups

22 3

and that 1 f $. Proposition 2. If B pairs b with b and y,S e r, then the following are equivalent: (i) yF £ S adjacent to 6F afon^ yb; (ii) Y36"1 = 1; (iii) yh = 6b and Y * 6. Proof. If (i) holds then F is adjacent to y~l &F along b and so by F3 Y"!6 = B. If (ii) holds then y t 6, since 1 £ $, and 5b = YBb = yb. If (iii) holds then yb = 6b C yF n 6F, i.e. Y F is adjacent to 6F along Y ^ . We now note a simple method for determining the orientability of X/r. Proposition 3. X/r is orientdble if and only if the only orientation reversing elements of $ are reflections. Proof. Using F4 it follows from [2] that X/r is homeomorphic to the space obtained by identifying the corresponding points of paired sides of F. (The sides of F fixed by reflections give the boundary of X/r.) If B € $ is an orientation-reversing element which is not a reflection then, by the Remark it will pair distinct sides b,b of F and so there is a point x relatively interior in b such that Bx e b, and x ^ Bx. Join x and Bx by a simple path lying inside F. If we thicken this path slightly we get a strip which maps down to a Mobius band inside X/r. Thus X/r contains an embedded Mobius band and hence is non-orientable. Conversely, if the only elements of $ which reverse orientation are reflections, then only orientation preserving elements of r identify distinct points on the boundary of F. Hence the orientation of F induces an orientation on F/r a X/r. 2. SCHREIER GENERATORS If <|> is a set of generators for r and if A is a subgroup of r, then a right Schreier transversal E is a set of words in $ such that (i) Every initial segment of a word in Z is also in I. (ii) a i > ha is a 1 - 1 correspondence from E onto the cosets of A in r. (We use a to denote the word in * or the element of r it represents, depending on the context.) For each a € E and B e $ there is a unique T e E such that AT = AaB. The Sckreier generators of A are all elements O&T'1, for

Hoare § Singerman:

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a,T £ E, 3 e $ with AT = Aa$. We now consider Ff =

^ aF and show that F1 is a connected afcI polygon satisfying Fl, F2, F3 and F4 for the subgroup A. If T = a.3 as words in $ then, from Proposition 3, aF and T F are adjacent along ab. So by induction on the lengths of the words a in E, Ff is connected. The boundary of F1 consists of all sides ab of aF,a ^ z, which have no TF,T 6 z, adjacent along them. Thus F1 is a connected polygon with boundary consisting of all such ab. We take these ab to be the sides of Ff. Now AF1 = AZF = TF = X so Fl holds. If x e Ff n XF f , 1 t X G A, then x e aF n XTF for some a,x e z, with a ^ XT. Therefore a"1 XT = $ € $ and x e ab. By Proposition 2, ab is on the boundary of F 1 . Therefore F2 holds for Ff and A. F4 clearly holds for Ff since it hblds for F. The property F3, for F' and A, is contained in the following theorem. Theorem 1. The Schreier generator aftx"1 pairs the side xb with the side ab, possibly equal to T 6 , if and only if it is not the identity. Moreover it fixes ab pointwise if and only if a = T and & is a reflection. Proof. If TF, T e z, is adjacent to aF along ab then by Proposition 2 agx"1 = 1 and, a fortiori, Aa3 = A T . Thus ab is a side of F1 if and only if the Schreier generator a3T~! is not the identity, in which case T S is the, not necessarily distinct, corresponding side of F1 and a3T~! is the unique element of A pairing T 6 with ab. If ab is a side fixed pointwise by the Schreier generator 1 a3x"" then a3 = TB and a3T~* / 1. By Proposition 2 this implies a = T and 3 fixes b = b pointwise, i.e. by the Remark in §1, 3 is a reflection. Conversely if 3 e $ fixes b and Aa3 = Aa then the Schreier generator a3a~! is not the identity and ab is a side of Ff fixed pointwise by a3a~ 1 . 3. THE MAIN THEOREM If r is a group with generators $ and if A is a subgroup, then the Sohreier ooset graph ?C(r,A,$) is the graph with vertices the cosets of A in r and labelled directed edges at each vertex for each 3 G $ such that 3 : Aa i > Aa3. If 3 is a reflection and Aa3 = Aa, then the directed edge 3 : Aa i > Aa3 = Aa is a reflection loop. Let ft be the Schreier graph with the reflection loops deleted. Each path in M corresponds to a word in $ and hence to an element of r. A path is

Hoare § Singerman:

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22 5

positive (negative) if it corresponds to an orientation preserving (orientation reversing) element of r. Theorem 2. The following are equivalent: (i) X/A is orientable; (ii) the only orientation reversing Schreier generators are conjugates of reflections in $; (iii) all circuits in W are positive; (iv) the cosets of A in Y divide into two disjoint classes such that in the action 3 : Ay • > Ay3 orientation preserving elements of $ fix the classes and (apart from reflections fixing cosets) orientation reversing elements interchange the classes. Proof. F1 and A satisfy the conditions for Proposition 3. The equivalence of (i) and (ii) follows from Theorem 1 and the Remark in §1. If (ii) holds then all circuits with base A in W corresponding to Schreier generators are positive. All other circuits correspond to a conjugate of a product of such generators, so (iii) holds. Suppose (iii) holds. Divide the cosets into classes according to whether a path in W from A to the coset corresponds to an orientation preserving or an orientation reversing element. By (iii) this is independent of the choice of path in ffi. Clearly these classes satisfy (iv). Finally suppose (iv) holds. Let E be a Schreier system of coset representatives. By (iv) using induction on the length of o £ Z, Aa belongs to the same class as A if and only if a is an orientation preserving element of r. (If a e E and 3 fixes the coset Aa, then a.3 is not in E.) If 3 e $ is orientation preserving and Aa3 = AT then by (iv) Aa and AT belong to the same class, whereas if 3 is orientation reversing they belong to different classes unless a = T and 3 is a reflection. Thus every Schreier generator a3T~* is orientation preserving unless a = T and 3 is a reflection, i.e. (ii) holds. As special cases of this theorem we have the following three corollaries. The first two appear in [3] and [5] in the case of an N.E.C. group with compact quotient space. The third was given by Maclachlan (private communication) in the case of an N.E.C. group generated by reflections. Corollary 1. If X/r is non-orientable and |r : A| is odd then X/A is non-orientable. Proof, r has an orientation reversing generator 3 e * which is not a

Hoare § Singerman:

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226

reflection. If X/A is orientable then by (iv) of Theorem 2 the cosets divide into two disjoint classes which are interchanged by $, so the number of cosets must be even. Corollary 2. If X/r is orientable and A a normal subgroup of odd index then X/A is orientable. Proof. Since X/r is orientable, the only orientation reversing elements of $ are reflections. Since the number of cosets is odd each reflection must fix at least one coset, and so must fix all of them since A is normal. Therefore putting all the cosets in one class, with the other class empty, we have that X/A is orientable. (Alternatively all edges in 5C correspond to orientation preserving elements of $, so all circuits are positive.) Corollary 3. If all elements $ are orientation reversing then X/A is orientable if and only if M is bipartite. Proof. Every edge of ^C corresponds to an orientation reversing element of $, so all circuits are positive if and only if they have even length, which is a necessary and sufficient condition for W to be bipartite. Examples. Let r = < a,c | c2 = 1, ca2 = a2 c >. Then r is one of the 17 plane Euclidean crystallographic groups (c_ in in the notation of [4], p.44). It has fundamental region

with c a reflection in AB and a an orientation reversing non-reflection taking BC to CA and with $ = {c,a.,SL"1 }. Consider the following permutation representations of r and let A = Stab (1). (i) ci > (12) (34) a i > (1234) A has graph ft

Hoare § Singerman: Subgroups of plane groups

22 7

with the convention that each edge in this diagram represents two edges, a and a"1, or c and c"1 = c, of *C. The coset classes {1,3} and {2,4} satisfy condition (iv) of Theorem 2, or alternatively the graph W is bipartite, so X/A is orientable. (ii) c« > (13) (2) (4) a I > (1234) A has graph #

with the two reflection loops deleted and with the same convention as above. K has a cycle

which is negative so X/A is non-orientable. REFERENCES 1. N. Ailing $ N. Greenleaf, Foundations of the theory of Klein surfaces* Lecture Notes in Mathematics, Vol.219, Springer, Berlin (1971). 2. A.F. Beardon, Fundamental domains for Kleinian groups, Ann. Math. Studies 79_ (1974), 31-41. 3. E. Bujalance, Math. Z.* to appear. 4. H.S.M. Coxeter $ W.O.J. Moser, Generators and relations for discrete groups* 3rd ed., Springer-Verlag, Berlin (1972). 5. W. Hall, Automorphisms and coverings of Klein surfaces* Ph.D. thesis, Southampton (1978). 6. W.J. Harvey (ed.), Discrete groups and automorphic functions* Academic Press, New York, London (1977). 7. A.H.M. Hoare, A. Karrass $ D. Solitar, Subgroups of N.E.C. groups, Comm. Pure Appl. Maths. 2^ (1973), 731-744. 8. A.H.M. Hoare, Lectures on N.E.C. groups* York University, Toronto (1974), unpublished. 9. A.M. Macbeath, The classification of non-Euclidean plane crystallographic groups, Canad. J. Math. L9 (1967), 1192-1205. 10. A.M. Macbeath § A.H.M. Hoare, Groups of hyperbolic crystallography, Math. Proc. Comb. Phil. Soc. 7£ (1976), 235-249. 11. W. Magnus, Non-Euclidean tesselations and their groups* Academic Press, New York London (1974).

228 ON GROUPS WITH UNBOUNDED NON-ARCHIMEDEAN ELEMENTS A.H.M. Hoare University of Birmingham, Birmingham, B15 2TT, England D.L. Wilkens University of Birmingham, Birmingham, B15 2TT, England

INTRODUCTION Let I be a length function on a group G such that N, the set of non-Archimedean elements, is a proper subset of G. Then Theorem 5.3 of [4] shows that the lengths of elements of N are bounded if and only if &(ax) = £(x) for all a in N, x in G\N, in which case N is a subgroup of G and £ is an extension of a non-Archimedean length function on N by an Archimedean length function on G/N. Among other results, this allows the structure of any length function on an abelian group to be determined. This leaves the question of whether, if N is a proper subgroup of G, the lengths of elements of N are necessarily bounded. In this paper we show that the answer is no. Theorem 1 of §1 gives a relation between £(ax), &(a) and i(x), for a in N, x in G\ N, whenever N is a proper subgroup of G whose elements have unbounded lengths. This suggests that it may be possible to express such a length function in terms of a non-Archimedean length function lx = JlL on N and an Archimedean length function i2 on G/N. In §2 we give an example of an HNN-group G with a length function I, such that N is a proper subgroup with unbounded lengths. 1. UNBOUNDED LENGTHS FOR N A length function on a group G is a function £ : G > ]R such that for all x, y and z in G: Al\ £(1) = 0, A2. ilCx'1) = jt(x), A4. d(x,y) < d(x,z) implies d(x,y) = d(y,z), where 2d(x,y) = £(x) + It follows immediately that 0 < d(x,y) = d(y,x) < A(x), A(y) .

(1)

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229

Three real numbers a, b and c are said to form an isosceles triple if two of them are equal with the third no greater.

This is the

opposite of the definition in [1]. Axiom A4 is thus equivalent to saying that -d(x,y), -d(y,z), -d(z,x) is an isosceles triple.

We note the

following.

Property I.

If r, s and t are any real numbers then max{r,s}, max{s,t}, max{t,r}

is an isosceles triple.

Property II.

If a., b., c. is an isosceles triple for all i in I, then

so is sup{ai>, sup{bi>, sup{c.}. Property I is immediate and Property II is most easily seen by observing that a, b, c is isosceles if and only if whenever two are less than (or not greater than) a real number m, the so is the third. An element x in G is Archimedean if £(x2) > £(x), i.e. if 1

2d(x,x~ ) < £(x), and is non-Archimedean otherwise.

Let N denote the set

of non-Archimedean elements of G, then a length function is Archimedean if N = {1} and is non-Archimedean if N = G. The following result is proved by Lyndon in Lemma 6.1 of [2]. Only Axioms A2 and A4 are used in the proof; not Al - A5 as stated in Lemma 6.1.

Lemma 1.

If xy ,x2,... ,x

are elements of G such that

d(x i _ 1 ,x^ 1 ) + d(xi,x"1+1) < Jl(xi), for i = 2,3,...,n-l, then r

r

£(xix2 ... x ) = I A(x.) - 2 I d(x .,x?) r

i=l

X

i=2

1

"i

X

for r = l,2,...,n. We use Lemma 1 to give a proof of the following result which is somewhat simpler than that given in Proposition 5 of [3] and Proposition 3.3 of [4].

Lemma 2.

If xyz = 1, where x, y and z are in N, then &(x), £(y),

is an isosceles triple.

Hoare § Wilkens: Proof.

Unbounded non-Archimedean elements

230

Suppose not and suppose without loss of generality that

A(z) > A(x), Jt(y).

Then 2d(x,y"1) = A(x) + l(y) - A(z) < £(x), A(y) .

Since x and y are in N 2d(x~l ,x) > &(x) and 2d(y~1 ,y) > fc(y) . Therefore by A4 applied twice we have 2d(x,y" 1 ) = 2d(x" 1 ,y" 1 ) = 2d(y,x"1) < *(x), A(y) . Thus d f x ^ " 1 ) + d ^ x " 1 ) < A(y) and dfr.x" 1 ) + d ^ y " 1 ) < Applying Lemma 1 we get £(z a ) = *(xyxy) = 2£(x) + 2l(y) - 2(d(x,y"1) + dfr.x"1 ) + d(x,y-f)) > £(x) + £(y) - 2d(x,y" 1 ) = A(z), contradicting the hypothesis that z is in N. Theorem 1.

Let G /zaue a length function I such that N -is a proper sub-

group whose elements have unbounded lengths* G\ N.

Let x be any element of

Then either

A (ax), A(x), A (a) + &(x 2 ) - £(x) and U a x " 1 ) , A(x), A (a) + A(x) - il(x2) are isosceles triples for all a in N, or t/ze same foZas w^t/z x replaced by x"1 . Proof.

Since the lengths of elements of N are unbounded we can choose b

in N such that A(b) > 2£(x).

Moreover, interchanging x and x"1 if

necessary, we may assume that 2d(x -1 ,b) < 2d(x,b).

By A4 this implies

2d(x"1 ,b) < 2d(x"! ,x) < £(x),

since x f. N,

<

by the choice of b,

fc(b),

< 2d(b,b~ 1 ), since b e N. So by A4 again 2d(b" 1 ,x" 1 ) = 2d(x" ! ,b) < £(x) < Jt(b),

(2)

from above. Thus we may apply Lemma 1 to x"1 b"1 x giving = 2£(x) + i(b) - 2d(x"1 ,b) - 2d(b" 1 ,x" 1 ) , by (2). Now N is a subgroup of G by hypothesis and it is normal by Proposition 3 of [3]. Hence applying (3) and Lemma 2 to the elements x ^ b x b " 1 , b and x" ! b" l x in N we have £(x" l b" 1 x) = £(x" 1 bxb" 1 ), that is

(3)

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231

2d(x"1bx,b) = A(x"!bx) + Jt(b) - ilCx'^xb"1) = £(b). Moreover 2d(b,x) < 2£(x) by (1), giving 2d(b,x) < £(b) by the choice of b. So by A4 2d(b,x) = 2d(x'!bx,x) = &(x~lbx) 2d(x"1,b"1) = 2A(x) - 2d(b,x" 1 ), by (3), 1

> 2d(b,x" ), by (2). Hence by A4 2d(b,x"1) = 2d(x,x"1) = 2A(x) - £(x 2 ), and moreover, from above, 2d(b,x) = 2A(x) - 2d(b,x"1) = £(x 2 ). Now take any a in N.

Since the lengths of the elements of N

are unbounded we can choose c in N so that £(c) > £(a), 2£(x). 1

b" ,bc~

!

Then

and c are in N and 2d(b,c) = Z(b) + £(c) - Jtfbc"1) > 2A(x), by Lemma 2, > 2d(b,x" ! ), 2d(b,x), from above.

So by A4 2d(c,x~1) = 2d(b,x"1) = 2A(x) - £(x2) and 2d(c,x) = 2d(b,x) = £(x 2 ). By Lemma 2 Jlfa""1), Afac" 1 ), Jl(c) is isosceles, and by the choice of c £(c) > £(a). Therefore Jlfac"1) = A(c) and so 2d(a,c) = A (a) + A(c) By A4 the triples -2d(a,x" 1 ), -2d(a,c), -2d(c,x"1) and -2d(a,x), -2d(a,c), -2d(c,x) are isosceles.

Substituting from above and adding I(a) + £(x)

to every term gives that the triples £(ax), &(x), £(a) + Jl(x2) - Jl(x)

and

are isosceles for all a in N.

2.

AN EXM4PLE Let G be the group given by generators u and g. , for i: in 7L ,

and relations u"1 g.u = g-+1«

Let N be the subgroup generated by the g^

for all i e 7L . The subgroup N is then free with basis {g^ : i e 7L }, and G is an HNN extension of N with single stable letter u.

It follows

that each element of G can be expressed uniquely in the form au some a in N and some r in 7L . ei

£2

ek

If a = g. g. ...g. , e. = ± 1 , in reduced form, define 1 \ X2 k J

for

Hoare § Wilkens:

Unbounded non-Archimedean elements

m(a) = 2

232

k

and put m(l) = -°°. For all a in N and r in 2 follows that a

define a

is in N and that (a )

= u" r au r .

1

= (a" ) , (a )

= a

It immediately +

,

(ab) r = a r b r and l r = 1.

If a is in N then m(a) = mfa" 1 ) = m(a 2 ) and m(a ) = m(a) + 2r.

Lemma 3.

If a, b and c are in N with abc = 1, then the triple m(a), m(b), m(c) is isosceles. Proof.

If a / 1 then since N is the free group on the generators g., the

maximum subscript for generators appearing in the reduced form for a will also be the maximum subscript for those appearing in the reduced form for a2 and for a"1 , and so m(a 2 ) = m(a) = mfa" 1 ).

If a = g. g. ...g. , ii i 2

ik

in reduced form, then, by repeated application of the relations e

a

r

i

= sg.

e

*

g.

il +r6i2 +r

e

k

...g.

\+T

in reduced form.

Hence

m(a r ) = 2 max{i 1 +r,i2+r,.. .,ik+r} = 2 max {ix ,i2 ,.. - j i ^ + 2r = m(a) + 2r. If a = 1 then m(a) = mfa" 1 ) = m(a 2 ) = m(a ) = -°°. If abc = 1 then the generators with maximum subscript in the reduced forms of a, b and c must appear in at least two of them.

Thus

m(a), m(b), m(c) is an isosceles triple.

Define I : G

> TL by Jl(x) = max{m(a)+r,r,-r}, where

r

x = au , a € N.

Theorem 2.

If % is defined as above then it is a length function on G

such that the subgroup N has unbounded lengths., and consists of all the non-Archimedean elements of G. Before proving Theorem 2 we give some immediate consequences of the definition and prove a preparatory lemma. By Lemma 3 if x = au , a G N, then SL{x) = max{m(a)+r,r,-r} = max{m(a)+2r,2r,0} - r = max{m(ar) ,2r,0} - r. If y = bu

then xy"1 = au u

m(a b" ) - 2r, we have

b

= ab

u

and, since m(ab

) =

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233

l

) = max{m(ab~1 ) + r - s , r - s , s - r } = max{m(a b " 1 ) - r - s , r - s , s - r } max{m(a r b^),2r,2s} - (r+s) .

Hence 2d(x,y) max{m(a ),2r,0} + max{m(b ),2s,0} - max{m(a b ' ^ ^ r ^ s } . In the following lemma we will simplify this expression in various cases using the fact that &(a) + 2r = max{m(a),0} + 2r = max{m(a ),2r} and hence max{m(a ),2r,0} = max-U(a) + 2r,0}.

Lemma 4.

Let x = au , y = bu , with a, b in N, and r, s in 7L .

(i)

If 0 > £(a) + 2r > £(b) + 2s then 2d(x,y) = -maxCmCa^" 1 ) ,2r,2s}.

(ii)

If £(a) + 2r > 0 > £(b) + 2s then d(x,y) = 0.

(iii)

If £(a) + 2r > A(b) + 2s > 0 t^en 2d(x,y) > max{m(bs),2s}

2 s , with

equality

unless

m(a ) = m(b ) > 2 r , 2 s , m(a b " 1 ) , in which

2d(x,y)

= 2m(a ) - max{m(a b " 1 ) , 2 r , 2 s } .

Proof.

In case (i) max{m(bs),2s} < max{m(ar),2r} < 0, so

case

2d(x,y) = 0 + 0 - max{m(a b ^ ^ r ^ s } . In case (ii) max{m(a ),2r} > 0 > max{m(b ),2s}.

Thus

2d(x,y) = max{m(ar) ,2r} - max{m(a r b^ ) ,2r,2s}. If m(a ) > 0 then m(a ) > m(b ) = m(b -1 ) and, since by Lemma 3 m(a * ) , m(a b " 1 ) , m(b s ) is isosceles, m(a b" 1 ) = nifa^) > 0 > 2s. Thus 2d(x,y) = max{m(a ),2r} - max{m(a ),2r} = 0. If 2r > 0 > m(a ) then m(a ) , m(b ) < 0 and, again by Lemma 3, m(a b " 1 ) < 0 . r r s r s Hence 2d(x,y) = 2r - 2r = 0. In case (iii) max{m(ar),2r} > max{m(bg),2s} > 0 so that 2d(x,y) = max{m(ar) ,2r} + max{m(bs) ,2s} - max{m(a r b^ ) ,2r,2s}. If 2r > m(a ) then 2r > m(b ) , 2s and, by Lemma 3, 2r > m(a b" 1 ) and so 2d(x,y) = 2r + max{m(bs) ,2s} - 2r = max{m(bs) ,2s}.

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234

If m(a r ) > 2r then m(a ) > m(b ) , 2s and so by Lemma 3 m(a ) > m(a b" 1 ) with equality if m(a ) > m ( b ) .

Therefore

2d(x,y) = m(a ) + max{m(b ),2s} - max{m(a b"1) ,2T,2s} f2m(ar) - max{m(a r b^),2r,2s}, if m(a r ) = m(b s ) > mCa^b"1),2r,2s, |max{m(b ),2s},

otherwise.

Proof of Theorem 2.

Axiom Al' for a length function is satisfied since

£(1) = max{m(l),0} = max{-«,0} = 0.

If x = au r then x"1 = u"ra"1 = a"1 u" r

and so, £(x -1 ) = maxdnCa"1 )-r,-r,r} = max{m(a)+r,-r,r} = &(x), showing that I satisfies Axiom A2. To show that I satisfies Axiom A4 we let x = au , y = bu and z = cu , with a, b and c in N, and r, s and t in 2Z .

We assume,

without loss of generality, that A (a) + 2r > a(b) + 2s > £(c) + 2t. We consider four separate cases. Case 1.

0 > A(a) + 2r > £(b) + 2s > £(c) + 2t.

By part (i) of Lemma 4 2d(x,y) = -max{m(a r b^),2r,2s}, 2d(y,z) = -max{m(b c ' ^ ^ s ^ t } , 2d(z,x) = -max{m(c a"1),2t,2r}. By Property I max{r,s}, max{s,t}, max{t,r} is isosceles, and so is m(a b " 1 ) , m(b c " 1 ) , m(c a"1) by Lemma 3. It follows by Property II that r s s t t r -d(x,y), -d(y,z), -d(z,x) is isosceles. Case 2.

A(a) + 2r > 0 > i(b) + 2s > &(c) + 2t.

By Lemma 3, m ( b - 1 ) , m(b c " 1 ) , m(c ) is isosceles, and so m(b c"1) < S

max{m(b ),m(c )}.

S

L

X

5

Thus by Lemma 4, part (i),

2d(y,z) = -max{m(bsc"1) ,2s,2t} > -max{m(b ) ,2s,m(c ),2t} = -max{il(b)+2s,Jl(c) + 2t} > 0. Also by Lemma 4, d(x,y) = d(x,z) = 0, and so A4 is satisfied.

L

Hoare § Wilkens:

Case 3.

Unbounded non-Archimedean elements

23 5

A (a) + 2r > £(b) + 2s > 0 > A(c) + 2t.

By Lemma 4, 2d(x,y) > max{m(b ),2s} = £(b) + 2s > 0, and d(x,z) = d(y,z) = 0.

Axiom A4 is thus satisfied.

Case 4.

A(a) + 2r > A(b) + 2s > A(c) + 2t > 0.

We introduce three further conditions, (a)

m(c t ) = m(a r ) > 2r, 2t, m f ^ a " 1 ) ,

(S)

m(b s ) = m(c t ) > 2s, 2t, m O ^ c " 1 ) ,

(Y)

m(a r ) = m(b s ) > 2r, 2s, m ^ b " 1 ) .

Using the isosceles property of m(a b " 1 ) , m(b c " 1 ) , m(c a"1) it can be seen that any two of these implies the third.

If neither (a) nor (B)

holds then by Lemma 4, 2d(x,y) > max{m(bs),2s} = £(b) + 2s, 2d(y,z) = 2d(x,z) = max{m(ct),2t} = A(c) + 2t, and so A4 is satisfied.

If condition (a) holds then

m(a ) = A (a) + 2r > £(b) + 2s = max{m(b ),2s} 2t = max{m(ct),2t} = m(c t ) = m(a r ) . Therefore 1 2d(x, y) = 2m(a ) - max{iri(a r b g ),2r, 2s},

2d(y, z) = 2m(a r ) - max{iri ( b s c ^ ) , 2 s , 2t}, 2d(z, x) = 2m(a ) - max{iri ( c t a ^ ) , 2 r , 2t}. By the same argument as used in Case 1, it follows that A4 is satisfied. If condition (3) holds but (a) does not hold, then m(b g ) = m(c.) = max{m(b ),2s} = max{m(c,),2t}, and moreover (y) does not hold. L

S

t

Thus by Lemma 4 2d(x,y) = max{m(bs),2s} = m(c t ), 2d(z,x) = max{m(c ),2t} = m(c t ), 2d(y,z) > max{m(ct),2t} = m(c ) , and A4 is satisfied, completing the proof in Case 4. Axioms Al' , A2 and A4 and is a length function on G.

Thus I satisfies

Hoare § Wilkens:

Unbounded non-Archimedean elements

236

We now show that N is the set of all non-Archimedean elements of G.

Let x = au r , then x2 = au r au r = aa

2

&(x ) = max{m(aa_ )+2r,2r,-2r}.

u

and so

If r = 0 then by Lemma 3, &(x2) = £(a2)

= max{m(a2),0} = max{m(a) ,0} = £(a) = il(x).

If r > 0 then by Lemma 3,

m(a_ ) = m(a) - 2r < m(a) and so by the isosceles property m(aa 2

Thus £(x ) = max{m(a)+2r,2r} = max{m(a)+r,r} + r = £(x) + r. then by Lemma 3, m(a

) = m(a) .

If r < 0

) = m(a) - 2r > m(a) and so m(aa_ ) = m(a) - 2r.

Thus &(x 2 ) = max{m(a),-2r} = max{m(a)+r,-r} - r = &(x) - r.

Thus for all

x we have shown that £(x2) = &(x) + |r|, and £(x 2 ) < &(x) if and only if r = 0, that is, x is in N.

The subgroup N therefore consists of all the

non-Archimedean elements of G. If a is in N then £(a) = max{m(a),0}, and so clearly the lengths of elements of N are unbounded.

In fact if PL-, for i > 0, rs

defined to be the subgroup of N generated by the generators g. for j < i, then this gives a collection of subgroups {Ho. : i > 0} with I ) H«. = N. 21

i>0

Zl

The restriction of A to N is then the non-Archimedean length function described in Proposition 4 of [3], that is £(a) = inf{2i ; a € H ^ } .

REFERENCES

1. I.M. Chiswell, An example of an integer-valued length function on a group, J. London Math. Soo. 16_ (1977), 67-75. 2. R.C. Lyndon, Length functions in groups, Math. Scand., 12 (1963), 209-234. 3. D.L. Wilkens, On non-Archimedean lengths in groups, Mathematika, 23 (1976), 57-61. 4. D.L. Wilkens, Length functions and normal subgroups, J". London-Math. Soc. 22 (1980), 439-448.

237 AN ALGORITHM FOR THE SECOND DERIVED FACTOR GROUP

J.R. Howse University of Nottingham, Nottingham, NG7 2RD, England D.L. Johnson University of Nottingham, Nottingham, NG7 2RD, England

1.

Given a finite presentation < X|R > for a group G, there is a

simple and direct method for computing the rank and invariant factors of G/Gf = G a .

This is described in §6 of [9], of which the following is a

paraphrase.

If X = {xi ,...,xn>,

R = (n ,...,rm>,

(1)

let e. . be the exponent-sum of x. in r., so that E = (e. .) is an m x n matrix over the integers 2Z , called a relation matrix for G a .

It is a

standard result that unimodular matrices P and Q can be found such that PEQ = diag(di ,... ,d,), where k = min(m,n) and the d. are non-negative integers, each dividing its successor.

Discarding any l's and 0 f s,

occurring at the beginning and end respectively, the remaining dfs - say d

.,..., d

- comprise the invariant factors of G a , whereupon

rk(G ab ) = n-t. The aim of this lecture is to describe and justify a similar algorithm for N

, where N is any normal subgroup of finite index in G.

The method coincides with the above when N=G, and the important (and favourable) special case N=Gf is illustrated in §§5,6, below.

The main

idea is contained in [4] (see also [6]), together with a proof involving relative homology.

An explicit formulation appears in §3 below, followed

by a simple proof (§4) that depends only on the fundamental theorem described in §2. To avoid complications, notation will be abused in various harmless ways, of which the following are fairly typical.

While the

meaning of the symbols X, F=F(X), G and N is fixed throughout, R denotes a set of defining relators for G only in odd-numbered sections; elsewhere, it stands for the normal closure of such a set in F.

Secondly, any

mapping induced by the natural epimorphism <j> : F •* G, such as 2 F •>• 2 G and ZZ F

->• 2 G

$ : G H- G/N.

, will also be denoted by (f>, and the same goes for

Howse § Johnson:

238

The second derived factor group

2. Given a group G, let U be the augmentation ideal of the group -ring ZZG, so there is a short exact sequence (2)

ZZG^ZZ+o

of left ZZ G-modules.

If G is generated by a set X as in (1), then U is

generated by {x.-e|l < i < n} over ZZG, and there is a short exact sequence 0 + Q -> ZZG®n I U + 0,

(3)

y.(x.-e) and Q = Ker IT.

where IT(YI ,... ,Y ) n

X

It has long been known

1

(see [6],[7],[8] and [11], for example) that Q is 2Z G-isomorphic to the relation module R

, where the G-action is induced by conjugation within

F. Such an isomorphism can be constructed in terms of Fox derivatives.

For each x € x, there is a mapping 3/3x : F •> 2Z F computed \ +1 if w = yi...yv with each y. € X , then 3w/3x = \ a. ,

as follows: where

1

Vi ...yi_ a. =

0

,

when Yi = x

,

when y. = x

,

when y . # x ~

~

Letting 9:F -^ ZZ F d n send w to (3w/3xi ,. .. ,3w/3xn) and <\>: ZZ F®n ^ ZZG®n, the fundamental theorem can be stated as follows.

Theorem.

With the above notation, the rule K

: RRa ba rR 1

Q <()3(r)

defines an isomorphism of 7Z G-modules. Foxfs proof [5] involves results of Blanchfield and Lyndon, and may be found in [1], pp.103-110.

3.

The Jaoobian J = 3R/3X of the presentation G = < X|R > is the

|R| x |x| matrix whose (i,j) entry is 3r./3x..

As J is a matrix over 2 F ,

<J>(J) has entries in 2 G and, by the theorem, its rows generate Q as a left

Howse $ Johnson: ZG-module.

The second derived factor group

239

Here, however, interest centres on the matrix iJ><J>(J) over

ZZ (G/N). Any matrix with entries in a group-ring can be "blown up" by replacing each of its entries by the image of that entry under the left regular representation. matrix over 7L (G/N).

Explicitly, let G/N = (yi,...,y } and A be a £ Then any entry y € 2Z (G/N) of A defines an £-tuple

£ of integers by y = \ n.y.. The £-tuple corresponding to y.y(l < j < £) is a rearrangement of this, and we let m(y) denote the £ * £ matrix having this as its jth row.

Thus, m(y) is a relation matrix for

2Z (G/N)/7L (G/N)y as a TL -module.

Finally, m(A) is the matrix of integers

obtained by applying m to each entry of A. Proposition.

With the above notation^ D=m(i|/$(J)) is a relation matrix

for the group N a

® Z9^ " '

ab The invariant factors of N can thus be calculated from D by ab diagonalisation as in §1, and rk(N ) is the result of the corresponding computation for ranks, minus £-1. 4.

To prove this proposition, put S = K e r ^ , so that G = F/R,

G/N = F/S and R C S , as depicted in the Hasse diagram of Fig. 1. Now formulae (2) and (3), in conjunction with the theorem, yield the following commutative diagram with exact rows. 0

0

.ab

ZZG

ZZ (G/N)©n

U

0

-> V

0

Here, K is really K followed by inclusion (of Q in ZZG

) , i is induced

by the inclusion of R in S, and the lower row is the analogue of the

Fig. 1

Howse $ Johnson:

The second derived factor group

upper over 2Z (G/N), whose augmentation ideal is V.

240

The commutativity

follows from the fact that ty commutes with the formation of Fox derivatives. Since the rows of <\>(J) generate Im K over 7ZG (as noted above), those of M ( J ) generate ip(Im K ) = Im I/>K = Im K ! I over ZZ (G/N). n

define ZZ (G/N)® /Im K'I as a ZZ (G/N)-module.

It follows that D = m(iJ4(J))

is a relation matrix for this as an abelian group - call it C. !

out Imi = S R/S

f

Thus they

Factoring

!

= Im K I , this yields a short exact sequence

0 -*- S ab /(S f R/S f ) + C + V •* 0. Since V is a 2-free (of rank £-1), this splits over ZZ , and the group defined by D is, as required, just the direct sum of S 9 ^ " ' and

5.

The groups F a>

>c

= < x,y|x2 ,xyaxy xy C > were introduced by

Campbell, Coxeter and Robertson in [2]. The special case a + b + c = 1 will be used to illustrate the algorithm.

The second derived factor

group of this group will be computed. ^ 12 a b l-(a+b) >^ . f J = < x,y|x ,xy xy xy v > - Gn (say).

Ga

H

= G/G! i s c a l c u l a t e d and found t o be Ga

= < x,y|x 2 = 1, y =x > .

8rnThe relation y =x is needed to calculate T — ^ (mod G 1 ) (i.e. 3Xj T ^ - w i t h the notation of §§1,2), where ri = x2 , r2 = xyaxy xy " 9Xj

and xi = x, X2 = y, which we now proceed to do:

= 1 + x (mod G ' ) , xa+1

+

^

= 0 (mod G-),

xa+b

TIT E (x-x b ) ^ 3x" a -a 3x a — . because ___ = -x .

* (x a -l) ^

* xa+b+1

X

3x

Howse § Johnson:

241

The second derived factor group

There are three cases to consider, (i)

b even.

We have

~^- = 2 + x (mod G f ) , ax

2%- = x (mod G f ) . 3y

Thus the relation matrix for G f /G n x ZZ is

1 1 0

0

1

0

0

0

C 2 + C4

1

0

0

0

1 1 0

0

r 2 -ri

0

0

0

0

C2 - C 3

0

1

0

0

2

1

r 3 -2n

0

-1

0

1

c2 <-*C4

0

0

1

0

r 4 -ri

0

1

1

0

Ti

0

0

0

0

1 0

1 2

1 0

So Gf/G" s E, t h e t r i v i a l (ii)

a even,

b odd.

group,

We have

•y2- = 1 + 2x (mod G f ) ,

2 3 - = i (mod G ' ) .

The r e l a t i o n matrix f o r G'/G" x ZZ i s 1

1

0

1 1 0 1 2 2

0 ~ 0

1 0 1 0

1

which i s the same as that in (i) with (iii)

r4 , so G'/G" = E.

a,b odd. We have 2 3 - = 3 (mod G»),

y - H 2 x - l

(mod G ' ) .

f

The relation matrix for G /G" x 71 is 1 1 0 1 1 0

0 0

3

0 - 1 2

0

3

ca-ci

1

0

0

0

r 3 +2r2

1

0

0

0

xi-ri

0

-1

2

3

c 3 +2c 2

0

1

0

0

r3-3n

0

2

-1 -3

c 4 +3c 2

0

0

3

0

0

0

0

0

2 - 1

0

0

0

0

C4-C3

so Gf/GM a Z, . To summarise we have Z odd Gf/G' a* / 3 if ab ~ \ E if abeven.

6.

The Fibonacci Groups were introduced in [3]; a more general

class of groups is investigated in [10]. The group F(2,6) is used as a

Howse $ Johnson:

second example to illustrate the algorithm. <

242

The second derived factor group

F(2,6) has the presentation

Xl , X2 , . . . , X6 | Xi X2 = X3 , X2 X 3 = X4 , . . . , X 6 Xi = X 2

>

and is isomorphic to the group G Now G where ri

=

"^ x v xy2 x~* y2 yx2 y~* x2 ^

= < x , y | x 4 ,y4 , [ x , y ] > ,

and we p r o c e e d t o c a l c u l a t e -r^- (mod G 1 ) ,

= x y 2 x " ! y 2 , r2 = y x 2 y " ! x 2 ,

^-=

1-y

and xi = x , x2 = y . = x + x y + y 2 +y 3

(mod G ' ) ,

^- = y + xy + x2 +x 3 ox

(modG'),

(mod G ' ) ,

^ - = 1 - x2 (mod G'). dy

Let C(aia2...a ) and C[AiA2...A ] denote the circulant matrix and the block circulant matrix an an_i

a3

A, /M

A* «2 ....

A

m

Ai

A2

A3

A nm Am_ 1

. .'. .

Ai

respectively, where A. (i = l,...,m) is an £ x £ matrix. f C[C(1000)

0

C(-1000) 0]

[ C[C(0011) C(1100)

0

0] f

is a relation matrix for G /G

M

C[C(0100) C[C(10-10)

* 2Z

15

.

Then the matrix

C(0100) C(1000) C(1000)] 0

0

0

]

This matrix can be transformed,

by elementary row and column operations, into the diagonal matrix ' di

0

0 where di = d2 =

d32 ...

1 and

d32

=0.

Thus

G'/G" s* ZZ x 2Z x 7L .

Acknowledgement.

The first author is grateful to the Science Research

Council for its support, and the second to the Mathematics Institute of University College, Cardiff for its hospitality, during the preparation of this article.

Howse $ Johnson: The second derived factor group

243

REFERENCES 1. J.S. Birman, Braids, links and mapping class groups, Ann. Math, Studies 82. (1974). 2. C M . Campbell, H.S.M. Coxeter $ E.F. Robertson, Some families of finite groups having two generators and two relations, Proa. Royal Soc. London A 357_ (1977), 423-438. 3. J.H. Conway, Solution to advanced problem 5327, Amer. Math. Monthly 74 (1967), 91-93. 4. R.H. Crowell, Corresponding group and module sequences, Nagoya Math. J. 2£ (1961), 27-40. 5. R.H. Fox, Free differential calculus I, Ann. of Math. 57_ (1953), 547-560. 6. J. Gamst, Linearisierung von Gruppendaten mit Anwendungen auf Knotengruppen, Math. Z. £7 (1967), 291-302. 7. W. Gaschutz, Uber modulare Darstellungen endlicher Gruppen, die von freien Gruppen induziert werden, Math. Z. 60 (1954), 274-286. 8. P.J. Hilton § U. Stammbach, A course in homologicaT"algebray Graduate Texts in Mathematics, Vol.4, Springer-Verlag, New York Berlin (1971). 9. D.L. Johnson, Topics in the theory of group presentations> London Math. Soc. Lecture Notes, Cambridge University Press (1980). 10. D.L. Johnson, J.W. Warns ley $ D. Wright, The Fibonacci groups, Proc. London Math. Soc. (3) 2£ (1974), 577-592. 11. R.C. Lyndon, Cohomology theory of groups with a single defining relation, Ann. of Math. ,52 (1950), 650-665.

244 FINITENESS CONDITIONS AND THE WORD PROBLEM

V. Huber-Dyson University of Calgary, Alberta T2N 1N4, Canada

The purpose of this note is a generalization of the solution of the word problem for recursively presented simple groups, ([1],[5]). The results are pretty trivial but, like other easy observations in this field, they may have useful applications, ([1],[6]).

Most of the work is

accomplished by the choice of an appropriate terminology, some of which has been introduced in [4], [7] and [8]. Most ordinary decision algorithms consist of a reduction procedure to a normal form that yields the desired decision by a directly discernible typographical property. Thus triviality is reduced to zero length in free groups and the solutions of the word problem for individual finite or nilpotent groups are also of this kind.

A uniform solution for a class K_ of groups extends

to a solution for finitely presented residually-I^-groups, ([2]).

That

finite relatedness is indispensible here, is shown by examples of finitely generated, recursively related, residually finite groups with unsolvable word problems, [3]. On the other hand there are structural features of the lattice of normal subgroups that, in conjunction with a recursive presentation, will yield a decision algorithm for the word problem, while the class of all finite presentations with that property is not recursively enumerable.

Finiteness of the normal lattice is a typical

example and so is the property of being nearly critical in a sense that will be made precise below.

The algorithms consist of pairs of recursive

enumerations, one for the trivial words and one for the others, both procedures based on quantifier free predicate calculus with equality.

I

am looking for some interesting classes of groups to which these methods apply.

1.

NOTATION\ TERMINOLOGY AND A GENERAL PRINCIPLE The diagram A (A) of a structure A consists of the set A (A)

of all true atomic sentences of the language LA of the elementary type L of A augmented by a set "K of names for the elements of the domain A of A

Huber-Dyson: The word problem

245

together with the set A"(A) of the negations of all false ones. If A is generated under its basic operations by a set B, then the set FB of all well formed constant terms of the extension LB of L will do for A". The resulting diagram will be logically equivalent to the set A(A,B) = A+(A,B) U A~(A,B) of all L B-sentences that are true of A, where L B = L B U L B consists of all equations and instances of primitive relations between elements of FB and their negations. A is called decidable {constructive in Malcev's terminology) if A(A,B) is recursive in terms of some standard effective enumeration of the language LB. Strictly speaking one should talk about decidable LB-algebras, but if B is finite a constructive B-algebra will be decidable with respect to every finite set of generators and so the property is algebraic and mention of B can safely be omitted. Now, if A is compatible with a class K_ of L-type algebras, i.e. if A is embeddable in a model of the elementary theory T]C of jC, then it makes sense to call any subset of the diagram A(A,B) that has all of A(A,B) among its K-consequences a complete ^-presentation of A over B. So we write A = K_ < B|R;S > with R C A+(A,B) and S C A~(A,B), whenever TK U A(A) is consistent and TJC,R,S h y, for all ¥ G A(A,B) . Since the set of first order consequences of a recursively enumerable set is recursively enumerable and a set is recursive if and only if both it and its complement are recursively enumerable one has the following general criterion: Theorem 1. If TJC is axiomatizable by a recursively enumerable set, then a universal algebra that is embeddable in a model of TjC is decidable if and only if it has a complete presentation JC < B|R;S > with recursively enumerable sets R and S of relations and irrelations. If the type of A is purely operational one talks of word problems. The word problem for A is re or co-re over B according as A (A,B) or A~(A,B) is recursively enumerable, and its degree is the Turing degree of A (A,B) under a fixed enumeration of the set FB of constant terms of the language LB. It should be observed that, although knowledge of a complete presentation JC < B|R;S > may offer ready insight into the recursive structure of an algebra A, it may not contain any more information than an ordinary presentation. Indeed, if JC is closed under direct products, A is completely determined up to B-isomorphisms by R alone, for then all of A+(A,B) is derivable from T K U R alone and A~(A,B)

Huber-Dyson: The word problem

246

consists of the negations of all equations that are not so derivable. The usefulness of the theorem will depend on an understanding of structural properties that ensure the recursive enumerability of a complete presentation. Of special interest is the case of a finite set S of irrelations, which amounts to certain finiteness conditions on the lattice of congruences. On the other hand one may start with a class K^ of operational algebras that is dosed under substructures and direct products and consider pairs (R,S) consisting of a set R of equations and a set S of negations of equations between constant terms of the language of ]( enriched by a set B of constants. One obtains what B.H. Neumann calls absolute presentations in the case of groups, [7]. In answer to the question.what it might be that is presented let me suggest the following: Definition. A generalized ^-presentation over B is determined by a pair of sets R C L B and S C L ~B. It presents the poset K_< B|R;S > of canonical epimorphisms of L-structures that are generated by B and embeddable into some TK-model. It is inversely directed and (left) complete. A ^-presentation over B is determined by a set R of variable free equations over B. It canonically presents the algebra JK < B | R > that is obtained by subjecting the free h-algebra over B to the congruence induced by TJC U R. An isomorphism IT : JC < B | R > -^—> A is called a K-presentation of A bij_ < B | R >. A generalized presentation is fg if B is finite^ re_ if both R and S "aren recursively enumerable subsets of FB x FB and finite if B U R U S is finite. The diagrams of the algebras belonging to JC < B|R;S > are the L B-reducts of the consistent complete extensions of TjC U R U S. The diagram of JC < B|R > is the one with the unique minimal positive part, unless TjC U R U S is inconsistent and JC < B|R;S > is empty. If B U R is finite then JC < B|R > is residually a JC-structure, and if moreover the class of finitely generated algebras of JC is recursively enumerable with a uniformly solvable word problem, then JC < B|R > is re co-related, i.e. A~JC < B|R > is recursively enumerable. That the finiteness of R is needed follows from the existence of fg re presentations of groups that are compatible with finite groups but not residually finite and have an unsolvable word problem cf. [3].

Huber-Dyson:

2.

The word problem

247

GROUPS WITH SOLVABLE WORD PROBLEMS Now let ]( C £ be a class of groups closed under subgroups and

direct products.

Every equation of LB is equivalent in T£ to one of the

form w=l, where w e FB is an element of the free group FB generated by B. Using the abbreviations U_ and U, for the sets {u=l | u ^ U } and {u^l | u £ U } with U C FB we write K < B|U;V > and K. < B|U > for ]( < B|u_;V, > and jC < B|U_ >.

The consistent generalized presentation

jC < B|U;V > has the group jC < B|u > with the complete presentation jC < B|U;lT > = jC < B|u+;lf > for its left limit, where U + is the TKnormal closure of U in FB and U" is its complement in FB. (U;V)

+

If we set

= {w e FB|TK,U ,V, h w=l} and (U;V)" = {w e FB|TK.,U ,V, h w^l} = = + + ^ + ^

then (U;V)

= U

and is independent of V.

U

is disjoint from (U;V)

unless it intersects V non-trivially in which case they coincide and the presentation is inconsistent.

Completeness means that (U;V)

Note that (U;V)' = {w e FB| H v ^ V :

u (U;V)

= FB.

TK^,U_,w=l h v=l}.

Let jC=£, the class of all groups.

That the consistency

problem for generalized (^-presentations is recursively unsolvable follows from the unsolvability of the word problem for finitely presented groups, •while the undecidability of the completeness problem is a consequence of the unsolvability of the triviality problem.

By the word problem for

K < B|U;V > we mean the decision problem for (U+, (U;V) ")„.

When jC is

understood from the context and always when jC=£ we omit it as a subscript. The uniform solution of the word problem for one-relator groups yields a uniform decision procedure for zero-relator generalized presentations. Theorem 1 shows that a group C has a decidable word problem over a set B of generators if and only if it has a complete re K-presentation on B for some axiomatizable class j( of groups, in particular, as in [8]: Corollary 1.

A complete generalized presentation of groups has a solvable

word problem if and only if it is re* Dropping the condition of completeness it is possible to construct recursively inseparable re generalized presentations, and even finite ones, for which every group in G_ < B|U;V > has an unsolvable word problem.

If we are dealing with complete presentations we might as well

consider ordinary presentations and investigate the structural properties that reflect the existence of a finite set V of basic co-relators.

In

particular, if 6 = £ < B|U > has a completion G_< B|U;{v} > by a single co-relator, then the normal closure of v in FB presents a minimal normal subgroup of G that is contained in every non-trivial normal subgroup of

Huber-Dyson:

The word problem

G, in other words, 6 is monolithic.

If every word outside U

248

will do for

v we are faced with a degeneratedly monolithic group, namely a simple one.

Corollary 2.

Every re presentable monolithic group has a solvable word

problem. This innocuous observation has a significance for generalized presentations similar to the role that its corollary for simple groups is playing for presentations, ([1],[5]).

For, B.H. Neumann's Lemma 3.1 of

[7] can be used to construct yet another algebraic criterion for the solvability of the word probleml e: ]C < B|U;V >

An embedding

> JK < C|X;Y > of generalized presentations is a group

homomorphism e: FB

> FC that satisfies the conditions X + n e(FB) = e(U+)

and (X;Y)~ n e(FB) = e(U;V)~, so that e is an embedding of posets as well as of groups jC < B |U >

c

— > jC < C | X >.

Let me call e monolithic whenever

there exists a single word w e FC such that (X;{w})~ n e(FB) = e(U;V)~. If moreover e(B) is a recursively enumerable subset of FC then it is effective.

Corollary 2 is readily extended to effective monolithic

embeddability into an re presentation as a sufficient criterion for solvability of the word problem.

The necessity is obtained by starting

with an effective embedding into a two-generator group, applying B.H. Neumann's construction (Lemma 3.1 of [7]) and Higman-embedding the result into a finitely presented group. language we call an embedding 0 >

Translating into algebraic

> H monolithic whenever the set of

all normal subgroups of H that intersect 0 non-trivially has a unique minimal element, and observe that every embedding into a minimal normal subgroup is monolithic.

Without too much loss of generality we state the

result for finitely generated groups.

Theorem 2.

A finitely generated group has a solvable word problem if

and only if it has a monolithic embedding into a finitely presentable group. Note that an embedding into a simple subgroup is of course monolithic, so that one half of Theorem 2 is a corollary to Thompson's theorem of [5], while however the proof using B.H. Neumann's Lemma is quite a bit simpler.

On the other hand an embedding may well be monolithic

without factoring through a simple group.

Huber-Dyson:

3.

The word problem

249

GROUPS WITH A SIGNIFICANT SOCKEL No matter how satisfying an exact criterion may be, it will

not obviate the search for sufficient conditions that can be more useful in individual cases.

A few moments' reflection shows that the

appropriate generalization of a monolithic presentation is not a presentation with a finite set of basic co-relators but rather one with what, for want of a better term, I shall call a significant sockel.

The

socket of a group is the join of its minimal normal subgroups if there are any.

Otherwise let it be trivial.

It will be called significant if

it intersects every non-trivial normal subgroup non-trivially.

A set

S c (5 is a basis for the sockel of 0 if the normal closures <s>

in 6 of

its elements s are minimal non-trivial normal subgroups of 6 and the sockel is their direct product.

Now, if 6 = G^< B|u > has a significant

sockel with a basis TT(S), then w e FB is a co-relator of U if and only if, for some s e S and Sf = S\{s}, U = , SI, w=l h s=l. For, the normal closure of a non-trivial element of 0 will intersect the sockel non-trivially and so it will be a product of the form ax - •-\ with a. € < s^ > for distinct elements s. of the basis and any one of the s. will do for s.

Conversely, because of the 6-independence of the basis,

w=l cannot be a consequence of the relations U_ if the above entailment holds.

Therefore, if both U and S are recursively enumerable, then so

will be U

as well as its complement and $ will have a solvable word

problem over the set B of generators.

Of course this result is a

corollary to Theorem 1 because an re basis for a significant sockel will, under normal closure, yield an re set of basic co-relators.

However, the

proof given here gives a more meaningful procedure, and the existence of a finitely based significant sockel apparently is the correct generalization of monolithicity.

Observe that a group with such a sockel need

not have a finitely co-related complete presentation.

Theorem 3.

An re presentable finitely generated group with an re based

significant sockel has a solvable word problem.

4.

CONCLUDING REMARKS That the condition fg can be replaced by appropriate

requirements of effectiveness is again clear. comment.

More urgent seems another

It is customary to talk about recursive presentations rather

Huber-Dyson:

The word problem

than recursively enumerable ones.

250

And indeed any re set U of group-words

can be replaced by a set U* that is pointwise equivalent to U in the theory of groups while recursive as a set of typographical expressions. If f: ]N

> FB is an effective enumeration of U, replace f(n) by its

product with an n-fold product of symbols

f

l f . As algebraists, however,

we are interested in words qua elements of the free group FB, that is in •reduced words.

The question whether an element of FB is equivalent to a

word belonging to U* is just as decidable or undecidable as the question whether it belongs to the subset U of FB.

Moreover, the richness of word

problems hinges precisely on the fact that group-closure and normal closure do not preserve recursiveness, only recursive enumerability.

In

fact, applying the methods of §2 to a generalized presentation G_< B|U;V> with a recursively inseparable pair (U;V) one can construct finite generalized presentations Gm< A|R;S > that present posets consisting entirely of groups with unsolvable word problem.

The conjunc-

tion f\ (R_ U S,) together with the group axioms entails unsolvability of the word problem! The solvability of the word problem for re presentable groups with a finite normal subgroups lattice is a corollary of Theorem 3 as well as of Theorem 1.

Further finiteness conditions will be investigated.

Moreover, a connection with criteria for compatibility with the theory of finite groups will be discussed elsewhere.

Acknowledgement.

The author gratefully acknowledges support from the

Calgary Institute for the Humanities.

REFERENCES

1. W.W. Boone § G. Higman, An algebraic characterization of groups with soluble word problem, J. Austral. Math. Soc. ^8 (1974), 41-53. 2. V. Huber-Dyson, The word problem and residually finite groups, Amer. Math. Soc. Notices U_ (1964), 743. 3. V. Huber-Dyson, A family of groups with nice word problems, J. Austral. Math. Soc. l]_ (1974), 414-425. 4. A. Macintyre, On algebraically closed groups, Ann. of Math. 96^ (1972), 53-97. 5. R. Thompson, Embeddings into finitely generated simple groups which preserve the word problem, in Word 'Problems II; the Oxford Book* edited by S.I. Adian § W.W. Boone, North-Ho11and, Amsterdam, London (1980), 401-441. 6. C.F. Miller III, On group-theoretic decision problems and their classification* Ann. of Math. Studies 68^ Princeton Univ. Press, Princeton (1971).

Huber-Dyson:

The word problem

251

7. B.H. Neumann, The isomorphism problem for algebraically closed groups, in Word Problems> edited by W.W. Boone, F.B. Cannonito § R.C. Lyndon, North-Ho11and, Amsterdam, London (1973), 553-562. 8. H. Simmons, The word problem for absolute presentations, J. London Math. Soa. 6_ (1973), 275-280.

252 GROWTH SEQUENCES RELATIVE TO SUBGROUPS

W. Kimmerle University of Stuttgart, D-7000 Stuttgart-80, West Germany

INTRODUCTION Let G be a finite group.

The object of this article is the

comparison of the following growth sequences. r(G,H).

Let H be a subgroup of G.

Denote by d(G,H) the minimal

number of group elements needed to generate G together with H.

For the direct product of n copies of a group X we write

n

X .

The growth sequence r(G,H) of G relative to H is the

sequence (d(G n ,H n )) n G l N . r(I(G,H)).

For a group X the augmentation ideal of its integral group ring 2Z G is denoted by I (X). The quotient I (G)/I(H) tG , where tc is the augmentation ideal of H induced to G, is denoted I (H) by I(G,H) and is called the augmentation ideal of G relative to H.

The minimal number of generators of a 2 G-module M is

written as d G (M), the direct product of n copies of M as M n . The growth sequence r(I(G,H)) of the relative augmentation ideal is the sequence (d G (I(G,H) n ) n If H is the trivial subgroup of G, relative growth sequences reduce to ordinary ones. notation.

In this case we suppress the reference to H in all

Growth sequences r(G) were studied in [6], [7],[8] and [9], in

the case of finitely generated groups in [10]. It is clear that results for an arbitrary subgroup always contain the ordinary case and in this sense the results here extend those of the ordinary theory.

Note that,

if N is a normal subgroup of G,r(G,N) agrees with r(G/N) as well as r(I(G,N)) with T(I(G/N)). In the first part results of J. Wiegold on r(G) are generalized with respect to subgroups.

It should be mentioned that this is

basically contained in the author's thesis.

In the second part we compare

the two types of growth sequences that we have defined. If d(G,H) = d(G/H-[G,G]), then r(G,H) = r(I(G,H)).

The

Kimmerle:

Growth sequences

converse is valid if d(G,H) t 2.

253

In the case d(G,H) = 2 we need the

additional assumption that I(G,H) is a Swan module. H subnormal, I(G,H) is a Swan module. and only if d(G) = d(G/[G,G]).

If G is solvable or

In particular r(G) = r(I(G)) if

Moreover G is nilpotent if and only if

T(G,H) = r(I(G,H)) for every subgroup H of G, whereas G is perfect if and only if r(G,H) i r(I(G,H)) for every proper subgroup H of G.

This shows

how relative growth sequences may be used to obtain characterizations of finite groups.

1.

THE SEQUENCE T(G,H) As a f i r s t s t e p we e x t e n d a r e s u l t o f W. Gaschiitz

Proposition 1.1.

Let

Gi , . . . , G

be finite

groups

and H. < G^

[1] .

for

i=l,...,m.

Then d ( G i x . . . x G , H i x . . . x H ) m m = max {d(Gi,H! ) , . . . , d ( G m , H m ) , d ( ( G i , H i ) + x . . . x ( G m , H m ) + ) } . If X is a group and Y < X, then (X,Y) Let J

a

is defined as follows.

be the intersection of all maximal normal subgroups of X contain-

ing < Y,[X,X] > and J

the intersection of all maximal normal subgroups

of X containing Y but not [X,X].

Then (X,Y) + is X/(Ja n J ) .

For the

proof of Proposition 1.1 we need:

Propostion 1.2. Let l+N

+ G + G + lbea

H < G and Ha = H. Then any surjective

short exact sequence of groups*

group homomorphism

3 '• F * H -*• G,

where ¥ is a free group with d(F) > d(G,H) and 3| H = ou^, can be lifted to a surjective

group homomorphism

3' : F * H - > G i i ^ t 7 z B l | H = id and

3'a = 3. Proof.

Given T = {gi ,... ,g } C G such that < gia,... >gma>H > = G*.

Consider A(T) = { (gi m ,. . . ,g n ) ; n. G N} C G m . G containing H and E(S,T) = S

m

n A(T).

Let S be a subgroup of

Hence E(S,T) = 0 if and only if

S-N f G. If E(S,T) t 0, we obtain |E(S,T)| = |s n N| m . Let k V(T) = .U E(S.,T), where Si,...,S, are the proper subgroups of G containing H.

Since EfS^T) n E(S.,T) = E f S ^ S ^ T ) and S. n S. D H , it can

easily be seen that |v(T) | is independent of the choice of gi ,...,g m> as well as |A(T)|. there exists T

Of course, |V(T)| does depend on m. such that |A(T ) | - |V(T ) | > 0.

If m > d(G,H),

Consequently this holds

for all T. Let F be free on xi ,...,x . Then &:F * H -> G is determined m by xi $,...,x 3 and H3. We know now that there exist preimages

Kimmerle: gi »***>g

e

Growth sequences

254

G generating G together with H. Define 3 :F •*• G by x. •* g. ,

then by the universal property of free products, $ and id,, determine a surjection 8 f : F * H -*• G with the desired property. Proof of Proposition 1.1. then (KxL,UxV)

+

= (K,U)

+

If K and L are groups with subgroups U and V, x (L,V) + .

Consequently, if the proposition is

proved for m=2, it follows by induction. Let k = max{d(Gi,Hi),d(G2,H2),d((Gi ,H, ) + x(G 2 ,H 2 ) + )}.

Clearly

we have a surjective group homomorphism ir:F * (Hi x H 2 ) *> (Gi ,Hi ) x f ^ , ^ ) with iT|H

= id H

, where F is a free group of rank k.

Denote by K. the projection from (Gi ,H%) *(G2 ,lh)

onto

(G i ,H i ) . Then Kern C Ker(irK.) and TT maps Ker(7ric.) onto KerK. . Consequently KerffTKi)'Ker(TTK2) = F * (Hi x H 2 ) . Consider the diagram 1

> Kere

(Gi ,Hi) +

> Gi x H2 — >

>1

F* (Hi xH2 ) where e is the composite of the projection Xi :Gi x H2 •* Gi and the reduction map yi :Gi -*- (Gi ,Hi) . In particular (Hi x H 2 )e = 1. Since k > d(Gi,Hi) = d(Gi xH2 ,Hi xH2 ) , TTKI can be lifted to a surjection IT; :F*(HixH 2 ) -> Gi x H2 with *• | H ^

= id^ ^

by Proposition 1.2.

Analogously one obtains iM :F*(Hi xH2 ) -»• Hi x & . Let TTI = TTI1 Xi and TT2 = ir^ X2 .

We have TTKI = TTI yi and TTK2 = TT2 y2

For the first component we construct the following commutative diagram. 1

1

I

I

Ker(iTKi)

-

Jl

F * (HjxH 2 )

^

Gi

TTKl 1 (Gi,Hi)+

=

(Gi ,1

1 1

1

Kimmerle:

Growth sequences

Now TTI maps Ker(irKi) onto J .

255

Assume Keriri -Kern^ <1 F * (Hi xH2) .

Then

t h e r e e x i s t s a maximal normal subgroup T of F*(HixH2) containing Kerrri -Kem^ .

For the existence observe t h a t Kerfri -Kerfi^ has f i n i t e index

in F*(HixH 2 ). Hi x H2 C T. Hi C Tin .

Since 1 x H2 C Keriri and Hi x 1 C KerTr2 , we conclude t h a t Clearly Tin 2 Ji > s i n c e Hi x H2 C T implies (HixH2)in =

Consequently T D Ker(>Ki) .

Analogously we find T D Ker(TTK2 ) .

Hence T = F * (Hi xH 2 ), a c o n t r a d i c t i o n . Summarizing we obtain Kerrri *Kenr2 = F * (Hi xH2) and by [2, (6.16)] i t follows t h a t F*(Hi xH2 )/Kerm nKerir2 s F*(Hi xH2)/Ker7fi x F*(Hi xH2 )/Ker7r2 = Gi x G2 .

Moreover a maps Hi xH2 as

a subgroup of F*(HixH2) isomorphically i n t o Hi xH2 as a subgroup of Gi xG2 . Thus k > d(Gi xG2 ,Hi xH2 ) . On the other hand d(Gi xG2 ,Hi xH2 ) > d(G.. ,H.) for i = 1,2. F i n a l l y Hi xH implies d(Gi xG2 ,Hi xH2 ) > d ( ( G i , H i ) + x (G 2 ,H 2 ) + ). ri22 C Jj Ji x xjj22 imp] Hence k < d(Gi xGz ,Hi xH2 ) . Now we are in a position to extend results of J. Wiegold [8]. Theorem 1.3. a)

sample

m

Then d(G ,H ) = m-d(G/[G,G]-H) if m > d(G,H)/d(G/[G,G]-H).

Suppose that

images.

[G,G]*H ^ G and that G/< H > has nonabelian simple

Let s be the smallest

G/< HG >. c)

of H in G.

[G,G]-H ^ G and that G/< H > has only abelian m

images. b)

Denote by < H > the normal closure

Suppose that

order of a nonabelian simple image of

Then d(Gm,Hm) = m-d(G/[G,G]-H) if m > d(G,H)+l+log m.

Suppose that < H > 7* G and that

[G,G]-H = G.

Define

s as in b ) .

Then logsm < d(Gm,Hm) < d(G,H)+l+logsm for m > 1. d)

r(G,H) is constant

Proof.

if < HG > = G, i.e.

d(Gm,Hm) = d(G,H) for m > 1.

By Proposition 1.1 we know t h a t d(Gm,Hm) = max {d(G,H),d(((G,H) + ) m )}.

If < H G > = G, (G,H) + is trivial.

This proves d ) . Otherwise r(G,H)

agrees with r((G,H)+) as soon as d(((G,H)+)m) > d(G,H). the ordinary theory.

So we can apply

By [2,(6.16)] it follows that (G,H) + s G/J x G/J . a p

Moreover G/J is a cartesian product of nonabelian simple groups and G/Ja is abelian.

Note that d(G/J&) = d(G/[G,G]-H).

Now [8] applies to

give d(((G,H) + ) n ) = d(G/J )-n for n > 1 in case a ) , and d(((G,H) + ) n ) = d(g/J )*n for n > d((G,H) )+l+log n in case b ) , where s is the smallest order of a factor of G/J . Of course, s is also the smallest order of a P

Kimmerle:

Growth sequences

2 56

p nonabelian simple image of G/< H

>.

In case c) we obtain logsn < d((G n ,H n ) + ) < d((G,H) + ) + l + log s n for n > 1.

Now an easy computation yields the desired results. P

Remark 1.4. directly.

The fact that r(G,H) is constant if < H

> = G can be seen

If G = < H,gi,...,g H,gi,...,g >, then an elementary elementar calculation shows

that G n = < H nH, K n1 ) .K. .x, x n>i > ) where x. x = (g.,g^ ( g g (i) ,.. 7T 2 ,...,TT

are arbitrary permutations of deg degree m.

g

( i )

) and

Moreover it does not

make any difference, if G is finite or not.

We say a growth sequence (a )

G1N

is of linear

for some constant c ^ 0 and a l l n > n . If n oo linear. Corollary 1.5.

type i f a = n # c

= 1 we say (a ) ir

a) G is nilpotent if and only if r(G,H) is of linear type

for every proper subgroup H of G.

Moreover r(G,H) is always linear in

this situation. b) G is perfect if and only if r(G,H) is not of linear type for every subgroup H of G. We shall use the following elementary fact for subnormal subgroups.

Lemma 1.6. Assume H is subnormal in G. «

H

G

>,S > = G.

Proof. c < H

>.

Let S be a subset of G such that

Then < H,S > = G.

Observe that H is subnormal in G if and only if H is subnormal in c For a subnormal subgroup V of < H

> we define the length of V

as the maximal length of all possible chains V = V < H

< Vi < ... < V

=

>, where each V. is subnormal in < H >. Assume that H has length n and the lemma is proved for all P

subnormal subgroups of < H

> which have length less than n.

Let S C G

If H s = H for all s e S and H / < H G >, we can find P a subnormal subgroup Q of < H > such that H is normal in Q and the length of Q is less than the length of H. By assumption < Q,S > = G, so H has

with «

H G >,S > = G.

P

to be normal in G, a contradiction to H / < H >. Thus we can assume HS° ^ H for some s0 ^ S.

By [10] < H,H ° >

P

i s subnormal in < H >.

I t s length is less than the length of H.

Hence

Kimmerle:

< H,S > = «

Growth sequences

H,H ° >,S > = G.

257

Since the lemma is trivial if H has length

0, the result follows now by induction.

Proof of Corollary 1.5.

a) By Theorem 1.3(d) every maximal subgroup of G

has to be normal, i.e. G is nilpotent.

For the converse, if G is supposed p to Jbe nilpotent, every subgroup H of G is subnormal. Clearly d(G,< H >) = d(G,[G,G]«< H G > ) . By Lemma 1.6 we obtain d(G,H-[G,G]) = d(G,< H G >-[G,G]) = d(G,< H G >) = d ( G , H ) .

Now Theorem 1.3(a) completes the proof. b) If G is perfect, the statement is clear by Theorem 1.3(c) and (d). If G is imperfect, r(G,[G,G]) is linear.

2.

COMPARISON OF T(G,H) AND r(I(G,H)) A first result on r(I(G,H)) yields:

Proposition 2 .1 [4, (1.2) ].

Let 0 + A + TL G ^

+ B -> 0 be a short exact

sequence of TL G-lattices. Then the following are equivalent: (i)

A is not a generator;

(ii)

d G (B n ) = nm for all n e ]N;

(iii)

|Hom Z G (B,S)| = |s| m for at least one simple IG-module S. Thus all results on the generator problem of minimal relation

modules, ordinary or relative, can be translated to statements for r(I(G,H)).

An illustrative example is:

Theorem 2.2.

[G,G] is nilpotent if and only if r(I(G,H)) is linear for

all proper subgroups H of G. Proof.

Combine (1.2) and (2.4) in [4].

For further results we refer to [4], [5] and [12]. We compare now r(G,H) and r(I(G,H)).

Theorem 2.3.

If d(G,H) = d(G/H-[G,G]), then r(G,H) = r(I(G,H)).

The

converse holds if d(G,H) t 2, and in the case d(G,H) = 2 with the additional assumption that I(G,H) is a Swan module.

In particular r(G) = r(I(G)) if

and only if d(G) = d(G/[G,G]). For the proof we need the following:

Kimmerle:

Lemma 2.4. Proof.

Growth sequences

258

I(G,H)/I(G)-I(G,H) a G/H-[G,G].

Consider the known exact sequence of S G-modules G/[G,G] -> 0 Ap p A

where e is defined by g-1 H» g[G,G].. H#

[G,G]/[G,G].

I(H) I(H)

maps map under e onto

Hence the lemma follows immediately.

Proof of Theorem 2.3.

Assume d(G,H) = d(G/H-[G,G]).

By Lemma 2.4 we

obtain d G (I(G,H) m ) > d((G/H-[G,G])m) = m-d(G/H-[G,G]) for all m e ]N. By Theorem 1.3(a) we know that d(G m ,H m ) = m-d(G/H- [G,G]) for all m e ]N. Since always dG(I(G,H)) < d ( G , H ) , dG(I(G,H)m) < m-d(G,H) follows.

Putting

things together we conclude T(G,H) = r(I(G,H)). Conversely assume r(G,H) = r(I(G,H).

Set p(k) = d G (I(G,H) k ).

If I(G,H) is a Swan module, the kernel of a short exact sequence 0 -> A + TL G

y

^

«* I(G,H) -> 0

has no projective direct summand.

By [4,(1.1)] it follows that there

exists a simple TL G-module S such that |Hom2 G(I(G,H) ,S) | > |s|y(" >~ . Clearly |Homz G (I (G,H)n,S) | < |s| y(;n:) . n.(y(l)-l)

<

Consequently

y(n).

(1)

Observe that the same computation with a simple TL G-module T, which is an image of I(G,H), shows that r(I(G,H)) always has a lower linear bound. For this it is not necessary that I(G,H) is a Swan module.

By Theorem

1.3(c) and (d) we conclude that r(I(G,H)) - T(G,H) tends to infinity if H-[G,G] = G and H ^ G. By Theorem 1.3(a) and (b) we know now that for a sufficiently large n e W , p(n) = n-d(G/H-[G,G]) if r(G,H) = r(I(G,H)).

So we can

write (1) as n-(d(G,H)-l) < n-d(G/H-[G,G]). Thus d(G,H)-l < d(G/H-[G,G]), i.e. d(G,H) = d(G/H-[G,G]). is a Swan module, see [2,(7.8)].

Note that I(G)

By [2,(7.3)] and the fact that QG maps

onto Q «_I(G,H) it follows immediately that I(G,H) is a Swan module, if dG(I(G,H)) > 3.

Clearly I(G,H) is a Swan module, if dG(I(G,H)) = 1.

This

completes the proof.

Remark 2.5.

In [3] it is shown that I(G,H) is a Swan module if the n-th

Kimmerle:

Growth sequences

commutator subgroup of G is contained in H.

259

In particular, if G is

solvable, then T(G,H) = r(I(G,H)) if and only if d(G,H) = d(G/H-[G,G]). Also, if H is a subnormal subgroup of G, it follows easily from Lemma 1.6 that I(G,H) is a Swan module. As far as I know, there is no counterexample known to this question.

Observe that the proof of Theorem 2.3 actually shows that

T(G,H) agrees with the growth sequence of the semi-local relative augmentation ideal if and only if d(G,H) = d(G/H-[G,G]). Moreover, if there are r(G,H) and r(I(G,H)) which agree and are not linear, then it is easy to see that they have the form (2,2,3,4,...,n,n+l,...).

Furthermore suppose there exists a pair (G,H)

such that d(G,H) = 2, d(G/H-[G,G]) = 1 and I(G,H) is not a Swan module, then T(G,H) and r(I(G,H)) will agree.

Corollary 2.6.

a) G is nil-potent if and only if r(G,H) = r(I(G,H)) for

all subgroups H of G. b) G is perfect if and only if r(G,H) ^ r(I(G,H)) for every proper subgroup of G. Proof.

In the proof of Theorem 2.3 we saw that r(G,H) - r(I(G,H)) tends

to -~ if H f G and [G,G]«H = G.

Consequently, if G is perfect, this holds

for every proper subgroup of G.

If G is imperfect, r(G,[G,G]) =

r(I(G,[G,G])).

Hence b) is proved.

If r(G,H) = r(I(G,H)) for all subgroups H of G, any maximal subgroup of G has to be normal.

The converse is proved as in the proof

of Corollary 1.5 together with Theorem 2.3.

Remark 2.7.

Let G be an infinite group with a subgroup H of finite index.

Clearly H contains a normal subgroup N of G such that G/N is finite. Since d(G,H) = d(G/N,H/N) and dG(I(G,H)) = d G/N (I(G/N,H/N)), our results can be applied immediately to the pair (G,H).

REFERENCES

1. W. Gaschutz, Zu einem von B.H. und H. Neumann gestellten Problem, Math. Nachr. 1£ (1955), 249-252. 2. K.W. Gruenberg, Relation modules of finite groups* Regional Conference Series in Math. 25_, AMS, Providence, R.I. (1976). 3. W. Kimmerle, Uber den Zusammenhang der relativen Erzeugendenzahlen bei Gruppen und der Erzeugendenzahl relativer Augmentationsideale* Dissertation, Stuttgart (1978).

Kimmerle:

Growth sequences

260

4. W. Kimmerle, Relative relation modules as generators for integral group rings of finite groups, Math. Z. 172 (1980), 143-156. 5. W. Kimmerle, Relation modules and maximal subgroups, Arch. Math. 36^ (1981), 398-400. 6. J. Wiegold, Growth sequences of finite groups, J. Austral. Math. Soo. 17 (1974), 133-141. 7. J. Wiegold, Growth sequences of finite groups II, J. Austral. Math. Soo. 2£ (1975), 225-229. 8. J. Wiegold, Growth sequences of finite groups III, J. Austral. Math. Soo. 25A (1978), 142-144. 9. J. Wiegold, Growth sequences of finite groups IV, J. Austral. Math. Soo. 29A (1980), 14-15. 10. J. Wiegold § J.S. Wilson, Growth sequences of finitely generated groups, Arch. Math. 30^ (1978), 337-343. 11. H. Wielandt, Eine Verallgemeinerung der invarianten Untergruppen, Math. Z. 45 (1939), 209-244. 12. J.S. Williams, Trace ideals of relation modules of finite groups, Math. Z. 163 (1978), 261-274.

261 ON THE CENTRES OF MAPPING CLASS GROUPS OF SURFACES C. Maclachlan University of Aberdeen, Aberdeen, AB9 2TY, Scotland

0. Aj,A2,...,A

Let S denote a compact orientable surface of genus g and let be disjoint sets of distinct points on S with cardinalities

|A.| = n., i = 1,2,...,p.

The mapping class group of S with respect to

{Aj,A2,...,A } is defined here to be the group of homotopy classes of orientation-preserving self-homeomorphisms f of S such that f(A.) = A. for each i = 1,2,...,p.

As these classes only depend on the cardinalities

of the sets A ^ A j , . . . ^

the group is denoted by M(g,{nj ,n2 ,. .. ,n }). If

p = l, the notation is simplified to M(g,nj), including nt = 0, and if each n i = 1 , to P(g,p). In this paper, the centres of these groups are computed and shown to be trivial in almost all cases.

The proofs use the fact that

these mapping class groups are isomorphic to the outer automorphism group of a suitable Fuchsian group and the action of this group as the Teichmiiller modular group on the Teichmiiller space of the corresponding Fuchsian group. 1.

Let £ denote the group PSL(2,IR).

Every cocompact Fuchsian

subgroup r of £ has a presentation of the form Generators: x,,x2,...,x , a1 ,b, ,... ,a ,b Relations :: x.

= 1

i = 1,2,...,r;

1

In j=l

g [a.,b ] 3

J

n k=l

(U

x, = 1 . k

The isomorphism class of such a group is determined by its signature (g; m t ,...,m ) and the m t ,m 2 ,...,m

are called the periods of r.

Each

such group r has a fundamental region in the upper half-plane U whose hyperbolic area is given by y(r) = 2Tr[2(g-l) +

I (1 - -i-)] . i=l i

(2)

Maclachlan:

Centres of mapping class groups of surfaces

262

If, furthermore, r C rt and [Tt : T] = n, then p(D = n yCrj.

(3)

Every automorphism of r is induced by an automorphism of the free group on 2g +r generators and an automorphism of r is called orientation-preserving if the corresponding automorphism of the free group maps the word corresponding to the long relation in (1) into a conjugate of itself.

Let &(r) denote the group of orientation-preserving

automorphisms of r and M(r) the quotient group £(r)/Innr.

Then M(r) is

the outer automorphism group or Teichmiiller modular group of F. Let F have signature (g; mt ,11^ ,. .. ,m ) and partition the periods into sets A 1 ,A 2 ,...,A equal.

where all the periods in the same set are

If |A^| = n., then M(r) is isomorphic to M(g,{nj,1^ ,...,n }) via

the Nielsen isomorphism (see e.g. [9]). Also note that M(g,n) is isomorphic to the mapping class group of the n-punctured surface of genus g2.

In this section, known results (see e.g. [7], [9] and

references there) on the action of the Teichmiiller modular group M(r) on Teichmiiller space T(r) are assembled. For a cocompact Fuchsian group r, let
r -*- £\a is an orientation-preserving isomorphism with a(r) discrete}.

Topologise
1

for t G £ and all y e r.

> a

The quotient space,

whose elements we denote by [a], is the Teickrriuller space T(r) of r.

For

T as at (1), T(r) is known to be homeomorphic to a real cell of dimension 6g - 6 +2r. The group M(r) acts on T(r) as a group of homeomorphisms by ?[o] = [« ° •] where a €
(4)

If a subgroup G c M(r) has a fixed point [a] in T(r) then for every g = T

G

t

G, there exists t € £ such that afyjt"1 = a <> < K Y )

Taking Ti = < a(T), t

for every y € r.

: g ^ G >, the subgroup of £ generated by these

elements, then ri/a(T) ^ G is a group of conformal automorphisms of the surface U/a(T) and so is finite.

Thus

Maclachlan:

1

Centres of mapping class groups of surfaces

> r - 2 - » Ti — — >

G

> 1

263

(5)

is exact with G finite. If, conversely, there exists a short exact sequence as at (5) with r,Ti Fuchsian groups and G finite, then G can be embedded in M(r) as follows.

Define of : G

> M(F) by a(g) = a

where a (y) = a"1 (Yi BCYDYI" 1 )

where yi e Ti is any element such that TT(YI) = g. Furthermore, in this situation, the Teichmuller space of Ti embeds in the Teichmuller space of r via a :T(Ti) •* T(r) where a[B] = [6° a] for all [3] G T(ri).

Then the fixed points of T(r) under

the action of a(G) is precisely the set a(T(Ti)). A Fuchsian group r is called finitely-maximal if there is no Fuchsian group To such, that T C To , T ? To , and [To :F] < °° (see [6]).

Lemma 1.

M(T) fixes [3] e a(T(ri)) where

With notation as above, if~$€

[$] = [6 ©a ] and 6(ri) is finitely-maximal * then J e a(G). Proof.

Since J[6 © a ] = [6 °a], there exists t € £ such that

t 6 o aCyJt"1 = 6 o a © <j>(Y) for all y € r. 6a(T) in £.

for any cocompact Fuchsian group <Sa(r). maximality.

3.

But [r2 : 6a(r)] < «>

Thus r2 = 6(Fi) by finite-

Let t = 6(yi) so that t 6 oafyjt"1 = 6(ylaL(y)y~1) = 6(a(Y))

for all Y e r.

determined.

Thus t € r2 , the normaliser of

Since ot(r) is normal in Ti , 6(ri) C r2 .

Thus Y I O M Y I " 1

= a

° ^^

and ^" = aCirCy,)) € a(G).

In this section, the centres of the groups M(g,0), g > 2 are For g = 2 , the result is known (see e.g. [4]) and for g > 3

is apparently part of the folklore of mathematics. The methods of §2 are applied in a situation where Ti in (5) is a Fuchsian triangle group, so that T(Fi) is a single point.

This will

show that any element in the centre is necessarily of finite order and further must correspond to the sheet-interchange map of a hyperelliptic Riemann surface. Let Vi have signature (0; 2,4g,4), g > 2, and let Gi be the group of order 8g with the presentation {V,W|V2 = W 4 g = (VW) 4 = 1, Then

Z(Gi) = f < W 2 g > < W

g

>

VWV = )fi2g'1}.

if g is even, if g is odd.

MacLachlan:

Centres of mapping class groups of surfaces

Let ?r : Ti •* Gi be the obvious epimorphism.

264

Since TT preserves the orders

of the elliptic generators of Ti , the kernel of TT = r is torsion-free. Thus using (2) and (3), which combine to give the Riemann-Hurwitz relation, r has signature (g;-).

Thus we have an exact sequence -> 1.

(6)

Now let r2 have signature (0; 2, 2g+l, 4g+2) so that an epimorphism TT! can be defined from r2 onto a cyclic group G2 of order 4g+2 with torsion-free kernel of signature (g;-).

Again we have an

exact sequence

r2 -2LL> G 2 Theorem 2. a(< W

g

Proof.

> l.

(7)

The centre of M(r) = M(g,0), for g > 2, is a subgroup of

> ) , defined abovej which is a group of order 2. Let P be the point a(T(ri)) of T(r) obtained from (6). Let

c G Z(M(r)).

Since P is fixed by each element of a(Gi), so also is c(P).

But P is precisely the fixed point set of a(Gi) so that c(P) = P.

Since

Ti has signature (0; 2,4g,4), every 6(ri) for 6 G
By Lemma 1, c G a(Gi) and so lies in Z(a(Gi)).

if g is even, c G a(< W 2 g > ) . 1, 2 or 4.

However, an identical argument using the exact sequence (7)

shows that c has order dividing 4g+2.

Theorem 3. Proof.

Thus

If g is odd, c G a(< W g >) and c has order

Thus, in both cases c G a(< W

&

>).

Z(M(g,0)) is trivial for g > 3 and of order 2 if g = 2.

Suppose that c G M(r) is non-trivial, so that c = a(W

g

).

One can

now deduce how c acts as an automorphism of r from the exact sequence 1

> r — 2L -> To

> < W2g >

^> 1

where To is the subgroup of Ti in (6) corresponding to the subgroup < W

g

> of Gi .

Counting the number of conjugacy classes of elliptic

elements in To and applying the Riemann-Hurwitz relation ((2) and (3)) to To

c

Ti , one deduces that r0 has signature (0; 2,2,...,2) where there are

2g+2 periods of order 2.

Thus in the notation at (1), each x., x 2g 2g

i = l,2,...,2g+2 maps onto W

6

and a(W 6 ) acts on r by conjugation by xi .

Using the Reidemeister-Schreier rewriting technique, one obtains that r is generated by Vj,v2,...,v2 where v i = ^ i + 1 x i for i = l,2,...,2g and has the single defining relation vi v2 * V3 vi"1 .. .vo ,\>Zl v."1 v, .. .v"1 _v« =1. 2g-l 2g 1 2 2g-l 2g

Maclachlan:

Centres of mapping class groups of surfaces

26 5

Then the automorphism c acts by mapping v. to vT1 for i = l,2,...,2g (c.f. [4]). This automorphism is induced by a homeomorphism i which is described by Figure 1 in Lemma 4 for the case g = 2 (see also Figure 3 ) . In the case g = 2, the homeomorphism i commutes with the generating Dehn twists of M(2,0) up to homotopy and so i (or c) generates Z(M(2,0)) (see [4]).

To show that the centre is trivial in the cases g > 3, we describe

automorphisms ty which are induced by homeomorphisms which destroy the symmetry of the surface imparted by i. v2 v *

1#.

.vs1 v4

Precisely 1KV2} =

Now c commutes with ty up to inner automorphism. such that Y Q ^ - Y "

1

V2V

2g v 2g-l* " V s 1 V 4

Thus there exists y0 ^ T

= v. for i > 4

Now for g > 3, y0 commutes with more than two distinct hyperbolic elements of r and so y0 = 1.

But then oo commutes with two

distinct hyperbolic elements of r and so o) = l.

4.

But that is false.

In this section the centres of the groups P(g,n) and M(g,n)

are calculated, making extensive use of results of Birman ([1], [2], [3]). For g > 2, there is a short exact sequence 1

> P n (g)

> P(g,n)

> M(g,0)

> 1

where P (g) is the pure n-string braid group of the compact surface of genus g, which has trivial centre.

Thus for g > 3, each P(g,n) has

trivial centre.

Lemma 4. Proof.

Z(P(2,1)) is trivial. In the exact sequence 1

> P t (2)

> P(2,l) — £ _ > M(2,0)

> 1

the group Pt (2) is isomorphic to the fundamental group of a compact surface of genus 2 and so has trivial centre. Consider the surface of genus 2 with one puncture and the

Maclachlan:

Centres of mapping class groups of surfaces

generators of its fundamental group IT1 given by Fig. 1. 1

!

1

266

Then

1

Vj v^ v3 v4 vj* v2 vj v 4 p = 1 (c.f. [4]). Now the homeomorphism i induced by i (x,y,z) = (x,-y,-z) induces the automorphism i of nm given by i(v i ) = vT 1 , i = 1,2,3,4 and i(p) = (vj"1 v2 v3~x v4 )p(v1"1 v2 vj 1 v4 ) "l .

The

surface also admits the homeomorphism induced by the back and forth twist £ t about the curve t in Fig. 2, which, on an annulus about t, carries out Dehn twists in opposite directions on either side of t but leaves t pointwise fixed (see [2]). £ t induces the automorphism $ of ITj where

*P(v4) = v4

K p ) = Vjvf'pv^j1

If c e Z(P(2,1)) then TT(C) = ir(i).

Now i has order 2 and c

must have order 2 since Pl (2) has trivial centre. is torsion-free.

Thus c = i since P1 (2)

Thus i[> commutes with i up to inner automorphism.

But

!

considering the action on the elements v3 ,v4 ,v^" v2 , this implies that VjV^p" 1 commutes with vj"1 v2 in the free group TT1 .

This contradiction

shows that Z(P(2,1)) is trivial.

The exact sequence (see [2]) Fig.l

x

Fig.2

J

Maclachlan:

Centres of mapping class groups of surfaces

267

1 -* n, (T 2 \{x 1 ,x 2 ,...,x n _ 1 » -* P(2,n) -> P(2,n-1) -> 1 where T2 is the compact surface of genus 2 gives, inductively, that the groups P(2,n) have trivial centres for all n > 1. Since the mapping class group of an n-punctured surface contains classes of homeomorphisms which arbitrarily permute the punctures, the sequence 1 -* P(g,n) — is exact where S

M(g,n) i s

n

- 4 l Thus for g > 2, n > 3, M(g> n )

is the symmetric group.

has trivial centre as, of course, does M(g,l) = P(g,l).

The case n =2 is

dealt with below.

5.

In this section, the centre of the general mapping class groups

M(g,{nj,n2,...,n }) are determined.

For these groups, there is the exact

sequence P(g,n) -> M(g,{n1,n2,...,np})

f ^ ,n2 ,... ,n p })

where En. = n and S({nt,n2,...,n }) is the corresponding subgroup of S . This subgroup will be isomorphic to

S n

centre unless some of the n. are 2. 2

c

G Z(P(g,n)).

in M(g,n).

2

Thus c

and so will have trivial i

Now if c e Z(M(g,{n,,...,n })) then

= 1 and c G C(P(g,n)), the centraliser of P(g,n)

Now for any f e M(g,n), fcf" 1 e C(P(g,n)) and so C(P(g,n))

contains the subgroup H generated by all conjugates of c. mapped isomorphically onto the subgroup E of S of ir(c). Fig.3

Thus TT"1 (E) = P(g,n) © H.

Under TT, H is

generated by all conjugates

Maclachlan:

Centres of mapping class groups of surfaces

We first consider the cases n > 5. E = A

or S .

26 8

Since E is normal in S , n

Then H possesses a subgroup isomorphic to A* which

permutes 4 of the points and leaves the others fixed.

This subgroup then

has a fixed point in the Teichmuller space T(r) where r has genus g and n equal periods [8]. But then the corresponding surface admits a conformal structure on which A4 acts as a group of conformal automorphisms.

But

then the stabiliser of a point on a compact Riemann surface under a group of conformal automorphisms is necessarily cyclic, and this is false for any of the n-4 points fixed by A4. The cases n = 2,3,4 remain to be considered.

If, in the case

n = 4 , TT(C) is a transposition in S4 , then H will be isomorphic to S4 and we can argue as above using the subgroup S 3 . Case n = 2.

The argument to show that C(P(g,2)) is trivial, g > 2 is

almost exactly as in Lemma 4, but using Fig. 3. The automorphism i induced by (x,y,z) — • (x,-y,-z) is such that iT(i) = TT(C) where c e C(P(g,2)) and so i =c.

Thus i must commute up

to inner automorphism with the automorphism i|j in P(g,2) induced by a back and forth twist £

about a curve t running round the first hole and

through one of the punctures.

Calculating the effects of these auto-

morphisms gives a contradiction as in Lemma 4.

Cases n = 3,4.

In both these cases there is an element of order 2

commuting with every element of P(g,n) and a similar argument to that used above shows that no such element can exist. The accumulation of all the above results gives:

Theorem 5.

The centres of the mapping class groups M(g,{nj ,...,n }) for

g > 2 are trivial except that Z(M(2,0)) is cyclic of order 2. The results for g = 0,1 are known and can be gleaned from the literature ([3], [10], [5]).

We state these results for completeness.

M(g>{n! ,n2,...,n } ) , g = 0,1 have trivial centres with the following exceptions all of whose centres are cyclic of order 2 - M(l,0), M(l,l), M(l,2), M(0,2), M(O,{2,1}), M(0,{2,2».

REFERENCES

1. J.S. Birman, On braid groups, Comm. "Pure Appl. Math. £2 (1969), 41-72.

Maclachlan:

Centres of mapping class groups of surfaces

269

2. J.S. Birman, Mapping class groups and their relationship to braid groups, Comm. Pure Appl. Math. 22_ (1969), 213-238. 3. J.S. Birman, Braids, links and mapping class groups, Ann. Math. Studies £2 (1974). 4. J.S. Birman $ H.M. Hilden, On the mapping class groups of closed surfaces as covering spaces, Ann. Math. Studies 66_ (1971), 81-115. 5. E. Fadell § J. van Buskirk, The braid groups of E2 and S 2 , Duke Math. J. 29_ (1962), 243-258. 6. L. Greenberg, Maximal Fuchsian groups, Bull. Amer. Math. Soo. 6£ (1963), 569-573. 7. W.J. Harvey, On branch loci in Teichmuller space, Trans. Amer. Math. Soo. 15£ (1971), 387-399. 8. A.M. Macbeath, On a theorem of J. Nielsen, Quart. J. Math, l^ (1962), 235-236. 9. C. Maclachlan § W.J. Harvey, On mapping class groups and Teichmuller spaces, Proo. London Math. Soa. 3£ (1975), 496-512. 10. W. Magnus, Uber Automorphismen von Fundamentalgruppen berandeter Flachen, Math. Ann. JL09_ (1934), 617-646. 11. D. Singermann, Finitely maximal Fuchsian groups, J. London Math. SOQ. 6 (1972), 29-38.

270 A GLANCE AT THE EARLY HISTORY OF GROUP RINGS

C. Polcino Milies University of Sao Paulo, Sao Paulo, Brazil

1.

INTRODUCTION Group rings usually appear in courses on group representation

theory as a means to gain a broader view of the subject and connect it to the general

theory of algebras and their representations (e.g. Boerner

[1] or Curtis § Reiner [8]). This may suggest the misleading idea that it was precisely this point of view that motivated the definition and study of group rings.

In fact, this is explicitly stated by several

authors who attribute the idea to E. Noether [23]. Though both topics are closely related and representation theory was actually a motivation for much of the work done in group rings, the historical order of development was rather the reverse:

interest in

the structure of group rings led to the discovery of some of the earlier theorems on group representations.

This fact was pointed out in a most

interesting paper by T. Hawkins [14] but, perhaps due to the fact that it was published in a journal devoted to the history of science rather than to mathematics itself, it seems to have remained unnoticed by those working on the subject.

Recent books and surveys fail to credit either

A. Cayley or T. Molien, and some still attribute to E. Noether the creation of the theory, omitting even the influence of R. Brauer. In this note, we do not intend to give a full account of what was done before Noether's paper, but rather describe the successive "births" of the theory.

We have included first a section describing the

mathematical circumstances in which the theory was born as an attempt to show that these ideas were a natural consequence of the mathematics of the time.

Our main sources are [5], [7], [14],[23] and, for the more

general aspects, the well-known treatise by M. Kline [19].

2.

"PRE-HISTORY" As we shall try to show, the roots of the notion of group

ring should be sought in the theory of hypercomplex systems and these, in

Polcino Milies:

Early history of group rings

271

turn, developed from the concepts of quaternions, around the second half of the past century.

Of course, all these ideas are also closely related

to the theory of matrices, which was created at approximately the same time.

The reciprocal influences among these theories are shown in Fig. 1. Complex numbers were introduced in the 16th century as a

result of the work of Italian mathematicians while studying equations of the third degree.

A long controversy regarding their existence and

meaning was raised, and they gradually gained acceptance after a geometrical interpretation was given by Wessel, Argand and Gauss.

However,

though better understood, a need for an algebraic system in which the square of a "quantity" would actually be equal to -1 was still felt.

Such

a construction was given by Sir William Rowan Hamilton (1805-1865) in 1837, when he published his paper "Conjugate functions and on algebra as a science of pure time11 in the Transactions of the Royal Irish Academy. He pointed out that an expression such as 2+3i is not a genuine sum in the sense that 2+3 is, and introduced the idea of ordered pair (a,b) to represent the complex number a+bi, developing the theory on that basis, as it is now used. Hamilton was a man of many talents and had a special interest in physics.

Thus he was well aware that his "ordered pairs" gave an

algebraic system that could be represented as plane vectors and it was clear to him that, if he could develop a similar system with "ordered triples", he would be able to deal with space vectors.

Needless to say,

this would be an invaluable tool for the study of the physical world. After several failures, he realized that he actually needed to work with "ordered quadruples" and defined a quaternion to be an element of the form a + bi + cj + dk. It was natural to define the sum of two such elements by adding corresponding coefficients.

Since Hamilton implicitly assumed that the

Fig. 1

Matrix Algebras

Complex Numbers 4Quaternions \ Hypercomplex Systems 4> Group Rings <

>

Group Representation Theory

Polcino Milies:

Early history of group rings

272

distributive law should hold, to define the product of quaternions he only needed to decide how to multiply the symbols i, j , k among themselves. The rules he gave seem quite reasonable from a modern point of view, since they closely resemble vector products: i2 = j 2 = k2 = -1; ij=k=-ji; jk=i=-kj; ki=j=-ik. However, at the time, this was a revolution, since it was the first algebraic system where multiplication was not commutative. Though quaternions never had the importance that Hamilton expected them to have in physics, this was to be a decisive discovery for the future development of mathematics and, specially, of algebra, see for example [3, Ch.32] and [20]. Relevant to our present interest is that, once quaternions were discovered, it was only natural to consider "algebraic quantities" of the same type, but of higher dimensions.

Thus

a hypercomplex system H (a finite dimensional associative algebra over the field of real or complex numbers in present-day terminology) was naturally defined to be the set of all elements of the form: x. e, + x. e. + ... + x ne 11 22 n

where xt ,Xj ,. . . ,x

are real or complex numbers and el ,e2 ,. . . ,e

which were called the units of the hypercomplex system.

are symbols

As in the case

of quaternions, the sum of two such elements is defined by adding corresponding coefficients and, to define the product, it suffices to decide how to multiply the units among themselves. Since the product of two such units must be another element of H, it is possible to write it in the form:

e. e . = £ a, (i, j)e, . 1 J

k=l

The multiplicative structure of H was determined by giving the values of the coefficients a,(i,j), which were called the structural constants of H.

Of course, this should be done so that the associative law of multi-

plication holds (though sometimes non-associative systems were considered such as the octonions, defined by A. Cayley shortly afterwards). Though Hamilton himself began the work on hypercomplex systems in a paper in the Transactions of the Royal Irish Academy in 1848 and considered biquaternions (i.e. quaternions with complex coefficients) in 1853, it was mainly his work on quaternions which raised interest in algebras.

It was precisely at this early stage when most of the basic

Polcino Milies:

Early history of group rings

273

concepts were not yet formulated, that group rings were implicitly considered for the first time, in a paper by Arthur Cayley (1821-1895).

3.

ABSTRACT GROWS

AND GROUP RINGS

Interest in permutations was first focused by the work of Joseph Louis Lagrange (1736-1813) on algebraic equations, followed by P. Ruffini (1765-1822) and N.H. Abel (1802-1829).

E. Galois (1811-1832)

was the first to consider permutation groups, using the term "group" with its actual sense in his classical work of 1830.

A. Cauchy (1789-1857)

was a pioneer in understanding the importance of permutation groups in their own right; he wrote a series of important papers in the period 1844-1846.

Influenced by Cauchyfs work, Cayley realized that the notion

of group could be formulated in a more abstract setting.

In 1854 he

published a paper entitled "On the theory of groups as depending on the symbolical equation 6 =1" in the Philosophical Magazine.

This paper is

usually regarded as the first work on abstract groups (for example in [2], [19] and [21]) and it is also there that the construction of a group ring is first given. Since the terminology of set theory was not current in Cayleyfs time, he starts his paper trying to make clear that he is working with abstract symbols rather than concrete objects such as permutations or numbers.

He states:

"... 0(f> is of course different from <(>e. But the

symbols 6,<j),... are in general such that 0 .<j>x=8(|>.x, etc.s so that Q<\>x> 8<J>X0), etc. * have a definite signification independent of the particular mode of compounding the symbols ...".

He also states:

"It is not

necessary (even if this could be done) to attach any meaning to a symbol such as 6±<j>, or to the symbol 0, nor consequently to an equation such as 6=0, or 0±<j>=O; ...". As we shall see, this remark was the key to his construction of a group ring. He proceeds to discuss some elementary properties, introduces what we now call the Cayley table of an operation and remarks: "The distinction between the theory of the symbolic equation 9 =1 and that of the ordinary equation x n =l, presents itself in the very simple case> n=4". By an analysis of the possible tables of operation he shows that there are two possible "essentially distinct" groups; in present-day terminology: the cyclic group of order 4 which "is analogous to the system of roots of the ordinary equation x 4 -l=0" and the Klein four-group.

He remarks that

this second group is "of frequent occurrence in analysis", and "it is only on account of their extreme simplicity that they have not been

Polcino Mi lies:

Early history of group rings

2 74

expressly remarked11. He then studies the possible groups of six elements showing again that there exists essentially two different cases and cites, as an example, the group of permutations of three letters.

At the end of the

paper, Cayley returns to the possibility of giving a meaning to the sum of two symbols:

"It is* I thinks worth noticing* that if* instead of

considering a, 3, etc.* as symbols of operation* we consider them as quantities (or, to use a more abstract term* 'cognitables1) such as the quaternion imaginariesj the equation expressing the existence of the group are* in fact* the equations defining the meaning of the product of two complex quantities of the form w + aa + b$ + ...". He illustrates this remark by showing explicitly how to multiply two elements of the form (w+aa+b3+CY+d6+ee), where {1 ,a,$,Y»<$,e} is the non-abelian group of order 6 he had just studied, and concludes with the remark:

"It does not appear that there is in this system any-

thing analogous to the modulus w 2 +x 2 +y 2 +z 2 , so important in the theory of quaternions". Then Cayley actually gave the formal construction of the group ring (CS3 in essentially the way we do today.

However, not even the

basic notions of the theory of rings and algebras were formulated at this time and this concept remained unnoticed.

Thus it is perhaps correct to

attribute the origin of the theory to the work of Molien (which we briefly discuss in §4) as is done, for example in [12].

4.

STRUCTURE OF RINGS AND GROUP RINGS We now return to hypercomplex systems, since their study led

naturally again to group rings.

We shall refer only briefly to Molienfs

work since it was covered by T. Hawkins in [14].

(For a general view on

the history of algebras, see also [13].) Soon after the introduction of this concept, many individual algebras were discovered and a need for a classification was felt. Benjamin Peirce (1809-1880) undertook this task in a paper which was read in 1870 and published in a lithographed form in 1871.

There, 162 algebras

of dimension less than or equal to 6 were determined.

In doing so he

introduced some of the important ideas of ring theory, such as the notion of idempotent and nilpotent element and the use of idempotents to obtain a decomposition of a given algebra.

This paper was properly published in

the American Journal of Mathematics in 1881 (which was then being edited by J.J. Sylvester) with notes and addenda by his son, Charles Sanders

Polcino Mi lies:

Peirce.

Early history of group rings

275

It might be worth noticing that in these addenda, the concept of

regular representation and a new proof of Frobenius1 theorem (showing that the only finite dimensional associative division algebras over the reals are those of complex numbers and quaternions) were given. Following the work of S. Lie and W. Killing in the study of Lie groups and algebras, E. Study and G. Scheffers, in the period 18891898, introduced some basic notions for the structure theory of algebras, though working in the non-associative case.

Finally, Theodor Molien

(1891-1941) in [22] and E. Cartan (1869-1951) in [6] obtained important results regarding the structure of finite-dimensional real and complex algebras, introducing the notions of simple and semisimple algebras and characterizing simple algebras as complete matrix algebras.

All this work

culminated in [25] with the beautiful results by J.H.M. Wedderburn (18821948) describing the structure of finite dimensional algebras over arbitrary fields.

In the proof, techniques related to idempotent elements,

as suggested by the earlier work of B. Peirce, were used. Now we briefly turn our attention to matrices.

The theory of

determinants began in the 18th century with the works of Maclaurin (1729) and Cramer (1750) in connection with the resolution of linear systems of equations, thus preceding the explicit formulation of the notion of matrix. It was again Cayley who first defined matrices in a paper in the Jour, fUr Math, in 1855 as a convenient notation to express linear systems and quadratic forms.

Only the product of matrices was of interest at first

and Cayley arrived at this concept by considering the effect of two successive linear transformations.

In a subsequent paper in the Phil.

Trans. Roy. Soo. London in 1858, he defined addition and multiplication by scalars, studying the properties of these operations without explicit mention of connections with hypercomplex systems.

It was in this latter

paper that Cayley announced the result now called the Cayley-Hamilton theorem, stating that he had verified if for 3x3 matrices and that further proof was not necessary.

Because of these papers, Cayley is

generally regarded as the founder of matrix theory.

For views contesting

his importance in the creation and development of the theory see [17] and [18]. However he observed a connection with quaternions.

He

mentioned that if M and N are 2x2 matrices satisfying M 2 =N 2 =-I and MN=-NM then, setting L=MN, the matrices M, N and L satisfy "a system of relations precisely similar to that in the theory of quaternions".

This remark was

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276

to arouse the interest of J.J. Sylvester (1814-1897) in matrices.

The

results in one of C.S. Peirce's addenda to his father's paper suggested to Sylvester in 1884 "the method by which a matrix is robbed of its cereal dimensions and represented as a linear sum".

He was referring, of course,

to the now well-known fact that a total nxn matrix algebra can be viewed as an n2-dimensional vector space.

By defining E.. as a matrix with 1 in

the i,j position and 0 elsewhere, a matrix A = (a..) can be written in the form A =

7

a..E...

Sylvester included a discussion on how 2x2 and 3x3

matrices can be viewed as linear combinations of the corresponding {E..} basis (he did not use this notation which, apparently, was introduced by Study in 1889) .

It should be noticed that this establishes a clear

connection between matrix algebras and hypercomplex systems; matrices form such a system, with units {E..}, whose structural constant are determined by the rules giving the product of these elements:

E. . E,0 = 6., E. . XJ

K>6

J K 136

Finally, let us see how this development led again, in a natural way, to group rings.

The work on the subject was undertaken by

Molien, as a setting for applying some of his earlier ideas.

He was born

in Riga, Latvia and studied at the University of Yurev in Estonia, where he submitted his thesis in 1892. Math. Ann.

He published these results in 1893 in

To obtain a structure theory for hypercomplex systems, he

considered a system H with basis {e^: 1 < i < n} and structural constants a, (i,j).

Given two elements x = £x.e., y = £y.e., their product can be

K

X 1

expressed in the form xy = I c v e v

wnere

1 X c

v

=

1 \ (i>J) 1 »3

x

- X••

Molien1s approach was to study the bilinear forms c, , 1 < k < n, which can be regarded as defining the product in H.

He also

considered the regular representation of the hypercomplex system; i.e. to each element u ^ H h e R (x) = xu.

assigned a linear function R : H •* H defined by

He used this concept to obtain a necessary and sufficient

condition for semisimplicity, showing that H is semisimple if and only if the bilinear form i|j(x,y) = trace(R

) is non-singular.

Since the

xy structural constants are particularly simple in the case where the basis {e.} forms a group, the attempts to apply his semisimplicity criterion naturally led Molien to consider group rings. In this way he was led to important results "relating to the representability of a given discrete group in the form of a homogeneous linear substitution group" in two papers published in 1897.

He concluded

that the group ring was a direct sum of complete matrix algebras thus

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Early history of group rings

277

showing that "a given substitution group can be decomposed into its irreducible components" and proposed to study "only the properties of irreducible groups".

As a consequence Molien obtained some of the basic

theorems in group representation theory, including the orthogonality relations for characters. Results in group representation theory were obtained independently by several authors.

G. Frobenius (1846-1917) studied group

determinants and group characters around 1896 and presented his first paper on matrix representations in 1897 at a meeting of the Berlin Academy.

W. Burnside (1852-1927) started to publish his work on the

subject in 1898 and H. Maschke (1853-1908) obtained his result on complete reducibility in 1898.

For more details on these discoveries the

reader should consult [14], [15] and [16]. Somehow the approaches by Frobenius and Burnside became better known than that of Molien.

However,

it should be noted that Frobenius himself refers to Molien in [10] and [11] where he introduces Frobenius algebras as a generalization of group algebras.

5.

FINAL REMARKS Group rings earned a definitive status after the connection

between group representation theory and the structure theory of rings and algebras was widely recognized.

This was mainly due to a most

influencial paper by Emmy Noether (1882-1935) which, as we mentioned in the introduction, is frequently quoted as the first work in the area. This paper was of central importance in the development of the whole subject.

In this regard, Bourbaki [2, p.156] states:

"... because of

the importance of the ideas that are introduced and the lucidity of the exposition^ it deserves to appear^ together with Stenitzt memoir on commutative fields3 as one of the pillars of modern linear algebra".

At

this point it might be interesting to recall that B.L. van der Waerden's famous book [24] is generally regarded as a landmark in the birth of modern algebra as it is understood today. participation in Noether's paper. states:

He also had a direct

In fact, in her first footnote she

"This is a free elaboration made by B.L. van der Waerden of my

winter semester course of 1927/28. publication.

We wrote this work together for

I should also acknowledge B.L. van der Waerden for a series

of critical remarks". In the introduction to her paper, Noether recalls that hypercomplex systems had received an arithmetic treatment from Wedderburn and

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278

that representation theory was developed independently of hypercomplex systems on an elementary basis by Burnside and Schur, by considering directly a given representation and working with theorems on matrices. She then proceeds to present a unified approach to the subject. article is divided into four chapters:

The

"Foundations of group theory",

"Non-commutative ideal theory", "Modules and representation theory" and "Group representations and hypercomplex systems". We do not intend to give here a detailed account of the contents of this paper.

However, we would like to mention that, in §6,

she defines a hypercomplex system as "a ring R which is at the same time a right module over a commutative field K (also known in the specialized literature as an "algebra over K") such that: 1) It is of finite rank (has a linearly independent basis u,,...,un;. 2) ab.x=a.bx=ax.b.

This is expressed as follows:

K is

commutatively linked to R. 3) The unit element e of K is also the identity operator: e

a =a for a in R". This section ends with the following remark:

An example of a

hypercomplex system is the group ring of a finite group which is obtained when we take the elements of a finite group as elements of a basis u. and we admit as a multiplication the group multiplication. an arbitrary field".

In this case K is

The concept is used again in Chapter IV where the

connection between representations of groups and hypercomplex systems is established. It should be noted that Richard Brauer (1901-1977) also played an important part in this stage of the theory, as is stated by W. Feit in [9]:

"In his years at Konigsberg his mathematical interests were centered

on the theory of representations of groups and also in the structure of algebras.

The intimate connection between these two subjects had only

recently been recognized.

This was at least partly due to his joint paper

with E. Noether [5]". We should mention that [5] was actually published before [23]. In [5] the close relationship between splitting fields and maximal subfields of a simple algebra is studied.

The paper starts with a reference

1

to I. Schur s earlier results on the subject and gathers results, without proofs, that had been obtained independently by both authors.

As is

mentioned in the second footnote to this paper, Noether had already obtained her construction of representation theory based on the theory of

Polcino Milies:

ideals and modules.

Early history of group rings

279

Further reference to this paper was made by R. Brauer

in [4] where he again credits Noether for a new approach to the subject. In §1 of [5] group rings over a field P are explicitly defined as formed "by all "group numbers" (Gruppenzahlen); i.e.* all linear combinations of the elements of the group with coefficients in P., where group elements are considered as linearly independent and multiplication is defined through the group multiplication and arithmetic rules". Since then, group rings have been an important tool in representation theory and have been used in other branches of mathematics such as homology, cohomology and algebraic topology.

More recently, they

have also been considered as interesting algebraic objects in their own right and have been the subject of active research.

REFERENCES

Boerner, Representations of groups* 2nd ed., North Holland, Amsterdam (1970). N Bourbaki, Elements dfhistoire des mathematiques* Hermann, Paris (1969). C Boyer, A history of mathematics* Wiley, New York (1968). R Brauer, Uber Systeme hyperkomplexer Zahlen, Math. Z. * 30. (1929) , 79-107. R Brauer § E. Noether, Uber minimale Zerfallungskorper irreduzibler Darstellungen, Sitz. Preuss. Akad. Wiss. (1927), 221-228. E Cartan, Les groupes bilineaires et les systemes de nombres complexes, Ann. Fac. Sci. Toulouse L2j^ (1898), 1-99. A Cayley, On the theory of groups as depending on the symbolical equation 6 n =l, Thilos. Mag. 1_ (1854), 40-47. C W. Curtis § I. Reiner, Representation theory of finite groups and associative algebras* Wiley, Interscience, New York (1962). W Feit, Richard Brauer, Bull. Amer. Math. Soc. 1^ (1979), 1-20. G Frobenius, Theorie der hyperkomplexen Grossen, Sitz. Preuss. Akad. Wiss. Berlin (1903), 504-537. G Frobenius, Theorie der hyperkomplexen Grossen II, Sitz. Preuss. Akad. Wiss. Berlin (1903), 634-645. W W.H. Gustafson, Topics in group rings by S.K. Sehgal (Review), Bull. Amer. Math. Soc. l_ (1979), 654-658. W.H. Gustafson, The history of algebras and their representations, preprint. T Hawkins, Hypercomplex numbers, Lie groups and the creation of group representation theory, Arch. Hist. Exact Sci. S_ (1972), 243-287. T Hawkins, The origins of the theory of characters, Arch. Hist. Exact Sci. ]_ (1971), 142-170. T Hawkins, New light on Frobenius1 creation of the theory of group characters, Arch. Hist. Exact Sci. Jj2_ (1974), 217-243. T Hawkins, The theory of matrices in the 19th century, Proc. Internat. Cong. Math. Vol.2* Canad. Math. Congress, Vancouver (1975), 561-570.

1. A 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

15. 16. 17.

w

Polcino Milies:

Early history of group rings

280

18. T. Hawkins, Cauchy and the spectral theory of matrices, Historia Math. 2_ (1975), 1-29. 19. M. Kline, Mathematical thought from ancient to modern times, Oxford University Press, New York (1972). 20. C.C. MacDuffee, Algebra's debt to Hamilton, Scripta Math. K)_ (1944), 25-36. 21. G.A. Miller, History of the theory of groups to 1900, Collected Works* Vol.13 University of Illinois Press (1935), 427-467. 22. T. Molien, Uber die Invarianten der linearen Substitutionsgruppen, Sitz. Vreuss. Akad. Wiss. Berlin (1898), 1152-1156. 23. E. Noether, Hyperkomplexe Grb'ssen und Darstellungstheorie, Math. Z.

30_ (1929), 641-692. 24. B.L. van der Waerden, Moderne Algebra* Springer-Verlag, Berlin (1930). 25. J.H.M. Wedderbum, On hyper complex numbers, Proc. London Math. Soc. 6 (1907), 77-118.

281 UNITS OF GROUP RINGS:

A SHORT SURVEY

C. Polcino Mi lies University of Sao Paulo, Sao Paulo, Brazil

1. INTRODUCTION Historically group rings appeared for the first time in a paper by A. Cayley [14] which is also considered by many authors as the starting point of abstract group theory (e.g. Bourbaki [5] or M. Kline [40]).

They were studied later by T. Molien [45], [46] and G. Frobenius

[27] and earned a definitive status, in connection with group representation theory, after the work of R. Brauer and E. Noether [12], [13], [51] (regarding the history of group rings see [32]). In recent times the subject gained impetus after the inclusion of questions on group rings in I. Kaplansky's famous lists of problems [37], [38].

Other important facts to stimulate the area were the

inclusion of sections on group rings in the books on ring theory by J. Lambeck [41] and P. Ribemboim [73] as well as the publication of the first book entirely devoted to the subject, due to D.S. Passman [55]. Since then several survey articles have appeared, namely those by A.E. Zaleskii and A.V. Mikhalev [84], D.S. Passman [56], [57], K. Dennis [23] and D. Farkas [25]. Also new books on the subject have been published in recent years:

A.A. Bovdi [8], I.B.S. Passi [53],

D.S. Passman [59] and S.K. Sehgal [76]. Considerable work has been done lately on the structure and group-theoretical properties of the group of units of a group ring.

In

fact, the survey by K. Dennis [23] covers most of the results obtained up to 1977 and two chapters in S.K. Sehgal's book [76] include a large amount of new material. We do not intend to give a full account of what has been done since then but rather describe briefly some of the progress made in connection with several specific problems and raise a few questions. Most of the results we shall cover here are either very recent or still unpublished.

Polcino Milies:

Units of group rings

We start by introducing some notation.

282

We shall denote by

RG the group ring of a group G over a commutative ring R with unity, by U(RG) the group of units of this ring and by V(RG) the subgroup V(RG) = {y G LJ(RG) | e(y) = 1} where e : RG -> R denotes the augmentation mapping.

As usual we shall be interested in the cases where R is either

7L , the ring of rational integers, 7L the ring of p-adic integers or a field K. Also, for an arbitrary group X, we shall denote by c(X) the center of X and by T(X) the set of all elements of finite order in X, which we shall call the torsion set of X (or the torsion subgroup, whenever this is the case).

''ALGEBRAIC11 QUESTIONS

2.

We start by considering the following question, which is listed as problem 35 in [76]:

Problem 2.1.

When is U(RG) n C c(U(RG))?

A closely related question is also stated in [17]:

Problem 2.2.

Determine those groups G such that U(RG)/c(U(RG)) is

torsion. It should be noted that it is relevant to consider these situations since several problems are either related to or lead directly to them (see [76]) .

Lemma 2.3. If G is a finitely generated group such that U(RG) is FC then U(RC)/c(U(RG)) is finite.

Lemma 2.4. Let char (R) = p, a rational prime.

If RG is either Lie-

nilpotent or Lie m-Engel then exists a positive integer n such that (RG) P

c C (RG).

Lemma 2.5. For a finite group G, it is easily seen that if G < U(RG) then there exists a positive integer n such that U(RG) n c £(U(RG)). In the case where R=2 several statements related to Problem 2.1 appear in [76, proposition II.2.14].

Also [76, theorem II.2.15]

which appeared originally as lemmas 2.2 and 2.3 in [77] gives necessary conditions.

Polcino Mi lies:

Theorem 2.6.

Units of group rings

283

Let G be a group such that U(RG) n c £(U(RG)) for some n.

Then G

c £(G) for some m, T(G) is either dbelian or a hamiltonian 2-

group.

Moreoverj for any abelian subgroup Ti of T(G) and x e G we have

that either: (i)

x centralizes Ti ^ or

(ii)

x"'tx = f 1

for all t G T , .

As a rule it is probably very difficult to give answers to most of the questions we shall consider here in the general case.

If we

do not introduce certain restrictive hypotheses, we have little information about units available.

It is likely that we would rapidly

get involved with conjectures that have been open for quite a long time now, like the possibility of the existence of non-trivial units in KG when G is torsion-free. If we make the additional assumption that G is an extension of T(G) by a torsion-free nilpotent group, it is easily seen that the conditions in Theorem 2.6 are also sufficient.

In fact in this case we

have a good description of the units [76, theorem VI.3.22]; we know that U(ZZG) = U(ZZT).G.

Thus, given y e U ( 2 G ) , we can write v = vg with

v e U(ZZT), g e G .

Then y m = v*g m , where v* e U(ZZT) and g m e c(G), and

[76, theorem II.4.1] shows that here U(ZZT) = ±T; hence y m e c(U(ZZT)). In the case where R=K a field, Problem 2.1 has been fully answered for solvable or FC groups.

If char(K)=0 the results are due to

G.H. Cliff and S.K. Sehgal [17] who also gave partial results where char(K) = p>0 discussing separately the cases where n=p r for some r or where pfn.

Theorem 2.7. Let G be a solvable or FC group and K a field of characteristic 0.

Then U(KG) n C c(U(KG)) for some n if and only if

G m c £(G) for some m and T(G) is central in G. For char(K) = p>0 a complete answer-for solvable or FC groupswas given by S.P. Coelho [19].

Theorem 2.8. Let G be a locally finite group.

Then U(KG) n C c(KG) for

some n if and only if the following conditions hold: (i)

G m c £(G) for some m;

(ii)

G contains a normal p-abelian subgroup of finite index;

(iii) either every ^-element in G is central or G is of bounded exponent and K is finite.

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284

Theorem 2.9. Let G be a solvable or FC group which is not torsion* Then, U(KG) n C £(U(KG)) for some n if and only if either KG is m-Engel for some m or the following conditions hold: (i)

G S c £(G) for some S;

(ii)

A, the set of p '-elements in T(G)., is an abelian subgroup of G.

If

A is not central, then it is of bounded exponent and for all a e A and all x ^ G there exists an integer r such that xt x"1 = tP . Furthermore, in the second case, K must be finite and [K : GF(p)] |r for all possible values r; (iii) P, the set of all ^-elements in T(G) , is a subgroup of bounded exponent centralizing A.

If P is not finite, then G contains a normal

^-abelian subgroup of finite index* We recall that those groups G such that KG is Lie m-Engel, where char(K) = p>0, are completely described in [76, theorem V.6.1}. Many of the conditions in the theorem above might look familiar to those who have worked with group rings with polynomial identities.

This is so because the techniques in [19] consist mainly in

showing that, in several situations, if U(KG) KG satisfies a polynomial identity.

C ^U(KG) for some n, then

Also, we have been told by S.P.

Coelho that these techniques have led to some progress in regard to problem 31 of [76].

Problem 2.10. Characterize groups G such that A(G) is nil. Finally, Problem 2.2 is solved in [17] for solvable or FC groups.

Theorem 2.11. Let G be a solvable or FC group and K a field of characteristic 0.

Then U(KG)/c(U(KG)) is torsion if and only if G/e(G)

is torsion and T(G) is central*

Theorem 2.12. Let G be a solvable or FC group and K a field of characteristic p>0.

Then U(KG)/c(U(KG)) is torsion if and only if

G/c(G) is torsion and one of the following conditions holds: (i)

U(KG) is torsion;

(ii)

every p '-element is central;

(iii) K is algebraic over GF(p) and every idempotent in KG is central* It should be noted that the theorems in [17] are stated for solvable groups, but the proofs of those results which we mention here are also valid in the FC case.

We remark that Problem 2.2 is still open

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285

for integral group rings and that no work has yet been done in the case where R is a ring of p-adic integers. It is shown in [75, theorem II.2.14] that if G is a finite group such that G
Hence it

seems reasonable, in this context, to consider the following. Problem 2.13. Determine all groups G such that G <3 U(RG). This appears, in a slightly more general form, as problem 8 of [23]. Problem 2.14. Which subgroups H of G are normal subgroups of V(RG)? Problem 2.14 has been considered by K.R. Pearson [61], [62], A.A. Bovdi [6], [7] and A.A. Bovdi and I.I. Khripta [9] in the case where G is a finite group.

These results were mentioned in [23]. An answer

to Problem 2.13 was recently given by G.H. Cliff and S.K. Sehgal [16]. Theorem 2.15. G is normal in U ( S G ) if and only if the following conditions are satisfied: (i)

T(G) is either abelian or a hamiltonian 2-group;

(ii)

U(Z G) = U(2ZT).G;

(iii) for any abelian subgroup Ti of T(G) and x e G either x centralizes Ti or x"*tx = t"1 for all t ^ T i . Theorem 2.16. Let G be torsion free and R an integral domain.

Then G

is normal in U(RG) if and only if every unit of RG is trivial. Theorem 2.17. Let K be a field of characteristic p > 0.

Then a non-

abelian group G with torsion is normal in U(KG) if and only if one of the following conditions holds: (i)

K = GF(2) and G=S3 , the symmetric group on three letters;

(ii) p=2 and |G'| = |T(G)| = 2; ( i i i ) T(G) is a subgroup of TL (q°°) for some prime q ± p , Gf is of order q and (a) whenever T(G) has an element of order q11., the q n -th cyclotomic polynomial $ is irreducible overl.Kj q (b) either T(G) is central in G or K = GF(2), |T(G)| = 3 or 5 and if x e G does not centralize T then t x = t"1 for all t e T.

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286

In the same line as Problem 2.14 one might ask:

Problem 2.18. Ifhich subgroups H of G are subnormal in V(RG)? As far as we know this question has been considered only by K.R. Pearson and T.E. Taylor [63], who answered it for subgroups of finite groups G over rings R such that char(R) £ 0.

Closely related is

the following: Problem 2.19« Determine the normalizers (oentralizers) of subgroups H of G in V(RG). D.B. Coleman has shown in [20] that if G is a finite p-group and char(K) = p>0 then the normalizer of G in V(KG) is N (G) = G.£(V(KG)). A reformulation of his argument actually gives a relation between normalizers and centralizers of certain groups.

Proposition 2.20. Let H be a finite normal ^-subgroup of a group G and let K be a field of characteristic p.

Then N y (H) = G.Z y (H), where Zy(H)

denotes the centralizer of H in V(KG).

3.

GROUP THEORETICAL PROPERTIES First we turn our attention to problem 37 in [76].

Problem 3.1. When is U(RG) an FC group! An answer to the question in the case where R=Z was given by S.K. Sehgal and H.J. Zassenhaus [77] and is included in [76] as theorem VI.5.3.

The same paper gave a solution when R=K, a field of

characteristic 0.

Subsequently, the case where K is an infinite field

with char(K) = p ^ 2 was studied by C. Polcino Milies [67], [68] and a final solution was obtained by G.H. Cliff and S.K. Sehgal [15] where a complete answer is stated as follows:

Theorem 3.2. If K is a finite field of characteristic p and G a group which has no p-elementSj then U(KG) is an FC group if and only if one of the following conditions holds: (i)

G is finite;

(ii)

G is abelian;

(iii) G is an infinite non-abelian FC group with finite abelian torsion and such that every idempotent in K.T(G) is central in KG;

Polcino Milies:

(iv)

Units of group rings

287

G is a non-abelian FC group with central torsion subgroup

T(G) = 2Z (q°°) X B, | B | < «>, q + p , such that Gf C ZZ (q°°) . Theorem 3.3. If K is a field of characteristic p>0 and G contains an element of order p, then U(KG) is FC if and only if one of the following conditions holds: (i)

KG is finite;

(ii)

G is abelian;

(iii) p=2, |G!I = 2 and T(G) = G1 X A where A is finite, central, of odd order.

Theorem 3.4. If K is an infinite field of characteristic p > 0 and G contains no p-elements (when p>Oj^ then U(KG) is FC i/ and only if one of the following conditions holds: (i)

G is abelian;

(ii)

G is a non-abelian FC group with finite central torsion;

(iii) G is a non-abelian FC group with T(G) central of the form T(G) = TL (q°°) X B, |B| < », G1 C I (q°°), q ^ p. field obtained by adjoining all q

1

Moreover, if k^ is i?Z£

- th roots of unity to the prime

sub field k of K t/zen [k^ n K : k] < «. The case where R = Z. . , a localization of 71 at a prime ideal (p), was studied by H. Merklen and C. Polcino Milies [43] and extended to other coefficient rings in an unpublished note by H. Merklen.

We give a

sketch of his arguments below.

Theorem 3.5. Let R be an integrally closed domain of characteristic 0, with quotient field K such that R n Q = z

for some rational prime p.

Then U(RG) is FC if and only if one of the following conditions holds: (i)

G is abelian;

(ii)

G is an FC group such that T(G) is central the subgroup ofipr-

elements in T(G) is either finite or of the form T 1 = TL (q°°) X H, q ^ p, Gf c 7L (q ) , H is finite and there exists a natural number k such that K does not contain primitive roots of unity of order q . Proof.

The proof is quite similar to the one given in [43]. To show the

necessity of the conditions, by using the arguments in [43], one needs only to show that (ii) holds. q

m

for all m>0.

It is then possible to determine, by induction, two

£ e R, such that £o = 1 and £ q - = £ ; Z e ZZ (q°°), such m m+i m m = 1 and Z q , = Z . m+1 m

sequences: that Zo

If not, K contains roots of unity of order

Polcino Milies:

Units of group rings

For £ of order q

< q

288

consider the idempotent

Following [43], it can be seen that zi is a commutator of the form zi = (x,y) and considering the units y

= e(£ ,z )x + (l-e(£ ,z ) ) ,

an easy computation shows that (y ,y) = (y f,y) if and only if e(£ ,z ) = e(£

f ,z

,) and hence all conjugates of the form y y y'1 are different

from one another, a contradiction. The proof of the sufficiency also follows as in [43] with some extra computations.

We would like to mention a closely related problem. group G, its FC-subgroup <£>(G) is defined as follows: a finite number of conjugates in G}.

Given a

$(G) = {g G G | g has

Clearly G is FC if and only if

G = *(G) and it is easy to see that U(RG) is FC if and only if G C $(U(RG)). Now we consider the group S (G) = {g £ G|g has a finite R number of conjugates in U(RG)} = G n $(U(RG)), which has been called the R-superoenter of G. This subgroup of U(RG) might be of relevance to the isomorphism conjecture for integral group rings of finite groups since it will follow from Theorem 3.8 that in the finite case, S_(G) is precisely the intersection of all the group bases of ZZG. This group was first studied, for integral group rings, by S.K. Sehgal and H.J. Zassenhaus [79] who gave a description in the case where G is either torsion, nilpotent or FC and obtained results on the torsion subgroup T(S2(G)) for arbitrary groups G.

We shall not cover

these results here since the statements are rather long and technical. See also A. Williamson [83]. In the case where R=K is an infinite field, partial results have been given by C. Polcino Milies and S.K. Sehgal [71].

It is shown

there that, for any group G, T(Sr(G)) is central in T(G). As a consequence it follows: Theorem 3.6. Let G be a torsion group and K on infinite field. S K (G) = Also it is shown:

Then

Polcino Mi lies:

Units of group rings

289

Theorem 3.7, Let K be an infinite field with char (K) = p>0 and G a group which contains a normal p-group.

Then either S, (G) is central or p=2 and

T(SK(G)) = T(c(G)) = < t > XA, where t is of order 2 and A is a finite group of odd order.

Furthermore> for the commutator subgroup (G, S^(G))

we have that (G, S (G)) = . K. It seems more difficult to describe SK(G) and T(SK(G)) if G contains no p-element.

In [71] some examples are given to illustrate the

situations that may occur. Other coefficient rings were studied by H. Merklen [42]. In the following theorems, we shall denote by B the intersection of all the group bases of the group ring RG. Theorem 3.8. Let G be a torsion group. (i)

If R is an integral domain with char(R) = 0 and such that

{0(g)|g e G} n U(R) = {1}, then SR(G) = B. (ii)

If R is a commutative ring containing a subring R1 which is a

domain of characteristic 0 with non-zero Jacobson radical and {O(g)|g e G} n u(R') = {1}, then SR(G) = C(G). (iii)

If R=R! then also T($(U(RG))) = S R (G).

Theorem 3.9. Let R be an integral domain such that no rational prime is a unit in R.

Then T(SR(G)) = T(B). As a whole, the groups T(S (G)) are not well-known yet, K

particularly in the case when R=K, a field of characteristic p>0, and G has no p-elements (not even for special classes of groups G ) . Also no information about S^fG), when K is finite, has been given.

Hence, we can

still consider: Problem 3.10.

Determine S R (G).

If G is a finite group, it is easy to see that several grouptheoretical properties are equivalent for U(2 G) : FC, solvability, nilpotence and the fact that the torsion units are trivial or that they form a subgroup.

Any of these holds if and only if G is either abelian

or a Hamiltonian 2-group (see [76] and [52]).

It has been announced

recently [29] that these equivalences are also valid if the coefficient ring is the ring of algebraic integers of a totally real algebraic number field. B. Hartley and P.F.Pickel [31] made the following conjecture, which we state as a separate problem.

Polcino Milies:

Problem 3.11.

Units of group rings

290

Show that if G is a group such that U(Z G) does not

contain a free subgroup of rank 2 then every subgroup of G is normal in G and T(G) is either abelian or a hamiltonian 2-group. It was shown by S.K. Sehgal and H.J. Zassenhaus [76, theorem VI.4.2] that this is so for solvable groups and extended by B. Hartley and P.F. Pickel [31] to solvable-by-finite groups. Nilpotent or FC groups such that the torsion units form a subgroup have been studied by C. Polcino Milies [69] and [70] . The nilpotence of the group of units was studied by J.M. Bateman and D.B. Coleman [3], P.B. Bhattacharya and S.K. Jain [4], I.I. Khripta [39], J. Fisher, M.M. Parmenter and S.K. Sehgal [26], K. Motose and H. Tominaga [48], C. Polcino Milies [65], [66] and S.K. Sehgal and H.J. Zassenhaus [78]. Since most of these results are included in [76], we shall not discuss them here. The residual nilpotence of U(Z G) was studied recently by I. Musson and A. Weiss [50] who proved, for finite groups G, that U(Z G) is residually nilpotent if and only if G is a nilpotent p-abelian group. The authors also consider the question when G is finitely generated nilpotent or finitely generated FC, but in this case the answer is rather technical and some particular cases are not covered. The study of those groups G such that U(RG) is solvable was initiated independently by K. Motose and H. Tominaga [49] and J.M. Bateman [2], who considered mainly the case where R=K, a field, and G a finite group.

Some oversights of [2] were corrected by K. Motose and

Y. Ninomiya [47] and an alternative characterization was given by A.A. Bovdi and I.I. Khripta [10]. A nice exposition of these results was later given by D.S. Passman [58].

Theorem 3.12. Let K be a field and let G be a finite group.

Then U(KG)

is solvable if and only if one of the following occurs: (i)

char(K) = 0 and G is abelian;

(ii)

char(K) = p>0 and G/(D (G) is abelian;

(iii)

K = GF(2) and if we set G = G/(D2 (G) , then G = A x

<x>

is the

semidirect product of the elementary abelian 3-group A by the group < x > of order 2, where x acts on A by inverting its elements; (iv)

K = GF(3) and G = G/©3 (G).is a 2-group having an abelian subgroup

A of index 2 so that G = (A,x).

Furthermore> either (a) A is elementary

abelian; or (b) A has period < 8 and conjugation by x maps each element of A to its cube* or (c) [G : C(G)] = 4 and c(G) is elementary abelian.

Polcino Milies:

Units of group rings

291

A first step towards the study of the solvability of the group of units of not necessarily finite groups over fields was given by S.K. Sehgal [75] who proved that if K is a field of characteristic p > 0, p j- 2,3, and G is either a nilpotent or an FC group which contains no elements of order p then U(KG) is solvable if and only if every idempotent of KG is central and T(G) is abelian. Torsion groups were studied by A.A. Bovdi and I.I. Khripta [11] who showed that if char(K) = p > 0, p i 2,3, then U(KG) is solvable if and only if Gf is a finite p-group.

This readily implies that, for

torsion groups G, U(KG) is solvable if and only if KG is Lie-solvable [76, theorem V.4.6]. These statements are not equivalent in general, not even if p > 0 and G contains a p-element, as is shown by the following example, due to J.Z. Goncalves [30]. Let G = < a,b,c|apq = 1, [a,b] = 1, [a,c] = 1, [c,b] = a P > and let K be a field of characteristic p. natural projection.

Since G/< a

q

position 4.5] shows that U(KG/< a

Let IT : KG •> KG/< a q > be the

> contains no p-elements [75, proq

>) is solvable.

Ker(ir) = 1+A»(G; < a q >) (where A U (G; < S

Since

>) denotes the ideal of G

generated by the set {x-l|x ^ < a q >}) is nilpotent it follows that U(KG) is solvable.

On the other hand Gf C < a P > is not a p-group. The known results in the case where R = Z are due to S.K.

Sehgal and are given in [76, theorem VI.4.8].

They were extended to p-

adic group rings by J.Z. Goncalves [30].

Theorem 3.13. Let R = 7L, , and assume that U(RG) is solvable.

Then T(G)

is an abelian group such that every subgroup of T(G) is normal in G. Conversely, if T(G) is as above and T(G) is a p-group with G/T(G) nilpotentj then 11(0 G) is solvable, where Q

denotes the p-adic

completion of the field of rational numbers. Also the following appears in [30]:

Theorem 3.14. Let G be a group such that T(G) is a non-abelian subgroup and G/T(G) is nilpotent and let R be the ring of integers of a totally real algebraic number field. equivalent: (i)

U(RG) is solvable;

Then the following conditions are

Polcino Milies:

(ii)

Units of group rings

292

T(G) is a hamiltonian 2-group such that every subgroup of T is

normal in G; (iii)

T(U(RG)) is a subgroup of U(RG).

Theorem 3.15. If G is a torsion group and R is a ring of algebraic integers which contains a complex root of unity> then U(RG) is solvable if and only if G is abelian. Theorems 3.14 and 3.15 show that the structure of the ring of algebraic integers does have an influence on the unit groups.

It might

be interesting to complete the study of solvability, nilpotence and the FC property in this case.

Also one might study Problem 3.11 in this

context.

4.

EXPLICIT COMPUTATIONS AND NORMAL COMPLEMENTS G. Higman in his famous paper [33] determined those groups

G such that every unit in Z G is trivial, i.e. such that U(Z G) = ±G.

As

a rule it would be useful to know explicit examples of groups of units of group rings; in other words it is natural to consider the following:

Problem 4.1.

Describe the group of units of the (integral) group ring of

a given group (or family of groups). In each particular case, once a description has been obtained, several other questions may be considered.

Problem 4.2.

Decide if every unit of finite order in V(2Z G) is conjugate

to a trivial unit.

Alternativelyj how many conjugacy classes are there

in V(Z G) of subgroups of V(Z G) conjugate to G? According to [34] this question was raised by H.J. Zassenhaus. It should be noted that it is closely related to problem 21 in [76] which, in turn, is relevant to the isomorphism conjecture and the study of automorphisms of group rings.

Problem 4.3.

Let G be finite.

Given an automorphism y : 2 G •> 2 G , does

there exist an automorphism X : G -*• G and a unit a £ U(QG) such that Y(g) = ±a g X a"11 ,, for all g £ G ? Still another interesting question is the following:

Problem 4.4.

Does G have a normal complement in V(RG); i.e. does there

exist a normal subgroup F of V(RG) such that V(RG) = F.G?

Polcino Milies:

Units of group rings

293

Clearly if G has a torsion-free normal complement in V(Z G) then G is determined by its integral group ring.

This question seems to

have been considered for the first time by D.L. Johnson [35] (notice that this paper was only published in 1978, but was received by the editors in 1973) who considered the case where R = GF(p) and showed that finite abelian p-groups and the Sylow p-subgroups of GL(n,p) have normal complements. The problem was also studied by K. Dennis [22] and the results included in his survey [23].

It is to these two papers that

subsequent authors refer when considering the problem. We return to specific examples.

The group of units of TL S3

and Z D4 were studied by I. Hughes and K.R. Pearson [34] and by C. Polcino Milies [64] respectively, who gave, in each case, an answer to Problems 4.2 and 4.3. The group of units of TL A4 was described by P.J. Allen and C. Hobby [1] who showed that all elements of order 2 in V(Z A4) are conjugate in this group, and by K. showing the following.

Sekiguchi [81] who also studied TL S4

There are 4 conjugacy classes in V f Z A O

of

subgroups of VfZA*) isomorphic to A4 and there are 16 conjugacy classes in V ( Z S 4 ) of subgroups of V(ZZS4) isomorphic to S4 (these results were also announced in [80]).

Both authors showed the existence of torsion-

free normal complements. The units of Z G for certain families of groups G have been studied recently.

D.S. Passman and P.F. Smith [60] gave an interesting

characterization of the units Z D , p an odd rational prime, and used it to show that if G contains an abelian subgroup of index 2 then G has a torsion-free normal complement in V ( 2 G ) . Also T. Miyata [44] showed that D

admits a torsion free

normal complement in V ( S D ) and proved that there exist $(n)/2 conjugate classes in V ( 2 D ) of subgroups of V(Z D4 ) isomorphic to D of the locally free class group C(Z D ) of Zft denotes Euler's function.

is odd.

if the order

As usual, $

Also it is announced in [80] that Problem 4.2

was solved for D , where n is an arbitrary positive integer by S. Endo, T. Miyata and K. Sekiguchi [24]. See also K. Sekiguchi [82]. G.H. Cliff, S.K. Sehgal and A.R. Weiss [18] have shown that if G is metabelian, i.e. contains a normal subgroup A such that both A and G/A are abelian, and if G/A is of odd order or of exponent dividing 4 or 6 then G admits a torsion-free normal complement in V ( 2 G ) . They actually give a description of a family of normal complements in several

Polcino Milies:

cases.

Units of group rings

294

It should be mentioned that K. Sekiguchi [81] also had a proof of

the existence of a normal complement when G/Gf has exponent dividing 4 or 6. Some work has been done regarding the units of integral group rings of p-groups.

F.R. De Meyer and T.J. Ford [21] have considered

group rings of cyclic groups of order p idempotent such that p.l €= U(R) .

over a ring with no non-trivial

In this context they prove that G has

f.U(R(5 U(R(£ ))))where where? ± a normal complement in V(RG) if and only if 5£ ,f P P P denotes a primitive p ^ t h root of unity. This is also equivalent to the )) (and of U(RG)) has order p n .

fact that the Sylow p-subgroup of U(R(£

P

n

Also A. Jones [36] made some observations on the explicit computation of units of integral group rings of cyclic groups of order p . Recently J. Ritter and S.K. Sehgal [74] gave a characterization of the integral group rings of the two non-abelian groups of order p 3 .

In the special case p=3 they were able to describe

the units of the group ring as a group of 3x3 matrices over 2Z [to], where u>3=l. It might be worth mentioning that many of these papers (e.g. [36],[44],[74], [80], [81]) use an exact sequence derived from a "pull back" diagram as in I. Reiner and S. Ullom [72]. We conclude by recalling problem 16 of [23].

Problem 4.5.

Let G be a finite group.

Find generators and relations for

U(ZZG). K. Dennis himself solved the problem in [23] for V ( 2 S 3 ) and V(Z D4).

However no further attempts seem to have been made in the new

examples which are now available. REFERENCES

1. P.J. Allen § C. Hobby, A characterization of units in 2Z [A4], J. Algebra 66 (1980), 534-543. 2. J.M. Bateman, On the solvability of unit groups of group algebras, Trans. Amer. Math. Soa. 15^ (1971), 73-86. 3. J.M. Bateman § D.B. Coleman, Group algebras with nilpotent unit groups, Proo. Amer. Math. Soc. 1£ (1968), 448-449. 4. P.B. Bhattacharya § S.K. Jain, A note on the adjoint group of a ring, Arch, der Math. 21 (1970), 366-368. 5. N. Bourbaki, Elements d'Tristoire des mathematiques^ Hermann, Paris (1960). 6. A.A. Bovdi, Periodic normal divisors of the multiplicative group of a group ring, Sibirsk. Mat. Z. £ (1968), 495-498.

Polcino Milies:

Units of group rings

295

7. A.A. Bovdi, Periodic normal divisors of the multiplicative group of a group ring II, Sibirsk. Mat. Z. U_ (1970), 492-511. 8. A.A. Bovdi, Group Rings (in Russian), Uzgorod (1974). 9. A.A. Bovdi § I.I. Khripta, Normal subgroups of the multiplicative group of a ring, Mat. Sbornik \6_ (1972), 349-362. 10. A.A. Bovdi § I.I. Khripta, Finite dimensional group algebras having solvable unit groups, in Trans. Science Conf. Uzgorod State University (1974), 227-233. 11. A.A. Bovdi § I.I. Khripta, Group algebras of periodic groups with solvable multiplicative groups, Math. Notes Acad. So. USSR 27,..3 (1977), 725-731. 12. R. Brauer, liber Systeme hyperkomplexer Zahlen, Math. Z. 30. (1929), 79-107. 13. R. Brauer $ E. Noether, Uber minimale Zerfallungskorper irreducibler Darstellungen, Sitz. Preuss. Akad. Wiss. (1927), 221-228. 14. A. Cayley, On the theory of groups as depending on the symbolical equation en=l, Philos. Mag. 7_ (1854), 40-47. 15. G.H. Cliff $ S.K. Sehgal, Group rings whose units form an FC-group, Math. Z. 161 (1978), 163-168. 16. G.H. Cliff § S.K. Sehgal, Groups which are normal in the unit groups of their group rings, Arch, der Math. 33^, 6 (1979), 529-537. 17. G.H. Cliff $ S.K. Sehgal, Group rings with units torsion over their center, Manuscripta Math. 33_ (1980), 145-158. 18. G.H. Cliff, S.K. Sehgal § A.R. Weiss, Units of integral group rings of metabelian groups (to appear). 19. S.P. Coelho, Group rings with units of bounded exponent over their centers (to appear). 20. D.B. Coleman, On the modular group ring of a p-group, Proc. Amer. Math. Soc. _15, 4 (1964), 511-514. 21. F.R. De Meyer $ T.J. Todd, On units of group rings, J. Pure Appl. Algebra ]± (1980), 245-248. 22. K. Dennis, Units of group rings, J. Algebra 43 (1976), 655-664. 23. K. Dennis, The structure of unit group of group rings, Lecture notes in Pure and Appl. Math. 2£, M. Dekker, New York (1977) . 24. S. Endo, T. Miyata $ K. Sekiguchi, Picard groups and automorphism groups of integral group rings of metacyclic groups (to appear). 25. D. Farkas, Group rings: an annotated questionaire, Comm. Algebra 18, 6 (1980), 585-602. 26. J. Fisher, M.M. Parmenter § S.K. Sehgal, Group rings with solvable n-Engel unit groups, Proc. Amer. Math. Soc. 5£ (1976), 195200. 27. G. Frobenius, Theorie der hyperkomplexen Grossen, parts I and II, Sitz. Preuss Akad. Wiss. Berlin (1903), 504-537 and (1903), 634-645. 28. S. Galovich, I. Reiner $ S. Ullom, Class groups for integral representations of metacyclic groups, Mathematika JJ3 (1972), 105-111. 29. J.Z. Goncalves, Group rings over totally real fields with solvable unit groups, Proc. of the 13? Coloquip Brasileiro de Matematica (to appear). 30. J.Z. Goncalves, Group rings with solvable unit groups (to appear). 31. B. Hartley § P.F. Pickel, Free subgroups in the unit groups of integral group rings, Canad. J. Math. 32., 6 (1980), 13421352. 32. T. Hawkins, Hypercomplex numbers, Lie groups and the creation of group representation theory, Arch. Hist. Exact Sci. 8^ (1972), 243287.

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33. G Higman, The units of group rings, Proa. London Math. Soc. 2_, 46 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45.

46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56.

57. 58. 59. 60.

(1940), 231-248. Hughes § K.R. Pearson, The group of units of the integral group ring ZZS 3 , Canad. Math. Bull. \S_ (1972), 529-534. D L. Johnson, The modular group ring of a finite p-group, Proe. Amer. Math. Soc. 68, 1 (1978), 19-22. A Jones, Units of 2Z C n (to appear). I Kaplansky, Problems in the theory of rings, NAS - NRC Publ. 502, Washington (1957), 1-3. I , Kaplansky, "Problems in the theory of rings" revisited, Amer. Math. Monthly 77 (1970), 445-454. I I. Khripta, The nilpotence of the multiplicative group of a group ring, Math. Notes U_ (1972), 119-124. M , Kline, Mathematical thought from ancient to modern times, Oxford Univ. Press, New York (1972). J , Lambeck, Lectures on rings and modules, Blaisdell, Toronto (1966). H Merklen, On the supercenter of a group over domains of characteristic 0, Proc. Amer. Math. Soc. (to appear). H , Merklen § C. Polcino Milies, Group rings over 7Z , . with FC unit groups, Canad. J. Math. 3£, 5 (1980), 1266-1269. T , Miyata, On the units of integral group rings of a dihedral group, J. Math. Soc. Japan 32., 4 (1980), 703-708. T , Molien, Eine Bemerkung zur Theorie der homogenen Substitutionsgruppen, S'ber. Naturforscher-Ges Univ. Jurjeff (Dorpat) ja (1897), 259-274. T. Molien, Uber die Anzahl der Variablen einer irreductibelen Substitutionsgruppe, S'ber Naturforscher-Ges Univ. Jurjeff (Dorpat) JL1_ (1897), 277-288. K. Motose § Y. Ninomiya, On the solvability of the unit groups of group rings, Math. J. Okayama Univ. JJ5 (1972), 209-214. K. Motose § H. Tominaga, Group rings with nilpotent unit groups, Math. J. Okayama Univ. 1± (1969), 43-46. K. Motose $ H. Tominaga, Group rings with solvable unit groups, Math. J. Okayama Univ. Jj>_ (1971), 37-40. I. Musson § A. Weiss, Integral group rings with residually nilpotent unit groups (to appear). E. Noether, Hyperkomplexe Grossen und Darstellungstheorie, Math. Z: 30 (1929), 641-692. M.M Parmenter $ C. Polcino Milies, Group rings whose units form a nilpotent or FC group, Proc. Amer. Math. Soc. 6£, 2 (1978), 247-248. I.B 5. Passi, Group rings and their augmentation ideals, Lecture notes in mathematics Vol. 715, Springer-Verlag, Berlin (1979). I.B 5. Passi, D.S. Passman § S.K. Sehgal, Lie solvable group rings, Canad. J. Math. 25_ (1973), 748-757. D.S Passman, Infinite group rings, M. Dekker, New York (1971). D.S Passman, Advances in group rings, Israel J. Math. 2£ (1974), 67-107. D.S Passman, What is a group ring?, Amer. Math. Monthly 83^ 3 (1976), 173-185. D.S Passman, Observations in group rings, Comm. Algebra S_ (1977), 1119-1162. D.S Passman, The algebraic structure of group rings, Interscience, New York (1977). D.S Passman § P.F. Smith, Units in integral group rings (to appear). I

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6 1 . K.R 62.

Units of group rings

298 SUBGROUPS OF SMALL CANCELLATION GROUPS: A SURVEY S.J. Pride University of Glasgow, Glasgow, G12 8QW, Scotland

The aim of this note is to discuss what appears to be known concerning the subgroup structure of small cancellation groups.

I will

begin by defining what is meant by a small cancellation group. Let X be an alphabet, and let S(X) denote the set of words on X, that is, the set of expressions x £lx £2

i 2

e •'• x / >

r >

°> e i

= ±X

> x. G X (i = l,...,r).

Equality in S(X) will be denoted by =. An element of S(X) will be called reduced if it does not contain an inverse pair x"1 x or xx"1 (x G X) , and it will be called cyclically reduced if all of its cyclic permutations are reduced. A length function on S(X) is a function I : S(X) satisfying:

£(UV) = A(U)

+

> 7L+

1

A 0 0 , &CU" ) = £(U) for all U, V G S(X) .

Length functions are completely specified by their effect on the elements of X; given a set {n

: x G X} of non-negative integers one can define a

length function x ei x e2 . . •. x

> i=l

n

£ n X

. The particular case when

i

= 1 for all x G X gives the usual length function; this will be

denoted by L. Let < X ; r_ > be a group presentation.

Without loss of

generality we may assume that r_ is symmetrized, that is, each element of r^ is cyclically reduced and r_ is closed under taking inverses and cyclic permutations.

If PTi , PT2 are distinct elements of r_ then P is called a

piece (relative to r ) .

For positive integers p, q, positive real

numbers X, and length functions H we define certain hypotheses as follows. C(p) : T(q) :

No element of r_ is the product of less than p pieces. If 3 < h < q and Ri,... ,Rh G r with R± f R T ^ , i = 1,.. . ,h (where R, . is defined to be Ri), then at least one of R.R.,, i = 1,...,h is reduced.

Pride:

C^(X):

Subgroups of small cancellation groups

299

If R E PU is an element of r_ with P a piece, then £(P) < X£(R) .

Note that C^(X) implies C(p) for X < l/(p-l).

We will write C'(X)

instead of C'(X). By abuse of language, if r_ satisfies C(p) and G is the group defined by < X ; r_ >, then we often call G a C(p)-group. of language applies for the other hypotheses.

A similar abuse

The most common hypotheses

one considers are C(p) with p > 6, C(p) and T(4) with p > 4, C(p) and T(6) with p > 3, C^(X) with X < 1/6, C^(X) and T(4) with X < 1/4.

If r_

satisfies any one of these conditions then we will call G a small cancellation group. For a history of the development of small cancellation theory the reader should consult [11]. When one comes to consider the algebraic properties of small cancellation groups, a guiding principle is that the groups appear to behave roughly like free groups, at least in the case when r_ is finite. From now on, let G = < X ; r_ > where r_ is symmetrized.

I

will discuss aspects of the subgroup structure of G for r_ satisfying various small cancellation hypotheses.

FINITE SUBGROUPS If r_ satisfies C(6), or C(4) and T(4) , or C(3) and T(6), then the finite subgroups of G are ay olio. There is some confusion concerning this result, and it seems appropriate to clear this up.

In Theorem III of [9], Lyndon gives a

description of the relation module of small cancellation presentations. In [6], Huebschmann employs this description to calculate the cohomology of small cancellation groups, and the cohomology is such that he is able to use a theorem of Serre to say what the finite subgroups are.

The

description of the relation module given in Lyndon's paper is correct, but the proof has a gap in it, because Lemmas 3.2 and 3.3 of [9] are wrong.

In a forthcoming paper [3], Collins and Huebschmann will correct

Lyndon's work.

ABELIAN SUBGROUPS In [12] Schupp shows that if r_ satis fie s C(7) , or C(5) and T(4), then the abelian subgroups of G are locally cyclic. has pointed out that the 7 and 5 here are best possible:

D. Collins if

G = < a, b, c ; a"1b"1c"1 abc > then G is a C(6)-group and the subgroup generated by ba, be is free abelian of rank 2, while if

Pride:

Subgroups of small cancellation groups

300

G = < a, b ; a"1b"1ab > then G is a C(4), T(4)-group. Recently Huebschmann [7] has shown that if r_ satisfies C(6), or C(4) and T(4), or C(3) and T(6) then any abelian subgroup of G must be either cyclic* the direct product of two infinite cyclic groups* or a subgroup of the additive group of the rationals.

SUBGROUPS OF A GIVEN INDEX Comerford [4] has proved the following interesting result. Let r satisfy a condition C(p)., T(q)., or C f (A).

Let H be a

subgroup of index k (finite or infinite)* and let § be a free group of rank k-1.

Then H*$ has a presentation satisfying the same condition or C'(X).

TWO-GENERATOR SUBGROUPS A method for analysing the two-generator subgroup structure of small cancellation groups has been developed in [5]. This method leads to the following result. If r_is finite and satisfies C'(l/14), or C'(l/10) and T(4) , then G has only finitely many conjugacy classes of two-generator subgroups which are not free products of cycles. It follows in particular that under the stated hypotheses, there are only finitely many different isomorphism types of two-generator groups embeddable in G. The proof of the above theorem is rather intricate. method of proof produces a finite set {(u^, v . )

:

The

i = l,...,N(r)} of

pairs of reduced words such that if H is a two-generator subgroup of G which is not a free product of cycles then H is conjugate to sgp{u., v.} for some i.

For example, if H can be generated by two elements of finite

order then H is conjugate to a subgroup generated by a pair of the form (P, Z ^ Q Z ) , where P m , Q n e r_ with m, n > 1 and Z is a subword of an element of r_.

(Clearly there are only finitely many such pairs.)

When the

elements of _r are known explicitly one can often determine the structure of the subgroups generated by the pairs (u., v.) and thus obtain a total picture of the two-generator subgroup structure of G. One cannot obtain a theorem similar to the above for threegenerator subgroups:

E. Rips (to appear, outlined to me in a letter) has

shown that given X > 0 there is a finitely presented torsion-free C f (X)group having infinitely many conjugacy classes of three-generator subgroups which are not free.

Pride:

Subgroups of small cancellation groups

301

D. Collins [2] has shown that, apart from a few trivial exceptions, if r_ satisfies C(4) and T(4) then G contains a free subgroup of rank 2.

A similar result for r_ satisfying C(6) has been obtained by

M. al-Janabi [8].

BAD BEHAVIOUR The construction used by Rips in the example mentioned above has also been used by him to show that the finitely generated subgroups of small cancellation groups can be rather badly behaved in other ways. Rips shows that given any X > 0 and any finitely presented group H there is a short exact sequence of groups 1

> K

> L

> H

> 1

such that L is a finitely presented Cf(X)-group and K is finitely generated (as a group).

By lifting properties of H back to L he then

deduces that, for suitable H:

there are two finitely generated subgroups

of L whose intersection is not finitely generated; there is a finitely generated subgroup of L which is not finitely presented; for any d > 3 there is a strictly increasing sequence of ^-generator subgroups of L.

d-GENERATOR SUBGROUPS Rips (in a private communication) positive results.

has also obtained some

Let r_ be finite and satisfy C(7) .

For any d > 1 there

is a finite set of non-trivial elements {gi ,g2 ,... jEvr^n^ — ^ with the following properties.

Let
of G such that A is disjoint from every conjugacy class g. (i=l,...,k(d)). Then:

(i) every group in

... O^A

is finitely generated; (iii) there is no strictly increasing

sequence of subgroups with each term in the sequence belonging to
SOME OPEN QUESTIONS (1)

It has been asked [1, Problem 16, p.642], [11, Problem 3 ] , [13,

Problem F.5, p.386], whether G is residually finite if r_ satisfies a suitable small cancellation hypothesis.

However, as far as I am aware it

has not even been demonstrated that finitely presented small cancellation groups have proper subgroups of finite index.

Infinitely related small

cancellation groups need not have such subgroups - see [10, Theorem 3.6]. In Problem F.5 of [13] it is asked whether finitely presented small cancellation groups have torsion-free subgroups of finite index.

In

Pride:

Subgroups of small cancellation groups

302

[10, Problem 4] it is asked whether, apart from a few known exceptions, finitely presented small cancellation groups have subgroups of finite index which can be mapped onto the free group of rank 2. (2)

The groups satisfying a law which can be subgroups of small

cancellation groups have not been determined.

It seems reasonable to

suggest that if r_ is finite and satisfies C(p) for large enough p then the only subgroups of G satisfying a law will be abelian or infinite dihedral. (3)

In relation to Huebschmann's theorem on abelian subgroups mentioned

above, it is not known whether a non-cyclic subgroup of the additive group of the rationals can be a subgroup of a small cancellation group. In conclusion, I thank D. Collins, J. Huebschmann and E. Rips for their help in preparing this survey.

REFERENCES

1. W.W. Boone, F.B. Cannonito $ R.C. Lyndon (eds.), Word problems, North-Ho11and, Amsterdam, London (1973). 2. D.J. Collins, Free subgroups of small cancellation groups, Proa. London Math. Soc. (3), 26_ (1973), 193-206. 3. D.J. Collins § J. Huebschmann, Spherical diagrams and identities among relations, in preparation. 4. L.P. Comerford Jr., Subgroups of small cancellation groups, J. London Math. SOG. (2), 17_ (1978), 422-424. 5. P. Hill, S.J. Pride $ A.D. Vella, Subgroups of small cancellation groups, in preparation. 6. J. Huebschmann, Cohomology theory of aspherical groups and small cancellation groups, J. Pure Appl. Alg., IA_ (1979), 137-143. 7. J. Huebschmann, Homological methods applied to combinatorially aspherical groups, to appear. 8. M. al-Janabi, Ph.D. thesis, London University (QMC). 9. R.C. Lyndon, On Dehnfs algorithm, Math. Ann.,, 166_ (1966), 208-228. 10. S.J. Pride, The concept of 'largeness1 in group theory, in: Word problems II: The Oxford Book* North-Holland, Amsterdam, New York, Oxford (1980). 11. P.E. Schupp, A survey of small cancellation theory, in: Word problems* North-Holland, Amsterdam, London (1973). 12. P.E. Schupp, Quadratic equations in groups, cancellation diagrams on compact surfaces, and automorphisms of surface groups, in: Word problems II: The Oxford Book, North-Holland, Amsterdam, New York, Oxford (1980). 13. C.T.C. Wall (ed.), Homological Group Theory, London Mathematical Society Lecture Note Series, 42_, Cambridge University Press (1979).

303 ON THE HOPFICITY AND RELATED PROPERTIES OF SOME TWOGENERATOR GROUPS S.J. Pride University of Glasgow, Glasgow G12 8QW, Scotland A.D. Vella Oxford Polytechnic, Oxford 0X3 OBP, England

1. INTRODUCTION Let F be a free group of rank m, and let N be a normal subgroup of F.

Let $(N) denote the semigroup of automorphisms of F such

that N<J> C N, and let ir(N) denote the group of invertible elements of <3>(N). We say that N is stable [6] if TT(N) = *(N).

Let G = F/N.

Each element c|>

of $(N) induces an ependomorphism $ of G, where (wN)$ = wN (w £ F ) , and $ is one-one if and only if <j> €= TT(N). to be hopfian is that N is stable.

Thus a necessary condition for G

This condition is also sufficient if

the only normal subgroups of F with factor group isomorphic to G are automorphic images of N, that is, if G has one T-system (of generating mtuples) [5, §1.1].

(If G has more than one T-system then whether or not

G is hopfian depends upon whether or not a certain category $ is a groupoid.

The subgroup N is one of the objects of $, and $(N) is the

semigroup of maps in $ which start and end at N. below.)

This will be discussed

It is not difficult to show (see [2]) that if G has I Nielsen

equivalence classes (of generating m-tuples) then |Aut(G) : TT(N)| < A. Thus when I is finite "most" of the information about Aut(G) is contained in TT(N) .

The elements of $(N) [ir(N)] are often called the free ependo-

morphisms [automorphisms] of G (for the presentation F/N). Let $(N)/Inn(F) denote the set of cosets <J>Inn(F) with $ G *(N); this is a subsemigroup of Out(F).

It is easy to see that N is stable if

<3>(N)/Inn(F) is finite. In [6, Problem 2] the question was raised as to whether the normal closure of a small cancellation subset of F is stable.

We present

a theorem here which goes some way to answering this question in the case when F has rank 2.

Our theorem also gives information on ir(N).

Denote by F2 the free group of rank 2 on a, b.

A normal

subgroup N of F2 is said to be lower triangular if each element of $(N) differs by an inner automorphism from an automorphism a 1 b l

> a

b .

> a ,

(The name derives from the fact that the image of an

Pride § Vella:

automorphism a i

> ae, b •

Hopficity of two-generator groups

> a

b

304

under the natural homomorphism of

Aut(F2) onto GL(2,ZZ) is a lower triangular matrix.

We denote the group

of lower triangular matrices in GL(2,ZZ) by LT(2,ZZ).) lower triangular subgroups will be given below.

Examples of

An element of F2 is said

to be b-alternating if it has a cyclic permutation of the form a. B. "| \ a b"1 a b. If r is a subset of F2 and a is a subword of an element i of T_, then a is called an ^-exponent in r_.

Theorem.

Let r_be a symmetrized subset of F2 satisfying the C(9) small

cancellation condition, and suppose that the a-exponents in £ are bounded in modulus by k > 0. G = F2/N.

Let N be the normal closure of r_ in F2 , and put

If N is lower triangular then:

(i)

N is stable;

(ii)

7r(N)/Inn(G) is finite unless aN has infinite order and each element

of T_ is b-alternatingj in which case 7r(N)/Inn(G) is infinite cyclic-byfinite. It follows from (i) that if G has one T-system then G is hopfian.

Also, it follows from (ii) that if G has finitely many Nielsen

equivalence classes then Out(G) is finite, except when aN has infinite order and each element of r_ is b-alternating, in which case Out(G) is infinite cyclic-by-finite. A method of computing the Nielsen equivalence classes of twogenerator small cancellation groups is given in [2], In particular, it is shown in [2] that two-generator finitely presented C'(l/14) or C'(l/10), T(4) groups have finitely many Nielsen equivalence classes of generating pairs.

Many two-generator small cancellation groups turn out to have one

Nielsen equivalence class (and thus one T-system).

Using the techniques

of [2] we hope eventually to show that if G has a two-generator finite C ! (X) presentation for small enough X, and if all the relators of the presentation are powers greater than 2 (and no relator is a power of a primitive) then G has one Nielsen equivalence class. The following example shows that the condition on the aexponents in £ cannot be dispensed with in general in part (i) of the Theorem.

For i = 1,2,... let R. = ab"1 a^b" 1 aba 1 b, and let r_ be the

symmetrized closure of {R^,R^,...} (n > 9 ) . N be the normal closure of r_ in F2 . first paragraph after Lemma 1 below). b 1

> ab of F2 then R.\p = R.

Then £ satisfies C(9). Let

Then N is lower triangular (see the If ip is the automorphism a » 1

> a,

, so ip € $(N). However RJ f Nip by small

Pride § Vella:

Hopficity of two-generator groups

305

cancellation theory. A second example shows that the condition that N be lower triangular cannot be dispensed with in general in part (i) of the Theorem. For i = 1,2,... let R. = a(ab)

and let r_ be the symmetrized closure of

{R n ,R n ,...} (n > 18). Then r_ satisfies C(9), and the a-exponents in r_ are bounded in modulus. automorphism ai

Let N be the normal closure of r_.

2

If <J> is the

1

> ba , b i

> a" of F2 then R^cJ> is conjugate to R. +1 -

Thus <J> e $(N), so N is not lower triangular.

Moreover R n £ N, so N is

not stable. We now discuss the question of finding conditions on a subset £ of F2 which ensures that the normal closure of r_ is lower triangular. We use the connection between free groups and free lie algebras.

A good

reference for this connection is [4, Ch.5]. Let F be free of rank m on at ,8^ ,...,a , and let A = A(ZZ ,m) be the free lie algebra over TL on free generators E>1 ,£2 ,... ,£ m< Let Y n (F) be the n-th term of the lower central series of F, and let A be the n-th homogeneous submodule of A. There is an isomorphism y of A (under addition) onto y (F)/y + i(F) (under multiplication) defined by: if v = v(£j,...,£ ) is a lie monomial of weight n then vy is the coset of the commutator obtained from v by replacing £. by a. and replacing lie multiplication by commutation [4, Thm. 5.12]. Let ^ be a homomorphism of F into itself. Then m s. . a. i[> = IT a.1*J mod y2 (F) for integers s. . (i = 1,... ,m). Letty°: F •* F be 1 13 j=l 3 J m s. . defined by a. ** IT a.1-* (i = l,...,m). Then it is easily proved by 1 j=l 3 induction on n that if W e

Y

(F), Wip = Wi^0 mod y + i ( F ) -

Thus the

natural

homomorphisms ip , I/J° of y (F)/y +-.(F) into itself induced by if;, ip° respectively are the same. 1

y ^ y " : A •* A

It is then not difficult to establish that

is the restriction to A of the homomorphism S(i|;): A -*• A

m defined by ^. »->- J s..^. (i = l,...,m). 1 j=l 1J 3 of F then S(i|0 is an automorphism of A.

Note that iftyis an automorphism

Now suppose N^ C N, where N is a normal subgroup of F. Then C(N n yn(F))yn+1(F))i|; C (N n Y n (F))y n + 1 (F) and so ^ maps (N O Y n (F))Y n+1 CF)

z—rn

(N n y (F))y int0

v—rn

(F)

(n = i,2,...).

Pride $ Vella:

306

Hopficity of two-generator groups

rF) Let Y

y n

n+1 (F)

previous paragraph N

for n = 1,2,... . Then by the

S(ifO C N

(n = 1,2,...).

Define JL. to be the semigroup of all automorphisms T: q i

> I t±.Z. of A such that N ( n ) T C N ( n ) for

is a homomorphism from $(N) into C,.

C N is actually a group.

For i f N l

is an induced homomorphism o f A n / N

1,2,... .

Then S

We remark for future reference that

C~— N' 11 ' then since AnT

A there n

/Nv

onto itself with kernel

From the hopficity o f A n / N ( n ) w e deduce that N f n ) T

N((n)

It is easy to see that S gives information about maps in $(N) up to automorphisms in the kernel of the natural homomorphism of Aut(F) onto GL(m,2Z). rank 2. N4,

This makes S particularly useful in the case when F has

For then Ker(AutfF)

p. 169],

> GL(m,Z)) is precisely Inn(F) [4, Cor.

and it is clear that if if; e $(N) then ii|> e $(N) for any inner

automorphism i, and ifi is an invertible element of $(N) if and only if n|> is. We now restrict considerations to the case when F is the free group F2 of rank 2 on a, b. denoted by £, n.

An element £ i

identified with the matrix

Lemma 1. (ii)

(i)

The free generators of A(2,2) will be

If C

[I!]-

> p£+qn> n i

> s^+tn of C N can be

The following lemma is almost trivial.

is finite then N is stable.

If C N c LT(2,2Z) then N is louer triangular. It is clear that if N f l ) = < a£ > (a ^ 0) then C^ c LT(2,ZZ).

We give some higher dimensional analogues of this result. For a fixed I > 0, let p = n ° K ° J£ ° •«• ° C» ji

° n» and let w.(i e I) be the non-zero simple

fold lie products of weight greater than 1 in both £ and n. known [4, Cor. 5.12(v)], P,K and the co. generate A. 2a,8,Y^(i

e

As is well-

Moreover, if

!) are integers such that ap + Ey.a). + 3K = 0, then a = 3 = 0. Here is a simple, but illustrative result:

(a) if N^

+

^ =

< a p > (a ?* 0) then C^ is lower triangular; (b) if N ^ + 2 ^ = < a u> (a i 0)

Pride § Vella: Hopficity of two-generator groups

307

where w is one of the co. then C^ is finite. We give the proof straight away. Let v = n ° £ ° ... be a nonzero simple lie product of weight j +1 in £, k +1 in n. If T: £ «

> p£ +qn, n •

> s£ + tn is an automorphism of A(2Z ,2) then

i k i k vT = detT(p J s p + I y.m. + qJt K) for certain integers y..

Thus, for (a) taking v =p we deduce that if

p q] s

xj

[p q]

G C

i J

then p s

k

w

then q =0. Ji

For (b), taking v =o> we deduce that if

|_s

rj

€ C

= 0 = q t , so either p = t = 0 or q = s = 0. There are many other results along the same lines.

Here is an

example which generalizes (a) above. (c)

If<

N

k

ap + I Y-W- > C N(Jl+2:) c < ±

1 1

P|W.(i€I)

> with a t 0, then CKT is

1

N

either finite or is contained in LT(2,2Z). To see this, note that if T =

then

(ap + £ y.03.)T = Xp + J T.O). + 3x i i for certain integers a, T., 3.

3 = det T (aq

In particular,

+31q~t+...+30,qt

),

where 3, is the sum of those y-' s f ° r which w. has weight k + 1 in n. Now K

X I

if T G C then 3 = 0 . It thus suffices to show that if H is a subgroup of if T G C then 3 = 0 . It thus N GL(2,2) contained in the set

f::] •

q

p

o_1

+ Biq

p-1

t + ... + B ^ ^ t *

then H is either finite or is a subgroup of LT(2,2Z). This can be established without too much difficulty. The result (c) implies the following. Let r be a subset of Y (F 2 ), n > 2, and let N be the normal closure of r in F 2 . Each element R of r can be expressed in the form R = [b, a, a, .... a] j ( R ) C R [a, b, b, .... b ] k ( R ) V R •—— n-1 ' ' n-1 '

Pride § Vella:

Hopficity of two-generator groups

308

where CD is a product of n-fold commutators of weight greater than 1 in K

both a and b, and V R G Y n + 1 ( F 2 ) • j (R) ^ 0 for some R in r_, C

Then

if k

CR)

=

°

£or

all R e r and

is either finite or is contained in LT(2,2Z).

The result (b) obtained above implies that if C is a simple n-fold (n > 3) commutator of weight greater than 1 in both a and b, if r_ is a subset of F2 such that each element R of r can be expressed in the form C°^ ^WD with W D e Y (F ), and if not all the a(R) are zero, then R K n+1 2 the normal closure of r_ is stable. hopfian if it has one T-system.

Thus if G = < a, b; r > then G will be

There are many other conditions which

one can impose on a set r_ which ensure that if N is the normal closure of r_ then £__ is finite.

To obtain useful information about £.. it is often

necessary to look at more than one term of the sequence (N of computing the sequence (N

) . A method

) from r_ is given in [1].

It was indicated briefly in the first paragraph of this introduction that the idea of stability is a special case of a more general idea.

We now discuss this.

For A and B groups, denote by E(A,B)

the set of homomorphisms from A onto B.

Let G be an m-generator group

Two elements f, ff e E(F,G) will be said to

and let F be free of rank m. f

lie in the same T-system if f

= i|/f0 for some ip £ Aut(F), 9 €= Aut(G).

(Using the obvious one-one correspondence between E(F,G) and the generating m-tuples of G, it is easily established that the definition of T-system just given amounts to the same thing as the usual definition [5] in terms of generating m-tuples of G.) Let f.(i€l) be a non-empty set of elements of E(F,G), and let N. = Kerf..

Denote by f. the natural isomorphism of F/N. onto G.

\p € E(F,F) with N . f C N . , by (wN.)ij; = wi|;N..

Then ip induces a map iji : F/N.

Let

> F/N. defined

Thus we have an induced map $ = (f*)" 1 $ft in E(G,G) .

The following lemma is easily established.

Lemma 2.

Let x G E(G,G), ifj e E(F,F).

Then f±\ = i|>£. if and only if

N ^ c N. and x = $. Define a category $ = $(N.(i£I)) as follows:

the objects are

the N., and there is a map from N. to N. labelled i|> if i[> e E(F,F) and l

C N.. - 3

if N ^

Notice that a map N. — - — > F i

N. in $ is invertible if and only 3

= N..

Proposition, (i) If G is hopfian then $ is a grovqpoid. (ii) If f. (iGI) includes a set of representatives for the T-systems of

Pride $ V e l l a :

Hopficity of two-generator groups

G and $ is a groupoid^ then G is Proof.

(i)

309

hopfian. Then Ker j ; = (N.ifT1 / N . ) f t ,

Let N. —*—> N. be a map in $.

so i f G i s hopfian N.if; = N. and the given map i s i n v e r t i b l e . (ii)

Let x

e

E(G,G) and l e t i G I .

e

Then f.x

E(F,G) so t h e r e e x i s t s

such t h a t f.x6 = c|)f. for some <> f G Aut(F), 0 € Aut(G). i s a map in $, and since $ i s a groupoid N.<|> = N.. By Lemma 2, x9

=

$>

an<

*

so

X i-s

an

f.

Hence N. —^—

Thus <>j € Aut(G) .

automorphism.

If f. ( i ^ I ) includes a set of representatives for the Tsystems of G then it seems reasonable to call $ a hopficity category for G (relative to F ) . Thus if $ is a hopficity category for G, then G is hopfian if and only if $ is a groupoid. It is worth mentioning that we can generalize the homomorphism S: $(N) > C^ to a functor S from $ to a category C. Define C as follows: the objects are sequences (0 ) where 1 0n is a submodule of An , and there is a map F from v (0 ) to (0J ) labelled n n T if T: £. l

> £ t. .£. is an automorphism of A and 0 T C 0 f for

n = 1,2,... . Then we define S by

( N }

(

N

j

_

In a future paper we will discuss the hopficity of various groups by investigating the structure of hopficity categories. 2.

PROOF OF THEOREM In this section jr will satisfy the hypotheses of the Theorem. If U, V are words in a, b we write U = V to signify that U and V are the same word. If W is an element of F2 we denote the inner automorphism U i > W" ! UW (U € F 2 ) by x^. We need the following results. Lemma 3. Let W be a non-trivial freely reduced consequence of r_. Then for some cyclically reduced conjugate V of W, either V e £ or there is an integer m with 0 < m < 3, and V contains m + 1 disjoint m-rermants. This is a consequence of [3, Thm. V.4.4], using an argument similar to that used in the proof of [3, Thm. V.4.5]. We will be dealing with pieces, and for this it is convenient to define the following function (the 'piece function 1 ) P from the set of words in a, b to {0,1,2,...,»}: if W is empty P(W) = 0, and for W nonempty P(W) is the minumum number of pieces in which W can be written, or

Pride § Vella:

Hopficity of two-generator groups

310

is o°. The following lemma is easily established.

Lemma 4.

If W E WtW2

... W

n

n n then \ P(W.) + 1 - n < P(W) < \ P(W.). l

x

Now w e prove the Theorem. a i

> a, b i

x

i

We denote the automorphism

> ab o f F 2 b y \|/.

Firstly, if a is not a piece then r_ is the symmetrized closure o f a set {(ab ) , b ^ } , and the result follows in this case from the theory o f free products. Secondly, suppose a € r_ for some s > 0. Then Ni|/ = N. Let 0 b e an automorphism o f F 2 with N0 C N. 8

= ii|;

Since N is lower triangular,

for some integer t and inner automorphism i. Then N = Niif;

N02s C N 8 C N ,

=

Since i|>s e Ker", it is readily

so N 0 = N , and N is stable.

shown that ir?N)/Inn(G) is finite. From now on we assume that a is a -piece and that all relators involve b.

Suppose that there is an element U of r which is not b-

alternating.

To prove both parts of the theorem it suffices to show that

$(N)/Inn(F2) is finite. is infinite.

Assume by way of contradiction that $(N)/Inn(F2)

Taking a cyclic permutation of U~

if necessary, we may

assume that n

U = l^a U 2 a

. .. U a ,

e where U. begins with b

6. and ends with b

, the exponents of b in U.

alternate between ±1, |6.| = |e.| = -e t = -6 = 1 , and 6. = e . , e

= e j ) . Now i[>* is in $(N) for some X with |x| > 2k. o1+6tX U*

= Uj a

a2+62X U2 a

W e have

o +6 X n ... U n a .

Now Uip is cyclically reduced, and is a consequence of r_. Uip £ £.

(where

Since |x| > 2k,

Then, by Lemma 3, for some integer m with 0 < m < 3, Ut[>

when

written on a circle contains m + 1 disjoint m-remnants V. of elements R^ of £ .

In particular, P(V.) > 6 since P(R.) > 9. Now again because

a. ,+e.X a.+6.X |x| > 2k, V . must b e a subword o f a -1" J U . a J J for some j with 1 < j < n (where a 0 = a n ) . So V. = a B.a

where B. begins and ends

with b-symbols, and the B. are disjoint subwords of U when written on a circle.

Pride § Vella:

Hopficity of two-generator groups

311

Suppose first that no B. is a piece, so that P(B.) > 2. B. occurs in both R. and U, R. is conjugate to U.

Thus Rt = VlC with

P(C) < m and the other B., i ^ 1, disjoint subwords of C.

m > P(C) >

Since

m+1 I P(B.) + l - m > 2 m + l - m X 2

By Lemma 4,

= m+l,

a contradiction. Since P(aYl)> £(a Y l ) < 2,

Now suppose that Bj is a piece. £(Vi) ^ 5 by Lemma 4, again a contradiction.

To complete the proof of the Theorem, suppose that all elements of £ are b-alternating.

Then (i) holds (see the proof of [6,

Thm. 2]). To verify that (ii) holds, first observe that Tr(N)/Inn(F2) is infinite cyclic-by-finite (since ty € TT(N)).

We show that

Ker" = {i w : w e N}, from which it follows that ir(N)/Inn(G) = ir(N)/Inn(Fa) . When N is contained in the derived group of F 2 , the fact that Ker" = {i w : w e N} follows easily from the fact that G is centreless. However, to prove the result in general, we seem to require the fact (which can be shown without too much difficulty) that sgp{aN} is malnormal in G. Let a i belong to Ker".

> w-'aSv, b i

> w^a^'w

(|e| = |e'| = 1, w e

F2)

Then

w" 1 a £ w E a

mod N,

w^aV^'w = b

(1)

mod N.

(2)

By (2), ef = 1 (since ab(R) = 0 for all R G N ) , normality of sgp{aN} in G, w = a aN has infinite order in G.

a

By (1) and the mal-

mod N for some a.

Moreover, e = 1 since

Then (2) becomes

, -i -a+JL -a , b a b = a mod %T N. Now bN does not belong to sgp{aN}, so by malnormality, a"a = a" a+ mod N.

Hence w = a a = 1 mod N, and a

order in G, I = 0.

= 1 mod N.

= 1

Since aN has infinite

This completes the proof.

REFERENCES

1. K.-T. Chen, R.H. Fox § R.C. Lyndon, Free differential calculus IV,

Ann. of Math. 6£ (1958), 81-95.

Pride § Vella: Hopficity of two-generator groups

312

2. P. Hill, S.J. Pride S A.D. Vella, Subgroups of small cancellation groups, in preparation. 3. R.C. Lyndon § P.E. Schupp, Combinatorial group theory3 Springer-Verlag, Berlin, Heidelberg, New York (1977). 4. W. Magnus, A. Karrass § D. Solitar, Combinatorial group theory: Presentation of groups in terms of generators and defining relations3 Dover, New York (1976). 5. S.J. Pride, On the generation of one-relator groups, Trans. Amer. Math. Soa. 2_1£ (1975), 331-364. 6. S.J. Pride, On the hopficity and related properties of small cancellation groups, J. London Math. Soc. U_ (1976), 269-276.

313 THE ISOMORPHISM PROBLEM AND UNITS IN GROUP RINGS OF FINITE GROUPS K.W. Roggenkamp University of Stuttgart, D-7000 Stuttgart-80, West Germany

This is a report on joint work with L. Scott.

I would like to

take this opportunity to thank the University of Virginia for their hospitality in March 1981, when most of these investigations were done. Let G be a finite group and R a commutative noetherian ring with identity.

By RG we denote the group ring of G over R, and U(RG)

denotes the units in RG.

Whenever I is a two-sided ideal in RG we denote

by V,(RG) the congruence subgroup of U(RG) with respect to I ; i.e. Vj(RG) = {1 + x : x e 1} n U(RG) .

(1)

In particular, for the augmentation ideal g p , i.e. the kernel of the augmentation map e : RG -»- R, we have that V^ (RG) = V(RG) are the normalized units.

We shall write small german letters for the

augmentation ideals of the corresponding groups. A question raised by K. Dennis in 1976 [5] has recently obtained much attention, [1],[4],[7] and [8]: (2)

When is the natural injection \ : G -> V(RG) split, i.e. when does

there exist an epimorphism $ : V(RG) •*• G with i<|> = id G ? Apart from interest eo ipso* an epimorphism V(RG) •> G has a considerable impact on the isomorphism problem: (3)

}Ihen does RG = RH imply G - H? Note that there is no loss of generality if we assume that

q

= f) R i-

e<

RG = RH augmented - as we shall always do in the sequel. Let us start with a little observation from which we derive

the consequence (5). (4) Let <j> : V(RG) •> G be an epimorphism H n Kerc|> = 1, then G « H.

with i<j> = id G -

If RG = RH and

Roggenkamp:

(5)

Units in group rings

314

Let G be a nilpotent group and assume that no prime divisor of |G|

is a unit in R and charR = 0 .

If there is an epimorphism <\> : V(RG) •* G

with i(J) = id G then RG = RH (augmented) implies G - H. Proof.

We shall apply (4) and assume Ker(() n H ± 1.

Since H is nilpotent,

G being nilpotent, there exists 1 ± x e Kercf> n Z(H), where Z(H) stands for the centre of H.

On the other hand by a result of Passman [9], class-

sums of G and class-sums of H coincide [15] but - the identification RH = RG being augmented - even the lengths of the class-sums coincide, and so Z(G) = Z(H) inside RG, i.e. Ker<|> n z(H) = 1, a contradiction.

Remark 6.

Let G be nilpotent with centre Z (R as in (5)), then V

is torsion free ([6]) where c * ^1

G

= RG ®

D7CD,

KZ, K

°

t G(RG)

R

and so by induction V(RG)

has a filtration I < Vt < V2 < . .. V n = V(RG) with V ^ V ^

= Z± x T ,

where T. is torsion free and Z. is inductively defined as Z. = Z(G/Z. - ) . But it is not obvious to us, that a splitting <J> : V(RG) -> G necessarily has Ker<|> torsion free.

If so, this would very directly answer the

isomorphism problem in this case; and in general, a most elegant answer to the isomorphism problem is given, if there exists a splitting (J) : V(Z G) -> G with Ker(J) torsion free.

(7)

Hence it is natural to ask:

If there is a splitting

: V(RG) •* G, char R = 0 , when is Ker<J> torsion free? In this direction there is the following more or less satisfactory result.

Theorem 8 ([4], [6]).

Let G be a metabelian group with abelian normal

subgroup A and abelian G/A = G of odd order.

Then

(i)

there eodsts a "canonical" splitting $ : V(Z G) -> G,

(ii)

Kercf) is torsion free. The first part was proved by Cliff-Sehgal-Weiss [4] in 1980

and the second part by Jackson [6] in 1968.

(The paper of Jackson was

brought up to date by Miyata (unpublished) and in [11].) Later we shall elaborate on the results of [4]. We just remark here:

Roggenkamp:

Units in group rings

315

(i)

The splitting in Theorem 8(i) is by no means unique.

(ii)

If G is even, there still exists a splitting in case there exists

an abelian normal subgroup A with G = G/A abelian of exponent 2,4,6. (iii)

If G = A

^j G is the semi-direct product of two abelian groups

and the homomorphism y : G -> Aut(A) induced from the conjugation action of G on A has as image a maximal abelian subgroup of Aut(A), then there always exist splittings. (iv)

Note that the isomorphism problem for metabelian groups over 7L

has a positive answer (Whitcomb 1968 [17]), and so (8) does not contribute to the isomorphism problem, except that it presents a very instructive solution. From (8) it appears that a splitting of the units occurs quite frequently for metabelian groups.

The situation is quite different

for perfect groups, as the following shows.

Theorem 9 (Roggenkamp-Scott). Let G be a perfect group such that no epimorphio -image of G is isomorphio to PSL(n,q) for an integer n and a prime power q; assume in addition^ that the Sehur-indices of G are one. Then there is no splitting : V(Z G) •* G. We hope later to largely remove the assumption on the Schur-indices. I shall sketch the proof for the sake of simplicity only for G simple (e.g. G =31 , n > 5 ) . (a)

By passing to a big splitting field K - first one goes to a

splitting field and then one has to kill certain ideal classes - one showss using Bass-Milnor-Serre's congruence subgroup theorem [3], that for an assumed splitting cf> : V(2 G) -*- G :

N 3 V

| G | 2 Z G

(2

G)

i.e. N contains a genuine congruence subgroup. (b)

Let QG = n (IC)

, K. algebraic number fields with rings of integers

R. .

Then 7LG can, up to "ideal conjugation", be embedded into TT(R.)

So, in order to avoid technicalities, we assume 2GC (c)

H(Ri)n

and so V(2 G) C IT GL(n i> R i ) .

Since GL(n. ,R.)/SL(n. ,R.) is abelian and G is perfect induces an

epimorphism <j>i : SL(ZG) •+• G, where

Roggenkamp:

316

Units in group rings

SL(ZG) = V ( 2 G ) n n SL(n.,R.). (d)

Let o. = |G| 2 R. , and let a. = p » - ^

be its prime decomposition.

The

functoriality of SL then gives an epimorphism

ai moreover, the kernel of p.. is nilpotent. (e)

Now V i - p ^ p C Z G ) C Kerc(), and G has no nilpotent normal subgroups.

Hence if r(n. ,p. .) = Ker(SL(n. ,R.) •> SL(n. ,R./». .)) then n r(n. ,p. .) n G = 1.

So, i f SL(2 G) is the image of SL(ZG) in

n SL(n. yR^/p. .) then <>| factors

via SL(ZZG). (f)

Since u € Z G is a unit in 2 G if and only if it is a unit in any

maximal order containing it, we conclude that (J> factors via SL which is a subgroup of a product of certain SL(m. ,f.) such that the projection onto each factor is surjective. (g)

G is perfect and so in the above we can pass to the commutator sub-

group, and thus conclude that G must be isomorphic to a PSL(n,q) for some n,q. Let me demonstrate the situation for G =9I5 = PSL(2,4).

(10)

(Roggenkamp-Scott).

Let R = TL [—%y

and R/2R = F 4 . The group ring 7Z%

] then 2R is a prime in R,

has the form given in Fig.l.

The

blocks will be numbered 1 :

Q

3 : (K)3

(Q),

4 :

(Q)4 •

In block 2

, a,b,c,d e z ,

TL

We have in addition the following congruences.

The entries will be

labelled a.., k for the block. (a)

a = d(2), b = c(2) .

7L

Congruences modulo 2-^poWers: i\x = z2n mod(4)

z2n

= zSn mod(2).

Roggenkamp:

Units in group rings

317

Moreover, we have pullback diagrams

zz

'zz

R

zz zz TL

XTL

IF4

These congruences will be between the block 2 and 3 at the positions 3 2,2

'

3 2,3 '

3 3,2

;

3 3,3 *

We also have pullback diagrams: TL ®

TL =

ZZ

TL

R ©

Fig.l 1 •

ZZ

2

TL

TL

TL

TL

TL

6 TL

TL

TL

TL

TL

6 TL

TL

TL

TL

TL

6 TL

TL

TL

TL

TL

6 7L

TL

TL

Z

2

TL

3

TL

X X

TL

TL

5 TL

TL

TL

TL

5 TL

TL

TL

TL

5 TL

TL

TL

TL

X

X

TL

TL + 2R

R

R

2R

R

R

2R

R

R

Roggenkamp:

Units in group rings

These congruences will be at

1 2,

' and .

•

318 .,

. and

2 2

and at

the transposed positions. (b)

Congruences modulo 3: zln

E z2n mod(3).

The block 4 is at each entry congruent to the entries . ., i > 1, j > 1 1 >3 modulo 3. (c) Congruences modulo 5: 1 4 Z

E Z

4 The block 3 is at each entry congruent to the entries . ., i > 1, j > 1 i»3

modulo 5. Now one sees that Z9I5 has an epimorphic image - from the 3 block - to GL(2,4), and this induces a splitting V(Z9ls) -* PSL(2,4) -9ls . We shall now turn to our observation (4) and apply it to modular group algebras.

But before we do so, we shall make another

general observation. (11) Let <\> : V(RG) -* G be a splitting (R is again arbitrary), and assume that for no x e Ker(f>, x of finite order, RG as R<x>-module is free, then if RG = RH we have G - H. In fact, assume x e Ker n H, then RG = RH must be free as R<x>-module.

Whence the result follows from (4).

Theorem 12 (Roggenkamp-Scott). Let I = 2Z /pZ and let 31 be a finite dimensional nilpotent I-algebra satisfying (i)

9l p = o,

(ii) if x,y e Z(9fl, the centre of%

then x-y = 0. Put G = 1 +91, if

fG * lH, then G - H. Remarks.

(a)

It was proved by Passi-Sehgal [8] in 1972, that fG = IH

for G nilpotent of class at most 2 and exponent p, implies G - H.

But

such a group G is of the form G = 1 + 91 for 91 as in (12) (cf. [2]). Hence our result extends [8]. (b)

For groups of the form G = 1 +91, where 91 is a finite nilpotent ring,

the isomorphism problem was shown to have a positive answer over 2Z by Sandling in 1974 ([14]).

For such a G, there obviously is a splitting

: V(ZZG) + G and it was shown by Passman-Smith in 1980 [10], that Ker is torsion free, thus reproving Sandling1s result.

Roggenkamp:

Proof of Theorem 12.

Units in group rings

319

For rings, groups etc., we denote by Z(-) the

centre. Because of the special structure of G, we have a splitting 1

+

Vj(G)

•> V(lG) - ^ G

-> 1,

(12.1)

which is induced from the ring homomorphism

The sequence (12.1) induces by restriction to the centre, using (ii), the exact sequence 1

-> V

(!Z(G)) -> v

(IZ(G)) - A U

Z(G)

+

1.

According to Ward [16], the map PG :

ZCfG)

+

7 a K x where K

x

IZ(G)

I—>

x

I |KX|=I

a K , X

X

denotes the class sum of x, is a ring epimorphism with kernel

Z(lG) n [IG], and [tG] is the f-algebra generated by ab - ba, a,b e I G . Moreover, p G splits the natural injection fZ(G) -^ Z(lG). We now use condition (i) on 91 to show that *| v = 1, V 9(Z(lG) n [IG])

(12.2)

where g(Z(lG) n [fG]) = g f n (ZClG) n [ I G ] ) . Assume now IG = lH and let 1 / h e Ker<() n H. assume h is central, H being nilpotent.

Then we may

Because of (12.2), h

where

By applying (11), we get a contradiction.

Hence Kerifi n H = 1 and G = H

by (4).

Let me conclude with some remarks on the splittings for metabelian groups, as constructed in the integral case by Cliff-SehgalWeiss ([4] cf. also [12], [13]). 1 + A

+

G

+

Let

G"->1

(13)

Roggenkamp:

Units in group rings

320

be a metabelian group, with A and G abelian, and let R be a ring such that A and G are R-modules with R ®

A - A, and R &_ G* = G" (e.g. R = 2Z

or A and G are elementary abelian p-groups and R = 7L /ipTL ) . Then we have a split exact sequence 1

•+ V__

(RG) •• V_

2

9R

(RG) -> G" ->• 1,

(14)

9R

and so V(R<3)

=

V_

(RG)

H

«

G" x V_ 2

(RG).

9 R

One also has a natural anti-involution on RG * :

RG

->

Irgg

—>

Put V,*(RG) = {1+x e V

RG

Irgg-.

(RG); X = x*}, and V*(RG) = G" X V*(RG).

result of [4], if R = ZZ , then V*(ZG) = V ( 2 G ) .

By a

In particular, if |(f|

is odd, then V*(2G) has a natural complement to G. We shall next exploit the various ways of making A into a two-sided RG-module, such that the natural conjugation action of G on A coincides with the antidiagonal action.

(This is the crucial idea in the

paper of Cliff-Sehgal-Weiss,) Let .A. be the 2-sided RG-module with g °. a = g a g where 1 x J _ -!_-; the latter is the natural conjugation action, and a °. gf = g' J a g . We now consider only the modules J ^

=:

A(i),

i G ZZ/exp(G).

These have the property -1 g ° a ° g

=g

1-i i -i -(1-i) g a g g"-

-1 = g a g .

So the antidiagonal action is still conjugation.

(15) (c.f. [4]) The map

a

i

:

a

RfG

+

induced by (a-1) •—> a extends to a two-sided RG-homomorphism, The k e r n e l o f CK we d e n o t e by J ( i ) .

Roggenkamp:

Remark 16.

Units in group rings

Two of these maps are well-known:

kernel ClR9R

anc

^

a 0

with kernel 8nCtR«

321

ox : 0 R t

•* A, which has

These are the only homomorphisms

where the left-multiplication (right multiplication) on a D t K to the conjugation action on A.

is carried

These two maps each induce a natural

isomorphism H2 (G,A)

-

Ext RG (gR,A)

depending on whether g~R is viewed as left or as right module. The situation is quite different, if we pass to units.

Each

j. induces a homomorphism o i *:

H2(G,A)

H 2 (V(RG), A(i)}

+

in the following way:

gives the exact sequence + aJG

1

-> RE -* RG* -> 1

K

and so 1

•> V

(RE)

-*- V(RE)

+

V(RG)

+

1 .

Now each a. induces a group homomorphism V

r (RE)

-> A(i),

which, because of the construction of a., is G-equivariant, under the natural action of V(RG) induced from the conjugation action of V(RG) on V

G (RE),

whence the map

a *: I

H2(G,A)

+

H2 (V(RG), A ( i ) ) .

On the other hand the natural inclusion i: G" -> V(RG) induces a homomorphism i*:

H 2 (V(RG), A(i))

+

H2 (G,A),

such that

V l * = idHJ(G,A)The splitting of unit groups is closely related with these homomorphisms a., and the above observations.

The following result was obtained for

Roggenkamp:

Units in group rings

3 22

R = ZZ by Cliff-Sehgal-Weiss, and for R = Z / p Z by Scott and myself. Theorem 17. Assume G is abelian and odd3 and put s = ' I* . We have the natural sequence 1

+ V

r (RG)

+ V(RG) - ^ > V(RG) + 1,

V ana* pu£ = x"1 (V*(RG)).

V*(RG)

Then (i) V^fRG) acts trivially via conjugation on V we form the pullback 1

+

A

->

II E

o

:

1

->

A

G

-*

t •*

E,

G"

+

1

->

1

(RG). ( i i ) If

t ->

V*(RG)

t V,*(RG)

and the pushout 1

-> V

r(RG)

-> V*(RG)

|as Ef:

1

+

A

V*(RG)

| ->

E2

T/zen IE0 and Et are equivalent> and we get a <j>: V * ( R G )

+

-• 1

II -> V*(RG)

•> 1

splitting

-> G

induced from the homomorphism a g .

Remarks 18.

(a) Keeping in mind that for metabelian G, V*(ZG) = V(2 G ) ,

this gives for odd (T the splitting constructed in [4] . (b) The existence of splittings depends strongly on the choice of the homomorphism c^. (i)

For example (see [12], [13]);

for G = Cjj

^ cs > t n e Frobeniusgroup, a splitting can be achieved

in the above fashion (the important ingredient here is Theorem 17(i)) only by a i for i = 2,3,4, (ii)

for C 7 3

(iii)

for C 2 4 1 j C,o no o. satisfies Theorem 17(ii).

1 C8 no a. satisfies Theorem 17(i), That means that

for the above groups in (ii) and (iii) no splitting can be obtained using the above construction.

Roggenkamp: (c) Let G = C

xC

P

P

xC

Units in group rings j C , where the C

P ^ P

1

1

0

0

1

1

P

323 acts via

Then G is not of the form 1 + 9 1 for a nilpotent 7L /ipTL -algebra 9t; so there is no natural splitting.

But for p = 3, V*(ZZ/pZG) = V(Z /pZ G) and so

(17) yields a splitting of the units of IF G. (d) There does not seem to be any reason why an arbitrary splitting <\> : V(Z G) -> G should be related to a splitting coming from the homomorphisms a. as above.

More surprising is the following result.

Theorem 19 (Roggenkamp-Scott [12], [13]).

Let G = C

^ C be a meta-

oyolio Frobeniusgroup3 p a prime p > 5 and m > 2, m even. If there exists a splitting <J> : V(Z G) ->• G, then there also exists a splitting coming from a homomorphism a^ as in (17) for some i. Corollary 20. The groups C 73 ^ c 8 an(^ splitting of the unit group V ( 2 G ) . Remarks on the proof of (19).

c

24i

1 Cio

not

a ow

^

a

Let us write A = C , B = C . Then our

group G is given by the split exact sequence

Let R be the ring of algebraic integers in Fixr

Q(*Vl) , where C

is

m viewed as a subgroup of the Galois group of Q(*Vl); because m is even, R is real.

Let p be the unique prime above p R. We have the exact sequence 0

-* at

-> 2 G

*> 2 B

and if we put

P

*J=.K *

then e is an idempotent in QG, and

•> 0,

Roggenkamp:

Units in group rings

324

TL G e =

(i)

p

R^

R

(ii)

IP

PJ p

R

R

.' •' • J

(iii)

P P

2

* P P

P

V r (ZG) = {1+x, x e a t b , det(l+x) e u(R)} . at G

(iv)

It is easily seen that one has a G-equivariant isomorphism. (21)

(a)

V (2G)/V (2G) G at at 1 + x

moreover, 2 Ge/at Jt,...,J

c

at a

^ at / a t

«—>

atG/a2tG x ,

is semi-simple, and so we have ideals

and G-equivariant homomorphisms (cf. (15)) : V

(2G)

at G

A.

(b) The composition factors of at / at isomorphic to A. (c) If in ZZGe, we have

under B-conjugation action are

v,(g)

then v. v i + 1 modulo p is the conjugation representation of B on A. This is proved by computation.

Roggenkamp:

(d)

Units in group rings

325

Since F B i s semi-simple, v . ( x ) for x e F (B) (modulo p) are t h e

various non-isomorphic r e p r e s e n t a t i o n s , and so one may, more or l e s s , assume v. :

c -*• a , where C = < c :

c

= 1 > and a i s a p r i m i t i v e m-th

root of u n i t y in IF . An a n a l y s i s of t h e proof of (17) and (21) y i e l d s : (22)

There exists

if there exists

a splitting

coming from an ideal J. (17) if and only

an i 0 such that for every u e V(Z B),

v. (u) vT1 (u) 1

1

0

0

modulo p has order dividing m.

(23)

Now easy computations show that

(i)

for C 7 3

(ii)

for C 2 4 1 ^| C 1 0 , if v = c - c5 + c9 and if u = - 372099 + 114985c +

^ C8 , u = 2 + c - c3 - c4 - c5 + c7 does not satisfy (22);

301035c2 - 301035c3 - 114985c4 + 372100cs - 114985c6 - 301035c7 + 301035c8 + 114985c9, then their product does not satisfy (22). We shall now indicate why any splitting must essentially come from an ideal.

(24)

Assume $: V(G) -> G is a splitting.

Then (J> makes the following

diagram commute:

1

-*•

A

+

1

+ V r(7LG) at G

G

-> V(ZZG)

-*

B

-* 1

+ V(2B) -* 1

In f a c t , V(2ZG)/V r (2ZG) i s a b e l i a n , and so (j)' = <>j I n^ has image OtG atG A ^ Cj a Frobeniusgroup; but V G(2ZG) = Tf ^ A with Tt t o r s i o n f r e e . So A i s both t h e kernel and the complement of a Frobeniusgroup; i . e . Ct = 1, and <> f makes t h e above diagram commutative. (25)

We put W.(G) = {x e V r(7ZG) : det x = 1}. Then A atG

V r (ZZG) = U(R)«V/A(G). A at G

Moreover, W A (G) contains a congruence subgroup. A

In fact, since A is a cyclic p-group, it follows from (ii) that E (p 2 ), the subgroup generated by p2-elementary matrices, is

Roggenkamp:

contained in Ker<J>' .

Units in group rings

3 26

Since W.(G)Ker(}>' is odd and since r (p 2 ), the

congruence subgroup, is contained in W^(G), we can apply the Bass-MilnorSerre congruence subgroup theorem:

R is real and so the only units of

finite order in R are ±1; i.e. |r (p 2 ): E (p 2 )| < 2.

Since A is odd,

r m (p 2 ) < Kercf.'.

(26)

finally

Using calculations inside SL (R/p2) one shows W

V

r (ZZG)

< Ker* 1 .

r (G)

< Ker! and

2

m

n 1

or

Because of (21) (a), Ker<j>" = To must act t r i v i a l l y on Ker!, but a l l quotients of V r (2G)/V r ( 2 G ) are obtained from one of the

OtG a tG ideals J. in (21)(a). Hence there must also exist an ideal J. such that 1 1Q ~ T Q acts trivially on V g/Vj > and we can use (17) to construct a o splitting by an ideal. Thus C 43 their unit groups. C

24 3

3 C2

eacn

^ C8 and C 2 4 3

^ Ci0 do not allow splittings of

It should be noted that C 2 4 3

^ C 2 4 1 , C243

^ Cs and

allow splittings.

REFERENCES

1. P.J. Allan $ C. Hobby, A characterization of units in 7L [A4 ] , Abstract AMS 1_, 773-20-10. 2. J.C. Ault § J.F. Walters, Circle groups of nilpotent rings, Amer. Math. Monthly 8£ (1973), 48-52. 3. H. Bass, J. Milnor § J.P. Serre, Solution of the congruence subgroup problem for SLn(n>3) and Sp ? (n>2), Inst. Routes Etudes Sci. Publ. Math. 33_ (1967). 4. G. Cliff, S.K. Sehgal § A.R. Weiss, Units of integral group rings of metabelian groups, to appear. 5. K. Dennis, Structure of the unit group of a group ring, in Proc. Ring Theory Conference, Univ. Oklahoma, Marcel Dekker (1976). 6. D.A. Jackson, The groups of units of the integral group rings of finite metabelian and finite nilpotent groups, Quart. J. Math. Oxford 20_ (1969), 319-331. 7. T. Miyata, On the units of the integral group ring of a dihedral group, J. Math. Soc. Japan* 32_ (1980), 703-708. 8. I.B. Passi § S.K. Sehgal, Isomorphism of modular group algebras, Math. Z. 129_ (1972), 65-73. 9. D.S. Passman, Isomorphic groups and group rings, Pacific J. Math. jL5_ (1965), 561-583. 10. D.S. Passman § P.F. Smith, Units in integral group rings, preprint. 11. K.W. Roggenkamp, Units in integral metabelian group rings I, Jackson's unit theorem revisited, Quart. J. Math. Oxford, 32_ (1981), 209-224. 12. K.W. Roggenkamp § L.L. Scott, Units in metabelian group rings II: Nonsplitting examples for normalized units, Quart. J. Math. Oxford, to appear.

Roggenkamp:

Units in group rings

327

13. K.W. Roggenkamp § L.L. Scott, Non-splitting examples for normalized units in integral group rings of metacyclic Frobenius groups, C. R. Math. Rep. Aoad. Soi. Canada* 3_ (1981), 29-32. 14. R. Sandling, Group rings of circle and unit groups, Math. Z. 140 (1974), 195-202. 15. S.K. Sehgal, Topics in group rings* Marcel Dekker, New York, Basel (1978). 16. H.N. Ward, Some results on the group algebra of a group over a prime field, Harvard seminar on finite groups (1961-62). 17. A. Whitcomb, The group ring problem* Ph.D. Thesis, Univ. Chicago (1968).

328 ON ONE-RELATOR GROUPS THAT ARE FREE PRODUCTS OF TWO FREE GROUPS WITH CYCLIC AMALGAMATION G. Rosenberger University of Dortmund, Dortmund, West Germany

INTRODUCTION Let G = < ai ,. . . ,a ,bi ,. . . ,b

| wv = 1 >, 2 < p, 2 < q, where

1 ^ w = w(ai ,...,a ) is not a proper power nor a primitive element in the free group Hi = < ai ,...,a ; > and 1 ± v = v(bi,...,b ) is not a proper power nor a primitive element in the free group H2 = < bi,...,b ; >.

The

group G is of great interest both for group theory and for topology (see [2] and [8]). We are concerned with the one-relator presentations of G and the solution of the isomorphism problem for G. prove Theorem 3.19:

In this paper we

If p = q = 2 and {xi,...,x4} is a generating system

of G, then {xi,...,X4} is freely equivalent to a system {yi,...,y4) with {yi > • • • >V4 } c Hi u H2 .

Moreover, for {xi ,... ,x4 } there is a presentation

of G with one defining relation.

Also, G has only finitely many Nielsen

equivalence classes of minimal generating systems, and we can decide algorithmically in finitely many steps whether an arbitrary one-relator group is or is not isomorphic to G. This result stands in contrast to the corresponding results in [4] and [19]. An important but apparently extremely difficult question arises:

to what extent this result holds generally for any p,q > 2.

we can only prove a weaker proposition.

Here

The proof of Theorem 3.19 is

based on an investigation of the subgroups of G of rank < 4.

In [1]

B. Baumslag has shown that in G every subgroup of rank 2 is free. According to Karrass and Solitar (see [5]), every finitely generated subgroup U of G has a finite presentation.

For the problem of the one-

relator presentations of G we need a certain refinement and extension of these results, and we prove Theorem 3.3: If U is of rank < 3, then U is free; if U is of rank 4, then either U is free or for every generating system {xi,...,x4} of U there is a presentation of U with one defining relation.

Here, too, a

generalization of the results would be desirable. Finally, in the paper several problems are mentioned that are

Rosenberger:

One-relator groups

3 29

closely connected with the question of the one-relator presentations of G.

1.

PRELIMINARY REMARKS In the paper we use the terminology and notation of [10],[12]

and [18]; here < ...|... > indicates a description of a group in terms of generators and relations. By < ai ,. .. ,a > we denote the group generated by an ,... ,a . Frequently we obtain from one system {xi ,...,x } a new one by free (Nielsen) transformations and then denote the latter by the same symbols. If G is a group and H a subgroup, then we call H molnormal 1

if from gHg-

(A)

n H t {1} it follows that g e H.

Let G = Hi * H2 be the free product of Hi and H2 with amalgam

A = Hi n H2 .

We assume that in G a length L and an order are introduced

as in [12] and [18]. From Theorem 1 of [18] and Corollary 2 of [12] we obtain the following lemma.

Lemma 1.1.

If {xi ,...,x } c G is a finite system of elements of G, then

there is a free transformation from {xi ,...,x } to a system {yi ,...,y } for which one of the following cases holds: y i=l e

i

= ±lf

e

i

= e

i+l

if V

i

= V

i+1' W^th

L( y

- v.) <

L

^

f°r

i =

1

V

, i

»-"^-

e

i , a i 1, with y G A (i=l,...,q), V i=l V i i and in one of the factors H. there is an element x f A with xax"1 e A.

^

(1.4)

y

Of the y. there are p, p > 1, contained in a subgroup of G conjugate

to Hi or H 2 , and a certain product of them is conjugate to a non-trivial element of A. (1.5)

There is a g e G such that yi ^ gAg"1 , but for a suitable k e

1

we have yx € gAg"1 . The free transformation can be chosen so that {yi ,...,y } is smaller than {xi ,... ,x } or the lengths of the elements 0/ {xi ,... ,x } are preserved.

Remark.

If {xi,...,x } is a generating system of G, then in the case

(1.4) we find that p > 2, because then conjugations determine a free

Rosenberger: One-relator groups

330

transformation. If we are interested mainly in combinatorial description of < xi,...,x > in terms of generators and relations, we find again that p > 2 in the case (1.4), possibly after a suitable conjugation. (B) Let K = < B,t | rel B, t"1Kt t = K_, > be the HNN-extension with basis B, stable letter t, and conjugate subgroups Kx and K_t . Suppose that a length L and an order are introduced in K as in [10]. From [10] we deduce the following lemma. Lemma 1.6. If {xi ,...,x } is a finite system of elements of K, then there is a free transformation from {xi ,... ,x } to a system {yi ,... ,y } for which one of the following cases holds. (1.7) For every w e < yi ,... ,y > there is a presentation w = | | y 1 , n i=l v i e = ±l> e = e v = v (y ) i i i+l ^ i i+l' v. (1.8) Of the y. there are p, p > 1, contained in a subgroup of K conjugate to B, and a certain product of them is conjugate to a non-trivial element of K (e =±1). The free transformation can be chosen in finitely many steps so that {yi ,...,y } is smaller than {xt ,...,x } or the lengths of the elements of {xi ,... ,x } are preserved. 2. AN EQUATION IN FBEE GROUPS Let F = < a,b,c,...; > be the free group with the basis {a,b,c,...}. Let 1 / r G F be a cyclically reduced word in F that is not a proper power in F. Lemma 2.1. (a) Let x e F and let Ui = < ra,x >, a > 1, be the subgroup of F generated by r a and x. Let gr g"1 e Ui for a g € F and a 3 > 1. Then one of the following two cases holds: (i) g = gi r Y for a y > 0 and a gi £ Ui . ( i i ) There is a free transformation from {r a ,x} vo a system {r ,hr h"1 } for an h e F , h f < r > , and a 6 > 1. (b) Suppose, in addition, that r is not a primitive element of F; that x,y € F, that U2 = < x,y > is not cyclic; that r € U2 for an a > 1; and that there is no free transformation from {x,y} to a system in which an element is conjugate to a power of r. Then r e U2. From gr^g"1 e u2 for a g e F and $ > 1 it follows that g e U2 .

Rosenberger: Finally* a » <|>(U2)

> a, b i G

One-relator groups

let G = < a , b , c , . . . | r = 1 > and let <> j : F

> b, c i

>> G,

> c , . . . be the canonical epimorphism. Then

G is not cyclic* and Klfe) = < x,y | n = 1 > is a one-relator

group* where ri arises from r by representation Proof.

331

as a word in x,y.

If F is cyclic, then there is nothing to prove.

F is not cyclic.

Suppose then that

We prove Lemma 2.1 by induction on the (free) length of

r.

If Ui is cyclic, then 2.1 (a) holds, of course. Also when r only involves one generator of F. For let r = a, say. We regard F as free product F = < a > * < b > * < c > * ... of cyclic groups. Now 2.1 (a) follows by means of Lemma 1.1. If r is not a primitive element of F, then the length of r is at least 4 (because r is not a proper power in F); and r contains at least two generators. Now if r is of length 4 and not a primitive element of F, then 2.1(b) follows immediately from an elementary extension of Theorem 1 in [14], provided that there exists a free transformation from {x,y} to a system in which an element is conjugate to a power of r. Suppose now that Ui is not cyclic and that r involves at least two generators, say, a and b. Just as in [8, pp.198-200] and [9] we may confine ourselves to the case when the sum of the exponents a (r) for a generator t of F occurring in r is zero. Without loss of generality we may assume that t =a (and henceforth we replace a by t). For i £ 2Z we set b. = t b t , c. = t" 1 ct 1 etc. Since a (r) = 0, we see that r lies in the normal subgroup N of F generated by b,c,..., that is, r can be expressed in terms of the b.,c.,... . Let s be the cyclically reduced word obtained by writing this expression for r. then the length of s is smaller than that of r. Observe that r and s are equal as elements of F. Let m be the smallest and M the largest i for which b. occurs in s. We set B = < b m ,... ,bM,c.,d.,...; >, Kt = < b m ,... ^ ^ c ^ d . , ...;>, and K ^ = < b m+1 ,...,b M ,c i ,d i ,...;>. Then F = < B,t | f 1 ^ = K ^ > is an HNN-extension with base B, stable letter t, and conjugate subgroups Ki and K_1 . Both Kj and K_i are malnormal in B. We remark to begin with that s e B, but by the Freiheitssatz no s , y > 1, is conjugate to an element of K , e = ±1. We choose a left transversal R£, e = ±1, to K £ in B (K£ being represented by 1). We can choose R so that s V € R for all v e 7L and e = ±1. Suppose that relative to this description as HNN-extension and to the choice of R £ , e = ±1, a length L and an order are introduced in F as in [10]. With

Rosenberger:

One-re1ator groups

332

respect to this we carry out the cancellations in the systems under discussion. We investigate the two assertions separately. We cancel in {s a ,x}; since s a has the length

First 2.1(a).

zero relative to the description as HNN-extension, it remains unchanged. If the case (1.7) of Lemma 1.6 holds, then 2.1(a) is valid, of course. Hence we may assume that one of the following two cases holds: (2.2)

L(x) > 2 is even and x has a reduced symmetric normal form

x = hj t ' ... t n kt

n

hj"1 , e. = ±1, with k = hkjh"1 , \

... t

kj G K , e = ±1, h G B, k f K

, h. € R

(see [10]). (2.3)

Si

x

e

t 1,

, and e. = e.+- if h. = 1 i

L(x) = 0, that is x G B. (i) va

First let ht ^ 1 and hi £ < s a >.

Suppose that (2.2) holds.

Then s hi £ K

for all v G ZZ , because from svahi G K Va

v / 0 , it follows that hi = s" Jl for some £ G K

for some v G ZZ,

, that is, hj = s~VOt

. Next, hi"1 svcthi £ K

according to the choice of the transversal R

for

all v £ Z , v / 0, by the remark above. Finally, from hkih"1 = k £ K k

it follows that hk^h'1 =

£ K for all v G 2Z , v £ 0, because K is malnormal in B. ~en "Si

Hence,

there are no larger cancellations between s a and x, and now 2.1(a) follows immediately. Suppose now that ht G < s a >.

Then we may assume that hj = 1

(possibly after a free transformation), and 2.1(a) follows immediately, as above. (ii)

Suppose that (2.3) holds.

Then x € B and Ui = < sa,x > C B.

From gs g"1 e U, C B for some g G F and $ > 1 it follows that gs^g"1 = gj s g~

for some gt G B, by the conjugation lemma for HNN-extensions (see

[8], p.185). Now the discussion reduces to the group B, and 2.1(a) follows from the inductive hypothesis.

(Observe that F is free.)

Thus, 2.1(a)

is proved. Second 2.1(b). primitive element of F.

Suppose now, in addition, that s is not a

(Henceforth we write s instead of r, for r and s

are equal as elements of F.) The group Ife = < x,y > is freely generated by x and y.

In

what follows we keep in mind that there is no free transformation from {x,y} to a system in which an element is conjugate to a power of s. [3], we then have s G l^ . Now we cancel in {x,y}. Lemma 1.6 occurs, then x,y G B because s G B.

By

If the case (1.7) of

Hence, by Lemma 1.6 and

Rosenberger:

One-relator groups

333

[11], we may assume without loss of generality that one two cases occurs: (2.4) x ^ Ki; y has a reduced normal form y = tli! ... h. e Ri for i = 1,... ,n-l, h e B, and y - 1 xy G B. (2.5) x,y G B. (i) Suppose that (2.4) holds. Then, of course, cyclic in G, that is, <J)(U2) is of rank 2 in G. Let H = m > 2, and let I^J

:F

ty2 :H

> H, a 1 > G, a 1

> a, b 1 > a, b 1

> b, c 1 > b, c I

of the following th n , n > 1, with

4>(U2) is not < a,b,c,... | r m = 1 >

> c,... and > c,...

be the canonical epimorphisms. We obtain the commutative diagram

Since there is no free transformation from {x,y} to a system in which one element is conjugate to a power of s, it follows from [11] that t^ (U2) = < x,y \ st = 1 > is a one-relator group, where si arises from s by representing s as a word in x,y. Now the two remaining assertions of 2.1(b) follow immediately. (Remark. If in the cancellation of {x,y} the case (1.8) of Lemma 1.6 occurs, then possibly a suitable conjugation is performed in [11]. But this does not affect the argument in the case (2.4).) (ii) Suppose now that (2.5) holds. Then U2 C B. If s is assumed to be a primitive element of B, then on account of s ۥ U2 there is a free transformation from {x,y} to a system in which one element is conjugate to a power of s. But this gives a contradiction. Hence, s is not a primitive element of B. Now the assertion 2.1(b) follows from the inductive hypothesis, and everything is proved. The following corollary is an almost immediate consequence of the proof of Lemma 2.1 (and we omit the simple proof). Corollary 2.6 (see [17]). Let F2 be the free group of rank 2 with basis {x,y}, let w(x,y) e F2, and let F be free, with r e F. Then it can be decided algorithmically in finitely many steps whether or not there is a homomorphism <{>:F2 > F with <j>(w) = r.

Rosenberger:

Remarks.

1.

One-relator groups

334

The method employed in the proof of Lemma 2.1 is quite

suitable for the explicit investigation of equations w(x,y) = r in free groups F. 2.

A generalization of Lemma 2.1 to subgroups of higher rank, in

conjunction with Lemmas 1.1 and 1.6, would be very desirable. special r £ F such generalizations can be made (see [14]).

For certain

But the

following example shows how difficult the problem is in general: 2

2

2

a 3

1

3

be of rank n > 3, r = (a b ) (ca b c" } . z = c~*xc, and U = < x,y,z >. 1

2

3

C" re = z x .

3.

2 2

Let F 1

Let x = a b , y = cxc" ,

Then U is of rank 3.

We have r = x2 y3 and

But in G = < a,b,c,... | r = 1 > we have x = y3 z~2 = z~2y3 .

ON ONE-RELATOR GROUPS THAT ARE FREE PRODUCTS OF TWO FREE GROUPS WITH CYCLIC AMALGAMATION Throughout what follows we assume that G = < ai ,...,ap,bi ,... ,bq | w(ai ,...,a )v(bi ,... ,bq) = 1 >,

2 < p, 2 < q, that 1 ^ w = w(ai,...,a ) is not a proper power nor a primitive element in the free group Hi = < ai ,...,a ; >, and that 1 ^ v = v(bi ,...,b ) is not a proper power nor a primitive element in the free group H2 = < bi,...,b ; >.

Then G is not a free group and can be

written as a non-trivial free product G = Hi * H2 with cyclic amalgam A = Hi °> H2 = < a> = < w > = < v"1 >.

Suppose that a length L and an order

are introduced in G, as in [12], [16] and [18].

If we regard G in this

manner as a free product with amalgam, then cancellations of systems {xi,...,x } always refer to the length L and the order. We remark that A = Hi n H2 is malnormal both in Hi and H2 . In [1] B. Baumslag has shown that every non-trivial two-generator subgroup U of G is free. 1.1.

This also follows at once from [6] or from Lemma

According to Karrass and Solitar [5], every finitely generated

subgroup of G is finitely presentable.

Here we begin by refining and

extending these results for subgroups of G of rank < 4.

Lemma 3.1.

Let U c G be a subgroup of rank n > 2.

If {xi,...,x } is a

generating system of U and if there is a free transformation from {xi ,...,x } to a system {yi ,...,y } with y^ e Hi for i = 1,...,n-l, then U is free of rank n. Proof.

By hypothesis we may assume that x,,...,x

€ Hi .

If x

also

lies in Ht , then U is free of rank n, being a subgroup of a free group. Suppose now that x

does not lie in Hi; then, in particular,

Rosenberger:

One-re1ator groups

33 5

L(x n ) > 1. Proposition 3.2. The group V = < ai,... ,a ,x > is free of rank p+1. Proof of (3.2). We consider the generating system {ai,...,a ,x } of V; here x has a unique representation x = hj ...h b, m > 1, in which b £ A and the h. are from a left transversal, lie alternately in distinct factors of G and are not trivial. Since Hi = < ai ,...,a > and x ? Hi , we may assume (possibly after a free transformation) that b=l, hi £ H2 and h 6 H2 . But then L(r) > L(x ) > 1 for every freely cancelled word r € V in which x occurs. Hence, V is a free group of rank p+1. Since U C v, as a subgroup of a free group is also free, and of rank n because V C G. This completes the proof of Lemma 3.1. Theorem 3.3. (a) Every subgroup U c G of rank 3 is free of rank 3. (b) Let U c G be a subgroup of rank 4. Then one of the following two cases occurs: (i) U is free of rank 4; (ii) if {xi,...,X4} is a generating system of U, then there is a free transformation from {xi ,... ,X4 } to a system {yi ,... ,y4 } with yi ,y2 e zHi z"1 and y* ,y4 £ zH 2 z" ! for a suitable z e G. Moreover* for {xi,...,X4} there is a presentation of U with one defining relation. Proof. Let n , 3 < n < 4, be the rank of U and {xi,...,x } C G a o generating system of U. Being of rank n , U cannot be generated by n -1 elements. We consider the factorization G = Hi * H2 and cancel in {xi,...,x }. We may assume that {xi,...,x } is Nielsen reduced (see n o o [16]). If the case (1.2) of Lemma 1.1 occurs, then, of course, U is free of rank n . The cases (1.3) and (1.5) of Lemma 1.1 cannot occur, because A is malnormal both in Hi and H 2 . Hence we may assume, in accordance with Lemma 1.1, the remark following it, and Lemma 2.1 that one of the following two cases holds: (3.4) Of the x. there are m, n -1 < m < n , that lie in a subgroup of G conjugate to Hi or H2 . (3.5) n =4, xi and x2 lie in a subgroup of G conjugate to Hi or W%9 and a product of them is conjugate to a non-trivial element of A; furthermores there is no free transformation from {xi,x2} to a system in which an element is conjugate to a power of a. If (3.4) holds, then U is free of rank n , by Lemma 3.1; in

Rosenberger:

One-relator groups

336

particular, 3.3(a) is proved. Suppose now that (3.5) holds.

Without loss of generality we

may assume that xi ,x2 € Hi and that a product in xi and x2 is conjugate to a non-trivial element of A; also that the conjugating factor is trivial.

For from gwag"1 e Hi with g € G and a > 1 it follows that

gwotg"1 = giwagr1 for a gi € Hi .

By Lemma 2.1, in fact, w e < xt ,x2 >.

If there is a free transformation from {xi,...,X4} to a system {xi ,x2 ,y3 ,y4 } with y3 £ Hi , then U is free of rank 4, by Lemma 3.1.

We

now assume that (3.6)

there is no free transformation from {xi ,...,X4} to a system

{xi ,x2 ,y3 ,y4 } with y3 e Hi. In particular, then x3 f Hi , X4 f Hi , and L(x 3 ), L(xO > 1. We consider the following freely cancelled words in which both j =3 and j = 4 occur. (3.7)

r = ui...ut,

Uiui+1

f 1, t > 2, u., =

j Xj

, e. = ±1, j e {1,2,3,4}.

If r / 1 for all such words r, then U is free of rank 4. Suppose now that r = 1 for at least one such word.

Proposition 3.8. (3.9)

Then U is not free.

One of the following two oases occurs*

There is a free transformation from {xi ,x2 ,x3 ,X4 } to a system

{xi ,x2 ,y3 ,y4 } with the properties (i)

L(y3) = L(x 3 ), L(y4) = L(xO and

(ii)

y3 or y4 is conjugate to a power of a.

(3.10)

The case (3.9) does not hold, and there is a free transformation

from {xi ,x2 ,x3 ,X4 } to a system {xi ,x2 ,y3 ,y4 } with the properties (i)

L(y3) = L(x 3 ), L(y4) = L(xO and

(ii) y3,y4 € zHiz"1 for a z e G, i = 1 or 2. Proof of (3.8). Suppose that neither (3.9) nor (3.10) holds. In particular, no non-trivial element of < x3,X4 > is conjugate to a power of a. Among the words r = ui ...u one for which t is minimal.

as in (3.7) with r = 1 we choose

Let this be r = ui...uf.

Since {xi,...,X4)

is Nielsen reduced, every x. occurs in r at least twice (any three elements of [xi,...,X4> generate a free group of rank 3 ) . In particular, t > 8. Let Ui .. .u^, 2 < £+1 < k < t, be a partial word of r, with u ,u, € < x3 ,X4 > and u o + i> • • • >ui<. -,G ^ xi ,x2 >. then have necessarily, by Lemma 2.1, u

, .. .u,

1

Since t is minimal, we

£ A = < w > , for otherwise

Rosenberger:

One-relator groups

33 7

there would be a decisive cancellation in ui...u of the elements u , ...,u, does not take part. 3

in which at least one

Therefore, r is of the

3

n form 1 = r = aibiw * .. .bnw ba2 , with n > 0, b e < x3 ,X4 >, b± e <

X3

,x4 >, b 4 1, bi t 1, 3 i e 2Z\ {0} if n > 1, and

aj

e <

Xi

,x2 >,

for i = 1,...,n, j = 1 or 2. Suppose that Ui = < w,X3 ,x4 > c U is a free group of rank 2 (of course, Ui is not cyclic).

Then there is a free transformation

from {w,X3,x4} to a system {yi,y2,l}. assume that {yi ,y2 } is Nielsen reduced.

We cancel in {yi ,y2 } and may If the case (1.2) of Lemma 1.1

occurs, then necessarily y. € < w > = A for i = 1 or 2, because w e Ui . But this contradicts the fact that the rank of U is 4 (observe that w € < xi ,x2 > ) . The cases (1.3) and (1.5) of Lemma 1.1 cannot occur because A is malnormal both in Hi and H2 . Hence by Lemma 1.1, the remark following it, and Lemma 2.1, we may now assume that the following case occurs:

yi and y2 lie in a subgroup G conjugate to Hi or H2 , and a

product of them is conjugate to a non-trivial element of A. But then necessarily the case (3.9) or (3.10) occurs, which leads to a contradiction. 3.

Consequently, Ui = < w , X 3 , x 4 > i s free of rank

In particular, r1 = aj^ra^1 = a"1 a,"1 = b x w

l

• ••t>nw

b £ A = < w>,

that is, r, G U, n (HiXA). Now we consider the system {w,x3,X4>.

Since w is of length 0,

Ui is free of rank 3, and w G < xi ,x2 >, we may assume that {w,X3 ,X4 } is Nielsen reduced and none of the elements w, x3 , X4 is trivial. If the case (1.2) of Lemma 1.1 holds, then X3 or x4 must lie in Hi, in contradiction to (3.6). Lemma 1.1 cannot occur.

Again, the cases (1.3) and (1.5) of

But then it follows from Lemma 1.1, the remark

following it, and Lemma 2.1 that the case (3.9) or (3.10) must occur, which leads to a contradiction.

Proposition 3.8 is now completely

proved. Next, we investigate the cases (3.9) and (3.10) and sharpen Proposition 3.8.

Proposition 3.11.

If theve is a word r = ut ...u

as in (3.7) with r = 1,

then the case (3.10) holds. Proof of (3.11).

Suppose that (3.10) does not hold.

then (3.9) must hold.

By Proposition 3.8,

From this we deduce a contradiction.

It is enough

to confine our attention to the situation X3 ,X4 € H ^ A and x3 = zvotz"1 for an a > 1.

By Lemma 2.1 and on account of (3.6), v

f < x3,x4 > for

Rosenberger:

all $ > 1.

One-relator groups

338

Suppose that Ui = < w,x3,X4 > is free of rank 2.

Then there

is a free transformation from {w,X3 ,X4 } to a system {yi,y2,l}. consider the system {yi,y2}.

We

If there is a free transformation from

{yi ,y2} to a system in which an element is conjugate to a power of v, then owing to w € Ui , by Lemma 1.1, the remark following it, and Lemma 2.1, there is also a free transformation from {yi ,y2 } to a system (zi,z2} with zi = v Y for a y > 1. that U is of rank 4.

This is a contradiction to the fact

Hence, there is no free transformation from {yi ,y2 }

to a system in which an element is conjugate to a power of v.

Consequently,

by Lemma 2.1, in the free product < a p ...,a |w=l > * < bt ,... ,b |v=l > the group Ui = < yi ,y2 > is of rank 2 (addition of the relation a = 1). On the other hand, Ui = < w,x3 ,X4 >.

Since x3 is conjugate

to a power of v, in the same free product Ui is cyclic or trivial. is a contradiction.

This

Hence, Ui = < w,x3 ,X4 > is free of rank 3.

Again, let r = ui ...u. as in (3.7) with r = l and t minimal. Since {xi,...,X4} is Nielsen reduced, every x. occurs in r at least twice; in particular, t > 8. Let u... ,u, , 2 < l+l < k < t be a partial word of r with u ,u, € < x3 ,X4 > and u « + 1 >• • • > u k , e < xi ,x2 >.

Since t is minimal, we

then have necessarily, by Lemma 2.1, u. + , ... u,

e A = < w > = .

Therefore, r is of the form 1 = r = aj rt a2 with a. e < xt ,x2 > c H,, j = 1 or 2, and ri £ Ui C H2 , ri f. A (Ui is free of rank 3 ) . From 1 = airia2 it follows that rt = a"1 a,"1 e H2 . which is a contradiction.

Altogether, ri e Hi n H2 =A,

This completes the proof of Proposition 3.11.

Now we investigate the case (3.10). 1

generality we may assume that x3 ,x4 € zH^z""

Without loss of

for a z e G and i = 1 or 2.

Since, by hypothesis, there is a word r as in (3.7) with r = l, we must have i = 2 , that is, x3 ,X4 £ zH2 z"1 for a z e G. Let x. = Jlj .. .JMc.JT1 .. .Jl"1 , j = 3,4, be the symmetric normal form of x., where k. € H2\A, I

€ Ht for m > 1, U

G < xi ,x2 > for

m > 1, and the £, are taken alternately from a transversal to A in Hi and H2 .

Proposition 3.12. Proof of (3.12).

m < 1. Suppose that m > 2.

Then, in fact, m > 3 if U

e Hi .

We first establish an auxiliary proposition.

Proposition 3.13.

The group V = < ai ,... ,a ,x3 ,x* > is free of rank p+2.

Rosenberger: Proof of (3.13).

One-relator groups

339

We consider the generating system {ai ,... ,a ,X3 ,X4 } of

V and the symmetric normal forms above of X3 and X4 . Since Hi = < ai ,...,a >, we may assume that l\ £ H2 (possibly after a free transformation); we still have m > 2 by the remark above. Then L(s) > L(x.) - 1 > 1 for every freely cancelled word s € V in which x., j = 3 or 4, occurs.

Hence, V is free and of rank p+2.

This proves

Proposition 3.13. Since U c V, U is free as a subgroup of a free group and of rank 4.

This

leads to a contradiction to the fact that there is a word r as in (3.7) with r = l.

Hence, m < 1, and Proposition 3.12 is proved.

We now investigate the case m < 1 and sharpen Proposition 3.12. Proposition 3.14. Proof of (3.14).

m = 0. Suppose that m ± 0.

Further, l\ e Hi and £1 ? < xi ,x2 >.

Then m = 1, by Proposition 3.12. From the existence of a word r as

in (3.7) with r = 1 it follows that v 5 e < k3,k4 > for a 6 > 1, because otherwise we would have L(s) > L(x.) > 1 for every freely cancelled word s £ U, in which x., j = 3 or 4, occurs.

By Lemma 2.1 and Proposition

3.11 we then even have v £ < k3,k4 >, and there is no free transformation to a system in which an element is conjugate to a power of v. x

transform by l\

Now we

and consider the subgroup U2 generated by zt = ^f1 xi ^1 >

z2 = ilj * x2 £j , k3 , and k4 . Note that Zi,z2 € Hi and that U2 clearly is of rank 4.

Since

U is free, so is U2 . But then V2 = < v,zi , z2 > must be free of rank 2, and there is a free transformation from {v,zt,z2} to a system {yi,y2,lh Since by Lemma 2.1 Ik is of rank 4 in the free product < ai ,...,a |w=l > * < bi,...,b |v=l >, there is no free transformation from {yi,y 2 } to a system in which an element is conjugate to a power of v. Again by Lemma 2.1, since v,l~1vll € u2 = < yi ,y2 >, we have £i € v2 C U2 . Consequently, U2 = U, because on account of U G Uj conjugation in U2 with £1 defines a free transformation from {zi ,z2 ,k3 ,k4 } to {xi ,x2 ,X3 ,X4 }. reduced.

This contradicts the fact that {xi ,... ,X4 } is Nielsen

Hence, m = 0 and Proposition 3.14 is proved. It follows that x. = k. G H2 for j = 3 or 4.

From the

existence of a word r as in (3.7) with r =1 we deduce that Ui = < w,x3,X4 > is free of rank 2, that is, relative to the system of generators {w,x3,X4}

Rosenberger:

One-relator groups

340

there is a presentation of Ui with a single defining relation ro(w,X3,X4) = 1.

We obtain a complete system of defining relations

among the generators xi,...,X4 of U from the relation ro(w,X3,X4) = 1, bearing in mind that w € < xi ,X2 >. and the relation w = wi (xi ,X2), where wi (xi ,X2) arises from w by expressing it as a word in xi and X2 . Hence we have for {xi,...,X4> a presentation of U with a single defining relation.

Remark,

This proves Theorem 3.3.

In the proof of Theorem 3.3 an important role was played by the

generation of the free group of rank 2 by three elements.

Let us extend

somewhat the relevant question. Let F = < ai ,...,a the basis {ai,...,a }.

>, n > 2, be the free group of rank n with

Let r € F be freely reduced and suppose that there

is no free transformation from {ai ,...,a } to a system {bi ,...,b } with r € < bi ,...,b

Definition 3.15.

^ >.

We consider generating systems {r,xi,...,x } of F.

We say that in a free transformation x from {r,xi ,...,x }

to a system {r,yi ,...,y } the element r is not replaced if in all the elementary free transformations of which x is composed r remains unchanged or is changed to r"1 or is put in a different place of the relevant (n+1) -tuples. We refer to such free transformations in which r is not replaced and to the corresponding Nielsen equivalence classes as r-stable. Now let H = < ai ,.. .,a |r=l > and let <J> : F — > > H, a. — > be the canonical epimorphism.

a. ,

If {r,xi,...,x } is a generating system

of F, then {xi,....,x } is a minimal generating system of H; and relative to {xi,...,x } there is a presentation of H with a single defining relation.

Distinct r-stable Nielsen equivalence classes of generating

systems {r,xi,...,x } in F yield distinct Nielsen equivalence classes of minimal generating systems in H. The following question arises:

let (yi ,...,y } be a

generating system of H for which there is a one-relator presentation of H.

Under what conditions (for example, for what r or what n) does there

exist an automorphism i|/ of H, a free transformation x from {*(yi)».- • >iKyn)} to (xiKyi).-..,XiKyn)h and suitable x . e F with <j>(xi) = x4'(xi), i = l,...,n, such that {r,xt ,... ,xn> is a generating system of F?

Rosenberger:

Conjecture 3.16.

One-re1ator groups

Such x* fe and xi ,... ,x

341

exist if r is a proper power

in F. So far only few results in this direction are known.

If n = 2

and r is a proper power in F, or if n > 2 and r=a

1 11

...a

[a , , a ] m m + l m + zo

• • • [a

, , a ] , 0 < m < n , a. > 2 , n - l n I

that is, if r is an alternating product in the sense of [14], then such if/, x , and xi,...,xn exist (see [4], [7], [11], [13], [14], [15], [16], [18] and [19]). Theorem 3.19

below and the remark following it show that such

if/, x» and xi ,... ,x

also exist for n = 4 and r = ri (ai ,a2 )r2 (a3 ,a4).

The problem is of great interest for the treatment of the isomorphism problem for H. For the solution of the isomorphism problem for the group G with p = q = 2 we need the following lemma.

Lemma 3.17. {a,b}. F.

Let F = < a,b; > be the free group of rank 2 with the basis

Suppose that 1 ^ r e F is not a power of a primitive element of

Then there are only finitely many r-stable Nielsen equivalence classes

of generating systems {r,x,y} in F. Proof.

We assume that in F the free length L and a suitable lexicographic

order are introduced relative to the generators a and b. Let {r,xi,X2} be a generating system of F.

Since r is not a

power of a primitive element of F, there is no r-stable free transformation from {r,xi ,x2} to a system {r,y,l}.

Now we perform r-stable

free transformations from {r,xi,X2} to other systems.

Thus, we can

achieve that L(x i n r £ )>L(x i ) and L f x A x ^ ) > L(x.,) - L(r) + L(x.), n,e = ±1, i,j = 1 or 2 Since F = < r,xi ,x2 > and r is not a power of a primitive element of F, we must have either at least once L(x. n r e ) < L(r) or else always L(x i n r e ) > L(r), but at least once L(r n x i r e ) < 2L(r) - L(x.,), n,e = ±1, i = l or 2.

If at least once L(x. n r e ) < L(r), then L(x.) < L(r).

8

If always LCx^r ) > L(r), but at least once L f r V r ^ < 2L(r) - L f x ^ , then all letters in x. are cancelled, and L(x.) < L(r).

Suppose now that

L(xi) < L(r). We consider the subgroup U of F generated by xi and r. Certainly, U ^ F.

Suppose that U is cyclic.

Then there is a free

transformation from {r,xi } to a system {y,l}; also U = < y > , F = < y,X2 >. In particular, r is a power of a primitive element of F, which is a contradiction.

Rosenberger:

One-relator groups

342

Hence, U i s free of rank 2 with the basis {r,xi>. There i s a free transformation from {r,xi } to a system {zi,z 2 } having the Nielsen property. Let us consider this system { z i , z 2 } . Since L(xi) < L(r), we have L(z.) < L(r) for i = 1 or 2. If hi and h2 are elements of U, then the transformation from {r,xi ,x2 } to {r,xi ,hix 2 h 2 } is r-stable, that i s , we can achieve that L(x 2 n z i e ) > L(x2) and L(x 2 n z i x 2 e ) > 21^X2) - Lfz^ , n,e = ±1; i = 1 or 2; furthermore, L(x 2 £ z i z. n ) > L(x2) - L(z i ) + L(z.) and L C z A . x / ) > L( Zj ) - Lfz^ + LCx,), n,e = ±1; i , j = 1 or 2, and n = 1 if i = j . Since U j^ F = < r,xi ,x2 > = < Zi ,z2 ,x2 > we must have either at least once LCxjV 6 ) < L(z.,) < L(r) or else always L ^ V 8 ) > L(z..), but at least once L(z i e x 2 z. n ) < Lfz^ - L(x2) + L(z.) < 2L(r) - L(x 2 ), n,e = ±1; i , j = 1 or 2. This means that L(x2) < L(r). This completes the proof, for there are only finitely many generating systems {r,xi ,x2 } of F with L(x.) < L(r) for i = 1 and 2. Remark. For specific r e F = < a,b; > Lemma 3.17 can, of course, be stated more precisely. Here we mention the following two cases. 3.18(a) If r = a a b , a,8 > 2, and {r,x,y} is a generating system of F, then there is an r-stable free transformation from {r,x,y} to a system {r,a Y ,b 6 } with y = 1, 1 < 6 < 6/2, (6,6) = 1 or <5 = 1, 1 < y < a/2, (Y,oO = 1. 3.18(b) If r - a a ba b" 1 , a,6 ^ ZZ\{0}, and {r,x,y} is a generating system of F, then there is an r-stable free transformation from {r,x,y} to a system

{ r , a Y , b a } with

1 < y < | a | , (y,6) = 1, 0 < 6 < y or

1 < Y < | B | , Y | 8 , ( Y , O ) = 1, 0 < 6 < y .

Theorem 3.19. Stcppose now that p = q = 2. If {xi , . . . ,X4 } is a generating system of G, then one of the following two oases occurs: 3.19(a) There is a free transformation from {xi , . . . , x * } to a system {ai ,a 2 ,yi ,y 2 } with 3.19(b)

There

is

yi ,y 2 e H2 , and H2 = < v,yi ,y 2 > . a free

transformation

from

{ x i , . . . , X 4 > to a

system

tyi ,Y2 ,bi ,b 2 } with yi ,y 2 € Hi , and Hi = < w,yi ,y 2 > .

For {xi,...,X4} there is a presentation of G with a single defining relation. Furthermore3 G has only finitely many Nielsen equivalence classes of minimal generating systems3 and we can decide algorithmically in finitely many steps whether or not an arbitrary onerelator group is isomorphic to G. Proof. Let {xi,...,X4> be a generating set of G. By Theorem 3.3, we may

Rosenberger:

One-relator groups

343

assume that xi ,X2 £ Ht and X3 ,X4 € H2 . Suppose that H2 ^ < X3 ,X4 >.

Then xi and X2 must take an

essential part in the generation of H 2 , that is, a product in xi and x2 is conjugate to a power of w, and by Lemma 2.1 we then must have Hi = < xi ,X2 >, because the free product < ai ,a2 |w=l > * < bi ,bi |v=l > is of rank 4.

Hence, 3.19(a) or 3.19(b) occurs.

The remaining claim of Theorem 3.19 is an immediate consequence of Theorem 3.3, Lemma 3.17, and an application of the Whitehead method (see [18]).

For if Hi = < xi ,x2 > and H2 = < v,x3 ,x4 >,

then every v-stable free transformation from {v,x 3 ,x 4 } to a system {V>V3 >y4 } can be extended to a free transformation from {xi,...,X4} to the system {xi ,x2 ,y3 ,y4 }.

Remarks

Let H = < ai ,a2 ,bi ,b2 | (w(ai ,a2))a(v(bi ,b 2 )) 3 = 1 >, 1 < a

1.

and 1 ^ 3 , and suppose that 1 ^ w = w(ai ,a2) is not a proper power nor a primitive element in < ai ,a2; > and that 1 ^ v = v(bi ,02) is not a proper power nor a primitive element in.

Let {xi ,...,X4> be a

generating system of H (which is obviously of rank 4, like G ) . Then vi £ < w > and xi f < v>. If xi ,x2 e Hi and w by Lemma 2.1, w G < xi,x2 >.

6 < xi ,x2 > for a 6 > 1, then, in fact,

Therefore, Theorem 3.19 also holds for H.

This stands in contrast to the corresponding results in [4] and [19]. However, Theorem 3.3 does not hold for H in general; for if 2 < a, 3, then the subgroup < w,v > is not free. 2.

Theorem 3.19 is likely to hold analogously for arbitrary p,q > 2.

prove this would be most desirable.

To

Our proof here for p = q = 2 is based

essentially on Lemma 2.1 for which a useful extension is lacking. In the general case p,q > 2 we can only prove the following proposition.

Proposition 3.20.

If {xi ,...,x

+

} is a generating system of G, then

there is a free transformation from {xi ,... ,x {yi > . . • > y p + q

} w i t h

y>»• • • >yv

e

Hi

or

} to a system

+

y* > • • • »^ q

€

H2

•

In any case, Theorem 3.19 holds for arbitrary p,q > 2, provided that w and v are alternating products in the sense of [14] (see [12] - [15]). The proofs in [12] can easily be put into more elegant form with the help of [14].

Rosenberger:

Acknowledgement.

One-relator groups

344

The author wishes to thank Kurt Hirsch for the English

translation.

REFERENCES

1. B. Baumslag, Generalized free products whose two-generator subgroups are free, J. London Math. Soo. 43> (1968), 601-606. 2. G. Baumslag, Some problems on one-relator groups, in Proa. Second Internat. Conf. Theory of groups (Canberra 1973), Lecture Notes in Mathematics, Vol.372, Springer-Verlag, Berlin, New York (1974), 75-81. 3. G. Baumslag, Residual nilpotence and relations in free groups, J. Algebra 2_ (1965), 271-285. 4. D.J. Collins, Presentations of the amalgamated free product of two infinite cycles, Math. Ann. 2]n_ (1978), 233-241. 5. A. Karrass $ D. Solitar, The subgroups of a free product of two groups with an amalgamated subgroup, Trans. Amer. Math. Soc. 150 (1970), 227-255. 6. A. Karrass $ D. Solitar, The free product of two groups with a malnormal amalgamated subgroup, Canad. J. Math. 2Z_ (1971), 933-959. 7. R.C. Lyndon, Quadratic words in free products with amalgamation, Houston Math. J. £ (1978), 91-103. 8. R.C. Lyndon $ P.E. Schupp, Combinatorial group theory, SpringerVerlag, Berlin, New York (1977). 9. J. McCool $ P.E. Schupp, On one-relator groups and HNN extensions, J. Austral. Math. Soc. 1£ (1973), 249-256. 10. N. Peczynski § W. Reiwer, On cancellations in HNN-groups, Math. Z. 158 (1978), 79-86. 11. S.J. Pride, The two generator subgroups of one-relator groups with torsion, Trans. Amer. Math. Soc. 234 (1977), 483-496. 12. G. Rosenberger, Zum Rang-und Isomorphieproblem fur freie Produkte mit Amalgam, Habilitationsschrift, Hamburg (1974). 13. G. Rosenberger, Zum Isomorphieproblem fur Gruppen mit einer definierenden Relation, III. J. Math. 2£ (1976), 614-621. 14. G. Rosenberger, Alternierende Produkte in freien Gruppen, Pacific J. Math.JJ>_ (1978), 243-250. 15. G. Rosenberger, Uber Automorphismen ebener diskontinuierlicher Gruppen, in Proc. Stony Brook Conf. Riemann surfaces, Princeton Lecture Notes (to appear). 16. G. Rosenberger, Gleichungen in freien Produkten mit Amalgam, Math. Z. 173 (1980), 1-12. 17. P.E. Schupp, On the substitution problem for free groups, Proc. Amer. Math. Soc. £3 (1969), 421-423. 18. H. Zieschang, Uber die Nielsensche Kurzungsmethode in freien Produkten mit Amalgam, Invent. Math. 2 £ (1970), 4-37. 19. H. Zieschang, Generators of the free product with amalgamation of two infinite cyclic groups, Math. Ann. 227 (1977), 195-221.

345 THE ALGEBRAIC STRUCTURE OF Ko-CATEGORICAL GROUPS

J.S. Wilson Christ's College, Cambridge, CB2 3BU, England

1.

INTRODUCTION AND BASIC PROPERTIES A countable group is said to be Ko-categorical if it is

determined up to isomorphism within the class of countable groups by its first-order properties.

Ko-categorical groups have received a certain

amount of attention in a model-theoretic setting during the last ten years.

My object here is to give an account of the basic properties of

these groups from an algebraic point of view and to describe the first steps in an investigation of their algebraic structure.

That this is

feasible is a consequence of the rather pleasing characterization provided by a theorem of Engeler [8], Ryll-Nardzewski [17], and Svenonius [19]:

(1) A countable group G is No-categorical if and only if Aut G has only finitely many orbits in its action on G

= G * ... * G, for each n > 0.

The statement of this theorem can be taken as a working definition.

For brevity, we write $• for the class of countable No -

categorical groups.

Thus all finite groups lie in 5-, and it is almost

immediate that if Gi , G2 <= &, then Gi x G2 G $•.

Using the result (3)

below and the fact that Abelian groups of finite exponent are direct sums of cyclic groups, one can prove fairly easily that a countable Abelian group lies in £• if and only if it has finite exponent.

A somewhat more

substantial result, proved by Cherlin and Rosenstein in [7], is that if A is a countable Abelian group and G is a finite extension of A, then G G t i f and only if A, considered as a ZG-module, is a sum of finite submodules of uniformly bounded orders.

Further examples of 5--groups are

the countable extra-special groups - this was essentially proved in Feigner [10] - and wreath products G wr F with G ^ f

and F finite.

More

elaborate examples are known, some very interesting ones having been constructed by my Cambridge research student, Mr. A.B. Apps in [1] and [3].

For the moment it will suffice to say that *•-groups appear in

Wilson:

Ko-categorical groups

346

general to have rather complicated structure, and that descriptions as complete as the one mentioned above for Abelian by finite groups are only likely to exist for very small subclasses of $-.

There is a certain

amount of evidence that the following may be true:

Conjecture 1.

Every It-group is an extension of a nilpotent group by a

residually finite group. Other statements of an algebraic nature that one might hope to make about 5--groups will be suggested by an examination of some of their basic (and well known) properties. If K is a characteristic subgroup of an 3E--group G, then consideration of the action of Aut G on K K and G/K is an £•-group.

and (G/K)

shows that each of

However each such subgroup K is a union of

orbits of Aut G, and so G can have only finitely many characteristic subgroups.

(2)

This proves

I/G^f

then G has only finitely many characteristic subgroups, and

for each characteristic subgroup K one has K e $- and G/K e 3h

Thus G has

a finite characteristic series 1 = Go < Gi < ... < G = G, m whose factors are characteristically simple 5— groups. The characteristically simple ^--groups are therefore the building blocks from which arbitrary ^--groups are constructed, and it will obviously be useful to examine them more closely. Let G again be an £--group and let n > 0. and x G < gi ,...,g

n

>, then we may write x in the form gf1 ... g.^ with ii

k > 0 and e. = ±1 for each i.

containing (gi ,...,g n ,

Since the number of orbits is finite there is an integer k n such

that whenever gi ,... ,g

£ G and x e < gi ,... ,g

in the form g.1

with k < k .

ii

(3)

iv

The minimal k for which this can be done

clearly depends only on the orbit of Aut G on G n x).

If gi ,.. .,gn G G

... g.

ii^

n

>, then x may be written

We may conclude

I / G ^ i - then for each n > 0 there is a bound on the orders of the

n-generator subgroups of G. In particular, 5—groups are locally finite and have finite exponent.

The conclusion of [3) is precisely equivalent to the assertion

that var G, the variety of groups generated by G, should consist of

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347

Among the varieties consisting of locally finite

groups are the Cross varieties; these may be defined to be the varieties of the form var F with F a finite group.

The following conjecture seems

plausible:

Conjecture 2.

If G £ $•, then var G is a Cross variety.

For unexplained facts about varieties of groups we refer to H. Neumann [16]. Another property which turns out to be very useful in the study of $•-groups is the following:

(4)

If G e $- then there is an integer d such that, whenever g ^ G and

x £ < g 1 >, then x may be expressed as a product of at most d conjugates

of g. The proof of (4) is an immediate conseauence of the fact that Aut G has only finitely many orbits on pairs (g, x) with x £ < g ' >. Feigner [9] has shown that d may be chosen so that whenever g e G and x £ < g

> then x is a product of exactly d conjugates of g, but we do

not need this refinement.

For many applications it is convenient to work

with a weaker property of £•-groups than the one given by (4) :

Definition.

A group G has finite conjugate spread if there is an

integer d such that, whenever g ^ G and x e < g gi > • • • >gj

G

G

with x e < g

gl

>, there exist

,. .. ,g d >; the conjugate spread of G is then

the least integer d with this property. The utility of properties such as this arises in part from the fact that they are preserved in homomorphic images.

Much of the

difficulty in working with ^--groups stems from the paucity of subgroups and quotient groups which are ^--groups.

Apps [3] has shown for instance

that a subgroup of index two in an 5— group can fail to be an $--group. However there is one important family of subgroups (wider than the class of characteristic subgroups) which inherit the condition £-, the family of definable subgroups.

A subgroup H of a group G is said to be definable

if there is a finite family gi ,...,g

of elements of G such that Ha = H

for every automorphism a of G fixing each g.. case and that G € $-, and let n > 0. hi i ....,h , ) ,

Suppose that this is the

Choose representatives (gi ,...,g ,

.... (gi ,...,g , h, ,...,h

) with each h. . £ H for all

orbits of Aut G on G m + n which contain elements of form (gi ,...,g m , hi ,...,h ) with each h. e H.

Thus if hi,...,h

e H, then there exist

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a ^ Aut G and j < r such that g.a = g. for i < m and h.a = h. for i < n. Since this a induces an automorphism of H, it follows that Aut H has only finitely many orbits on H n .

This shows that H is an *•-group.

A similar

argument shows that if H is also normal in G, then G/H is an £-group. Trivially finite subgroups are definable, and because for example Hi a = Hi and H2 a = H2 imply < Hi ,H2 > a = < Hi ,H2 >, the family of definable subgroups is closed under taking finite joins, finite intersections, normal closures, centralizers and normalizers.

2.

CHARACTERISTICALLY SB2PLE ^-GROUPS Before proceeding further with the structure of 9E--groups I

need to introduce some more examples. and let F be a finite group.

Let X denote Cantor's ternary set

We regard the set of functions from X to F

as a group in the usual manner by defining multiplication componentwise: given functions gi ,g2 we define gi g2 by x(gig2) = (xgi)(xg2)

for all x e x.

We give F the discrete topology, and write B(F) for the group of continuous functions from X to F.

We shall prove that B(F) if an $•-group.

If g G G then the preimages under G of the elements of F form a finite partition of X into closed-and-open subsets, and since X has only countably many closed-and-open subsets there are only countably many possibilities for g.

Therefore G is countable.

Before starting the

proof that B(F) is No-categorical we note that if Xi , Yi are non-empty proper closed-and-open subsets of X, then (i) Xi and Yi are homeomorphic to X and (ii) there is a homeomorphism a of X such that Yi a = Xi .

From

this it follows easily that if {Xi,...,X } and {Yi,...,Y } are partitions of X into non-empty closed-and-open sets, then there is a homeomorphism a of X such that Y.a = X. for each i. Now let n > 0, and write m = |F| . We choose a partition {Xi ,...,X } of X into m non-empty closed-and-open subsets, and write H for the subgroup of B(F) consisting of maps constant on each set X^. Thus H = Fm.

We show that each n-tuple of elements of B(F) is equivalent

under Aut B(F) to an n-tuple of elements of H.

Let gi ,...,gn G B(F), and

define for each n-tuple f = (fi ,...,fn) of elements of F X f = {x; xgi = £1 ,..., xg n = fn>. The non-empty sets X f with f £ F n form a partition of X into at most m closed-and-open sets.

Let {Yi ,...,Y } be a partition of X into m non-

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349

empty closed-and-open sets such that each X,. is a union of Y.'s, and let a be a homeomorphism such that Y.ot = X. for each i.

Define a £ Aut B(F)

thus: x(ga) = Cxa-1)g

for

x G X, g e B(F) .

It is trivial to check that each g.a lies in H.

This finishes the proof

that B(F) is No-categorical. We fix an element xo of X and define B~(F) = {g e B(F) ; xog = 1}. A similar argument to the one above shows that B (F) is also an $--group.

The No -categoricity of these groups is a very particular

case of a theorem of Waskiewicz and Weglorz [21]. What concerns us here is the fact that if S is a finite simple group, then B(S) and B (S) are also characteristically simple.

The proof of this is easier than the

proof of Ho-categoricity and it can be left as an exercise. The intrusion of Cantor's ternary set here may at first seem a little mysterious; however it should be expected to arise in the study of infinite, fairly homogeneous structures. been disguised as follows: direct square Fi = Fo

x

Fo , embed Fi diagonally in its direct square

F2 = Fi x Fi , and so on.

The union of the groups F

embeddings is precisely B(F). subset of [0,1], F

Its presence here could have

define Fo = F, embed Fo diagonally in its

with these

(In the standard representation of X as a

corresponds to the group of maps constant on each of

the 2 n non-trivial intersections X n [3~ n (i-l), 3~ni] with 1 < i < 3n.) The group B (F) can be defined in a similar manner.

The proofs that B(F)

and B (F) are No-categorical can be carried out in combinatorial language starting from these definitions but the notation becomes cumbersome. The importance of these groups in the structure theory of ^--groups is explained by the following result:

(5)

Suppose that G is a oountably infinite characteristically simple

locally finite group of finite exponent.

Then one of the following holds:

(a) G is an elementary Abelian p-group for some prime p,. (b) G = B(S), G = B~(S) or G is a direct product of #0 copies of S* for some non-Abelian finite simple group S 3 or (c) G is a perfect p-groupj for some prime p. With the additional hypothesis that G has only finitely many isomorphism types of finite simple sections, I could prove this result nine years ago.

This hypothesis can now be lifted, because the recently

completed classification of the finite simple groups implies that there

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350

are only finitely many non-isomorphic finite simple groups of any given exponent.

A full proof of (5) is going to be included in Apps' paper [2].

If G is an 3E--group, the last possibility in (b) cannot arise:

this one

can easily check using (4). Unfortunately, very little is known about case (c). No examples of non-Abelian characteristically simple locally finite p-groups of finite exponent are known, and in fact examples of nontrivial perfect locally finite groups of finite exponent have only recently been constructed, by Vaughan-Lee and Wiegold [20] and by L'vov and Khukhro (see [14], p.73).

Conjecture 3.

Concerning $•-groups, it seems reasonable to conjecture:

Every characteristically simple locally nilpotent %r-group

is Abelian. This is the weakest of our three conjectures.

If it could be

proved, we would have a very satisfactory description of characteristically simple ^--groups as either elementary Abelian or of type B(S), B~(S) or S n for some non-Abelian finite simple group S.

Moreover, as we shall see

later, it would follow that every locally nilpotent $--group is nilpotent.

3.

CHIEF FACTORS:

THE THEOREM OF APPS

The result (5) has other implications for the structure of £•-groups, apart from those for characteristically simple *•-groups.

It is

well known that locally nilpotent chief factors of locally finite groups are Abelian (see for example Kegel and Wehrfritz [13], Corollary 1.B.4). Therefore the only possibilities for the infinite chief factors of an *•group are those listed under (a) and (b) in (5). Apps has made an important contribution by eliminating case (b):

(6)

Let G be a locally finite group of finite exponent.

If G has

conjugate spread d,, then each non-Abelian chief factor of G is a product of at most d copies of a finite simple group. Thus if the exponent of G is e and g(e) is the largest of the orders of the finite simple groups of exponent dividing e, then each nonAbelian chief factor of G has order at most g(e) . The main point of this result [2] is its finiteness assertion; more precise information can be obtained as follows.

A finite non-Abelian chief factor H/K of G is a

direct product of isomorphic non-Abelian simple groups St ,...,S r > and G permutes these by conjugation.

If s'K is a non-trivial element of Si then

(< s G > K)/K = H/K; however if gi ,...,gd of the groups Si,...,S .

G

G then each (sK)

x

lies in one

Because H/K has elements which have non-trivial

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projections into all of Si,...,S , we can conclude that r < d. The result (6) has two immediate consequences which should be mentioned.

If G is as in (6) and H/K is arnon-Abelian chief factor, then

G/C (H/K) acts faithfully as a group of automorphisms of H/K.

From the

structure of H/K as a direct product of at most d simple groups of order at most g(e), it follows that G/C (H/K) has order at most h(d,e) = d!(g(e)!) .

Thus the intersection T of all normal subgroups of G of

index at most h(d,e) centralizes all non-Abelian chief factors of G, and therefore is locally soluble (see for example Proposition l.B.ll of [13]). This proves

(6a)

If G is as in (6) and the exponent of G is e., then G is an

extension of a locally soluble group by a group which is residually(finite of order at most h(d,e)J. Now suppose further that G has a minimal characteristic subgroup Gi such that Gi — B(S) or Gi — B (S) for some non-Abelian finite simple group S.

Since Gi is not locally soluble and since Gi n T is a

characteristic subgroup of G we have Gi n T = 1, and so G can be embedded in G/Gi x G/T.

(6b)

Clearly G/T lies in a Cross variety.

We therefore have

If G is as in (6) and G has a minimal characteristic subgroup Q\

with Gi s B(S) or Ci = B~(S) for some non-Abelian finite simple group S, then (i)

G is nilpotent by residually finite if G/Gi is3 and

(ii)

G lies in a Cross variety if and only if G/Gi does. In trying to establish Conjectures 1 and 2, one might try to

argue by induction on the length of a maximal characteristic series.

The

result (6b) can be thought of as supplying part of the induction step. It would clearly be useful to have information about the Abelian chief factors of £•-groups.

However these seem much harder to work

with than the non-Abelian ones, and so far all attempts to show that they are all finite have failed.

If they could be shown to be finite, then

the argument used in proving (6a) would yield that every $--group is locally nilpotent by residually finite; and since for each n the intersection of all normal subgroups of index at most n is characteristic, it would follow that every $--group is locally nilpotent by residually(finite of bounded order).

If moreover Conjecture 3 could be proved,

then the truth of Conjecture 1 would follow.

Whether the chief factors

of an $--group are finite is also directly relevant to Conjecture 2,

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because of the criterion of Kovacs and Newman [15] for a group to lie in (and so to generate) a Cross variety: (7) A group G lies in a Cross variety if and only if there are integers e, m and c such that (a)

g e = 1 for all g e G,

(b) all chief factors of G have order at most in, and (c) all nilpotent sections of G have nilpotency class at most c. 4.

LOCALLY NILPOTENT *-GROUPS For locally finite groups of finite exponent, condition (c)

of (7) is equivalent to the requirement that all locally nilpotent subgroups be nilpotent.

However, as Conjecture 3 shows, we cannot even

exclude the possibility of non-nilpotent locally nilpotent *•-groups.

If

Conjecture 3 is true, and if G is a locally nilpotent 5--group, then the factors in a maximal characteristic series for G will all be Abelian and G will be soluble; the nilpotence of G then follows from (8) If G is a soluble locally nilpotent $r-group then G is nilpotent, I shall show how this can be deduced from a lovely theorem proved by Cher1in in [5] and [6]: (81)

Every locally nilpotent No-categorical ring is nilpotent. For our purposes a countable No-categorical ring can be taken

to be a countable ring R whose automorphism group has only finitely many orbits in its action on Rx ... xR for each positive integer n; the theorem of Engeler, Ryll-Nardzewski and Svenonius already cited in the context of group theory shows that this condition holds if and only if R is defined to isomorphism among countable rings by its first-order properties.

Cherlin provided two separate proofs for his theorem, one

for commutative rings, and a much harder one for arbitrary rings. Powerful though Cherlinfs methods in [5] and [6] are, they seem inadequate for an attack on Conjecture 3.

Not surprisingly, No-categorical rings

are in most respects easier to handle than No-categorical groups. We mention in passing that the analogue of Conjecture 1 for No-categorical rings is true:

this follows from (8f) together with results of Baldwin

and Rose [4]. In proving (8), we can reduce immediately by induction on derived length to the case in which G is metabelian:

if G is not

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353

metabelian then each of G/G" and G1 is a locally nilpotent $•-group of smaller derived length than G, and if each of these is nilpotent then so is G by a theorem of Hall [12]. We therefore suppose G metabelian. P f

We choose a e G acting by conjugation. of A is commutative.

and regard A = < a "" > as a ZZG-module with G

Thus the image R of 2Z G in the endomorphism ring

We denote by g the image of an element g of G.

For

each g £ G there exists an m such that a(g - l ) m = 0 because G is locally nilpotent, and we have ar'(g - l ) m = a(~g - 1 ) V for every rf e R, so that (g - l ) m = 0.

= 0 The ring E generated by all

elements g - 1 with g s G is therefore commutative and generated by nilpotent elements, and so is locally nilpotent.

The result (4) yields

an integer d such that for all r € R there exist df < d and gi , ...,g,, e G d d' _ satisfying ar = £ ag. , and because for each r ! £ R 1

d?

d

' ar'(r - £ g.) = (ar - J ag.) = 0, i=l d1 we have r = £ g.. i=l x

i=l

Therefore each element of R is a sum of at most d

elements of form g with g e G. Let 0 be the group of automorphisms of G fixing a.

The

elements of 0 induce automorphisms of 2Z G which map the annihilator of a to itself, and so they induce automorphisms of R.

Clearly these auto-

morphisms map E to itself and so their restrictions to E are automorphisms of E.

One way of overcoming the slight inconvenience that not every

element need be a sum of exactly d elements of the form g is to extend the action of Aut G to the disjoint union of G and a one-element set {0} by letting Aut G fix 0. to be zero.

The image of 0 in R will of course be understood

Choose n > 0, write n 1 = nd, and let (a, g,,,..., g , . . ) , . . . ,

(tf*g-i >-«»>g t ) be representatives of all orbits of Aut G on (G U {0})

containing (n'+l)-tuples with first component a.

Then

(g 11 ,.••,g nfl ),«••,(g ls ,.••,g n , s ) will be representatives of the orbits nf of 0 on (G U {0}) .

Because every element of R is a sum of d elements

of the form g with g e G U {0} it follows that every n-tuple of elements of R is equivalent under the action of 0 to

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d i=l for some j < s. Ex

nf

2d 1J

i=d+l

354

J

J

i=n'-d+l

Therefore 0 acts with only finitely many orbits on

... x E for each n, and E is No-categorical.

We can now apply

1

Cherlin's theorem (8 ) to conclude that E is nilpotent.

This implies

that there is an integer c such that «(gi - 1) • • • (gc - 1) = 0

for all gi ,... ,gQ e G.

In multiplicative notation this becomes l>,gi , . . . ,g c ] = 1

for all gi , . . . ,g c e G,

which is the assertion that a lies in the c'th term of the upper central series of G.

The same holds for each element of G' in the same orbit of

Aut G as a; and since a could have been chosen, from any of the finitely many orbits of Aut G on G1 it follows that G1 lies in a term of the upper central series of G, and that G is nilpotent.

5.

METANILPOTENT BY FINITE *- GROUPS We now combine some of the earlier results and methods to

show

(9)

If G is a metanilpotent by finite ^Sc-group* then (i) G is nilpotent

by residually finite and (ii) G lies in a Cross variety. The first step in the proof is to reduce to metanilpotent groups.

It is easy to show that if G is a group and Go a subgroup of

finite index, then G is nilpotent by residually finite if and only if Go is and (by (7) for example) G lies in a Cross variety if and only if Go does.

On the other hand every metanilpotent by finite group G has a

characteristic metanilpotent subgroup Go of finite index (namely the join of all metanilpotent normal subgroups), and if G is an $--group then so is Go.

Therefore we need only consider metanilpotent groups.

Of

course the first sentence in the above reduction shows that in general in trying to establish Conjectures 1 and 2 one can restrict attention to ^--groups having no proper characteristic subgroups of finite index. To establish assertion (ii) of (9) we shall use the criterion (7) for a group to belong to a Cross variety.

Condition (a) is auto-

matically fulfilled for ^--groups, and to verify (c) for an £-group G it suffices to show that the p-subgroups of G are nilpotent for each of the

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primes p dividing the exponent of G.

355

Let p be such a prime, let Q be the

1

largest normal p -subgroup of G and let H/Q be the smallest normal subgroup of G/Q with quotient group a p!-group.

Then each p-subgroup of G

is isomorphic to a subgroup of H/Q, and since both H and Q are characteristic subgroups, H/Q is an £--group.

If G is metanilpotent then

H/Q will also be a p-group, and so will be nilpotent, of class c, say, by (8).

Thus all p-subgroups of G have class at most c, and condition (c)

of (7) is satisfied. To establish condition (b), it will suffice to prove the following result:

(10)

If G is a metaniVpotent group of conjugate spread d and exponent e,

then each chief factor of G has order less than e This result also implies that every metanilpotent 5--group G is nilpotent by residually finite, for if U is the intersection of the centralizers of the chief factors of an $--group G, then G/U will be residually finite by (10) , and U being a metanilpotent locally nilpotent characteristic subgroup of G, will be nilpotent by (8). Thus the proof of both (i) and (ii) will be complete if we establish (10). If G is a group and V a TL G-module, we say that G acts with spread at most d on V if, whenever v £ V and x £ v Z G , there exist gi »-«*»gj e G such that x lies in the subgroup of V generated by vgi ,...,vg,.

Thus a group with conjugate spread d acts with spread at

most d on each of its Abelian chief factors.

Lemma.

Let G act with spread at most d on a non-zero TL ^-module V.

If

V is induced from a TZH-submodule U for some subgroup H of G, then the index r of H in G (is finite and) satisfies r < d, and H acts with spread at most df on U, where d1 = [d/r]. Proof.

Let ti, t2 , ... be a right transversal to H in G, and choose an

element Uo / 0 of U.

Each element Uog with g e G lies in one of the

subgroups Ut., and so, because of the structure of V as an induced 1 s module, an element of the form J uot. can lie in no subgroup generated 1 il by fewer than s elements of the form uog. It follows that r < d. r Now let u e U and y e u S H , and write x = £ yt. . Since G 1 i=l acts with spread at most d on V, there are gi ,.. . ,g^ £ G and m ,... ,n^ e TL

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356

d £ n.ug.. One of the cosets Ht. must contain at most df x j=l 3 3 of the elements g.; after renumbering we may suppose Hti n {gi ,...,g,} = such that x =

{gi ,...,g } with c < d f .

Write g. = h.ti for j < c.

Comparing

coefficients in Uti for the equation d J

x =

n.ug.

we obtain

y =

c \

n.uh..

This shows that H acts on U with spread at most d f , as required.

We now return to the proof of (10). A chief factor of a metanilpotent group can be regarded as an irreducible module for a nilpotent image of the group, so it is more than enough to prove the following:

if p is a prime, G a nilpotent group of exponent dividing &

and V an irreducible F G-module on which G acts with spread at most d, j 2

then |v| < p

P

.

(F

denotes the field of p elements.)

suppose that G acts faithfully on V.

Clearly we can

Let H be a subgroup of largest

possible index such that V is induced from an F H-submodule U.

Thus H

acts primitively on U, and so by Clifford's theorem if K is a normal subgroup of finite index in H then U is a direct sum of isomorphic irreducible F K-submodules.

Since G acts faithfully on U, it follows

that every such subgroup K has cyclic centre, of order at most e. K to have centre C(K) of order as large as possible.

Choose

If K is non-Abelian

we can find an element k G K\c(K) such that kc(K) G C(H/c(K)); then < C(K), k > is Abelian and normal in H, and its centralizer in H has finite index in H and has centre strictly containing C(K). Thus K is Abelian, and so both K and H are finite. normal subgroup of H.

Let A be a maximal Abelian

We have |A| < e, and since A = C^(A) and A is

cyclic we also have |H/A| < e-1 < e.

Therefore |H| < e2 , and because U

is an irreducible F H-module we deduce that |u| < p

.

Finally, V is a

direct sum of at most d subgroups of the form Ut with t G G, and so

|v|< P d e \ 6.

CONCLUDING REMARKS It will have been noticed that the methods used above,

especially in §5, seem closely tailored to the application in hand - in the case of §5, to metanilpotent groups.

The proof of the nilpotency of

the p-subgroups of a metanilpotent $--group is a good example:

no

modification of it is likely to yield information about an t-group of

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Fitting length three.

35 7

It is not clear whether this reflects the fact

that Conjectures 1 and 2 are simply too optimistic.

If so, it would not

be the first time that results about groups of Fitting length two have given no indication of the chaos that can rule in higher Fitting length. What is clear is that, if these conjectures are true, then much more powerful techniques will be required to prove them. The characterization of Abelian by finite ^--groups given by Cherlin and Rosenstein in [7] has already been mentioned.

Apps [3] has

given a very satisfactory description of ^--groups which are finite extensions of groups of type B(S), with S a finite non-Abelian simple group.

No doubt groups in other small subclasses of $- can be

characterized in other ways.

For example, let G be a metabelian £--group

whose p-subgroups are Abelian for all primes p dividing its exponent. The methods of §5 are sufficient to show that G is residually finite, and it could be that G must have the structure of a filtered Boolean extension of a finite group, in the sense of Schmerl [18]. Apps [1] has shown the existence of nilpotent 5--groups of class two with a variety of properties. cases of his results.

We mention just two special

First, if G is a countable nilpotent group of

class two with finite centre, and if G is a central product of subgroups of bounded orders, then G £ 3K

Second, for every odd prime p, $• contains

p-groups G of class two such that (i) 1, G1 and G are the only characteristic subgroups of G, (ii) both G1 and G/Gf are infinite, and (iii) every subgroup of finite index in G contains G f .

Such groups are

obviously very far from being residually finite. Finally, I should say that this account of No-categorical groups is not intended to reflect all of the work which has been done on the subject.

An interesting account of categoricity in its model-

theoretic setting can be found in Feigner [11].

Acknowledgement.

The article was written while the author held an

Alexander von Humboldt Fellowship.

REFERENCES

1. A.B. Apps, On No-categorical class 2 groups, to appear. 2. A.B. Apps, On the structure of No-categorical groups, to appear. 3. A.B. Apps, No-categorical finite extensions of Boolean powers, to appear.

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4. J. Baldwin § B. Rose, No-categoricity and stability of rings, J. Algebra 45^(1977), 1-16. 5. G. Cherlin, On No-categorical nilrings, Algebra Universalis 10(1980), 27-30.

6. G. C h e r l i n , On No-categorical n i l r i n g s I I , J. Symbolic (1980), 291-301.

Logic 45_

7. G. Cherlin § J.G. Rosenstein, On No-categorical Abelian by finite groups, J. Algebra 5_3(1978), 188-226. 8. E. Engeler, A characterization of theories with isomorphic denumerable models, Notices Amer. Math. Soc. 6^(1959), 161. 9. U. Feigner, Stability and No-categoricity of nonabelian groups, in Logic Colloquium 76 (ed. R.O. Gandy § J.M.E. Hyland), North Holland, Amsterdam (1977). 10. U. Feigner, On No-categorical extra-special p-groups, Logique et Anal. (N.S.) 118(1975), 407-428. 11. U. Feigner, Kategorizitat, Jahresber. Deutsch. Math.-Verein. 82_ (1980), 12-32. 12. P. Hall, Some sufficient conditions for a group to be nilpotent, Illinois J. Math. 2^(1958), 787-801. 13. O.H. Kegel § B.A.F. Wehrfritz, Locally finite groups* North Holland, Amsterdam (1973). 14. Kourovka notebook, Unsolved problems in group theory, Novosibirsk (1978). 15. L.G. Kovacs § M.F. Newman, Cross varieties of groups, Proc. Roy. Soc. London A 292^(1966), 530-536.

16. H. Neumann, Varieties (1967).

of groups* Springer, Berlin Heidelberg New York

17. C. Ryll-Nardzewski, On the categoricity in power < N o , Bull. Acad. Polon. Sir. Sci. Math. Astronom. Phys. 7_(1959), 545-548. 18. J.H. Schmerl, On the categoricity of filtered Boolean extensions, Algebra Universalis 8^(1978), 159-161. 19. L. Svenonius, No-categoricity in first-order predicate calculus, Theoria 25(1959), 82-94. 20. M.R. Vaughan-Lee % J. Wiegold, Countable locally nilpotent groups of finite exponent without maximal subgroups, Bull. London Math. Soc. 12K1981) , 45-46. 21. J. Waskiewicz § B. Weglorz, On o)o-categoricity of powers, Bull. Acad. Polon. Ser. Sci. Math. Astronom. Phys. 17_(1969), 195-199.

359

ABSTRACTS

Several papers were submitted to the conference proceedings and would have been included in this volume but for lack of space. begin with abstracts of two such papers.

We

The first is of a paper by

R.T. Curtis, University of Birmingham, B15 2TT, England which has been submitted to J. Combinatorial Theory, while the second is of a paper by J. Perraud, University of Nantes, 44072 Nantes, France which has been submitted for publication in Proc. Roy. Soc. Edinburgh.

Eight ootads suffice.

A set of octads is said to define a particular

Steiner system if it is a subset of the special octads of that system, but of no other.

The paper shows that it is possible to produce a set of

eight octads which defines the Steiner system % (5,8,24), whereas any collection of seven octads is a subset of no Steiner system or of more than one.

Note that, given an element of S 2 4, one is now able to decide

whether it is in M24 by simply applying it to a defining set of eight octads.

If the images are in the system the element must be in M24 .

On small cancellation theory over HNN-extensions.

Small cancellation

conditions C'(l/6), Cf(l/4) and T(4) are defined for finite subsets of an HNN-extension F in such a way that it can be decided whether any finite subset of F satisfies these conditions.

The word problem is solved for

quotients F/N where N is the normal closure of any finite subset of F which satisfies the condition C'(l/6) or the conditions C'(l/4) and T(4). It is proved that the condition C'(l/6) for symmetrized subsets of F implies the condition Cf(l/6) for finite subsets, in the following way: for each finite subset X there is a symmetrized subset Xi with the same normal closure, and if Xi satisfies the condition C'(l/6) for symmetrized subsets, then X satisfies the condition C'(l/6) for finite subsets.

360 Finally we give a preliminary report of work by A.C. Kim, Busan National University, Pusan, Korea 607. Laws in groupoids derived from groups. Let y be a binary operation on ZZ defined by XYy = aX + bY for fixed a,b e 7L . Let G(a,b) denote TL with the operation y thus defined. Then G(l,l) is the additive group 7L . The main result of this work is that the one-variable laws of G(a,b) are not finitely based provided a / 1, b / 1. The proof is in two steps (i) finding a one-variable law in G(a,b) (ii) deducing infinitely many onevariable laws in G(a,b).

ADDENDUM TO AN ELEMENTARY INTRODUCTION TO COSET TABLE METHODS IN COMPUTATIONAL GROUP THEORY COLIN M. CAMPBELL*, GEORGE HAVAS^and EDMUND F. ROBERTSON* * School of Mathematics and Statistics University of St Andrews North Haugh, St Andrews, Fife KY16 9SS, Scotland Email: [email protected], [email protected] and ^ARC Centre for Complex Systems School of Information Technology and Electrical Engineering The University of Queensland, Queensland 4072, Australia Email: [email protected]

Even after 25 years the article [30] by Joachim Neubiiser remains the first source to which all three of us refer those who want to find out about the use of coset tables for studying groups. Our view is confirmed by the 14 Reference Citations from 1998 to 2005 which MathSciNet [1] reveals for this article. Here we loosely follow the structure of the original article and provide some updates on the area (oriented towards our own interests). First we point out that two newer books [35, 22] include comprehensive details on coset enumeration and related topics in works which are much broader studies. They give excellent coverage of the areas addressed in this article and, further, provide much additional material. They also provide some alternative points of view and many references (as do the other materials cited here). One of Neubiiser's aims in writing his survey was to provide a unified view on coset table methods in computational group theory. He addressed the way coset table concepts were developed, implemented and used. In [22] Derek Holt follows the same kind of approach, including a long chapter "Coset Enumeration" and a shorter one "Presentations for Given Groups". Charles Sims in [35] focuses on finitely presented groups and he takes a perspective significantly based on some fundamental methods from theoretical computer science, namely automata theory and formal languages. He includes three chapters specifically relevant to coset table methods: "Coset enumeration"; "The Reidemeister-Schreier procedure"; and "Generalized automata"; with some extra implementation issues covered in an Appendix. He concludes his coset enumeration chapter with a section which points out that the Knuth-Bendix procedure can sometimes be used more effectively to enumerate cosets than Todd-Coxeter methods. Among the available computer implementations of coset enumeration procedures are those in the computer algebra systems GAP [14] and MAGMA [5] and a standalone program, the ACE coset enumerator [18]. An implementation is also available via quotpic [23], a software package with a nice graphical interface. A particularly useful tool for small-scale experiments with coset enumerations is the Interactive 1

The second author was partially supported by the Australian Research Council

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Todd-Coxeter package, ITC [12]. Many aspects of implementation and performance issues are addressed in [16, 17], including some comparisons with Knuth-Bendixbased methods. Much work has been done on practical strategies for successful coset enumeration. Accessible introductions to readily-available strategies can be found in the documentation for the GAP package ACE [13] and in the MAGMA manual [29]. Neubiiser describes several kinds of information available from coset tables. Such information can be readily extracted from the various implementations. A more recent program, PEACE (Proof Extraction After Coset Enumeration), gives users the opportunity to uncover proofs from the workings involved in coset enumerations. It is based on much earlier work of John Leech [24, 25]. Details appear in the Groups St Andrews 2005 proceedings [19] with a significant application in [21]. Reidemeister-Schreier-based methods for finding presentations of subgroups are described in [35, 22, 3]. The systems GAP, MAGMA and quotpic all incorporate efficient implementations. Applications of such methods continue to be widely used to address problems in finitely presented groups; see for example [11, 4]. They have also been extended to work in other structures, such as semigroups (see for example [33]) and Lie algebras. Initial implementations computed presentations on a set of Schreier generators for the subgroup and followed by simplification techniques. Subsequent, more complicated, algorithms utilise an augmented coset table which enables the construction of presentations on user-given sets of subgroup generators. Ideas which allow such a modified algorithm to be implemented more efficiently are described in [2] and such ideas are incorporated in GAP and MAGMA implement at ions. Neubiiser already gives some information about computing presentations for a concrete group and methods based on [7] are included in GAP and MAGMA. A newer method for finding short sets of defining relations is given by [15], which utilises ideas from double coset enumeration. Double coset enumeration is covered independently in [26]. Search-based methods for finding presentations with nice properties are used in [20, 6]. In his survey, Neubiiser describes a method for computing all subgroups of low index by systematically forcing coincidences in larger coset tables. Now, more recent implementations of low index subgroup algorithms are available in GAP, MAGMA and quotpic. They use another method, due to Sims, which does a backtrack search through incomplete coset tables. Recent adaptations of the low index subgroup algorithm are described in [9]. The Schreier-Sims algorithm is now well covered in material on permutation groups, including chapters in [34, 22]. It is used extensively in GAP and MAGMA. Its application to matrix groups is outlined in a recent survey [32, §7.5]. Neubiiser wrote that applications of coset table methods to group theoretical questions are too numerous to be listed in his article and are often hidden. This is even more valid now, 25 years later. Thus, most applications of GAP or MAGMA to finitely presented groups are likely to implicitly invoke coset enumeration and many other applications also do so. Recall there are no algorithms for answering quite simple questions about finitely presented groups, as Neubiiser [31] reminds

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us: there are "proofs of the non-existence of algorithms that could decide if a finitely presented group is trivial, finite, abelian, etc.", referring to [3] "for a vivid description". Often an appropriate way to start addressing a problem about a finitely presented group is to find some kind of permutation representation for the group, which is just what coset enumeration attempts to do. Thus, ask GAP or MAGMA for the order of a finitely presented group; unless the group is obviously infinite they both embark upon a coset enumeration (attempting to find the index of a cyclic subgroup whose order they also try to determine). One way for finding further information about applications is to try looking on MathSciNet. For example, a MathSciNet search "Anywhere" for "coset table OR coset enumeration OR Todd Coxeter" gave 172 matches in July 2006. Another way is to follow citations provided by papers in our admittedly limited bibliography. Neubiiser also wrote "we may also hope that we have not yet seen the last variation" on coset table methods. We finish by citing some other work which we have not mentioned above. This includes vector enumeration, Kan extensions, and parallel coset enumeration; see, for example, [27], [28], [8], and [10]. Additional note. Joachim Neubiiser has informed us that on page 16 of the original article it says: "(i) A coset Ug is contained in the normalizer NG(U) iff g~1Ug = U, i.e. iff g~1Ug < U and gUg~l < U. These two conditions are satisfied iff Ug~lsi = Ug~1 and Ugsi = Ug, . . . " He points out that the first condition is always enough, see page 114, exercise 17 (quoting a theorem of Takahasi) of Magnus, Karrass & Solitar (reference [41] of the original article). References [1] American Mathematical Society, MathSciNet Mathematical Reviews on the Web; http://www.ams.org/mathscinet [2] D. G. Arrell and E. F. Robertson, A modified Todd-Coxeter algorithm, in Computational Group Theory, Academic Press (1984), 27-32. [3] Gilbert Baumslag, Topics in Combinatorial Group Theory, Birkhauser (1993). [4] Jennifer Becker, Matthew Horak and Leonard VanWyk, Presentations of subgroups of Artin groups, Missouri J. Math. Sci. 10 (1998), 3-14. [5] W. Bosma, J. Cannon and C. Playoust, The Magma algebra system I: the user language, J. Symbolic Comput. 24 (1997), 235-265. See also http: //magma. maths. usyd. edu. au/magma/ [6] Colin M. Campbell, George Havas, Alexander Hulpke and Edmund F. Robertson, Efficient simple groups, Comm. Algebra 31 (2003), 5191-5197. [7] J. J. Cannon, Construction of defining relators for finite groups, Discrete Math. 5 (1973), 105-129. [8] S. Carmody, M. Leeming and R. F. C. Walters, The Todd-Coxeter procedure and left Kan extensions, J. Symbolic Comput. 19 (1995), 459-488. [9] Marston Conder and Peter Dobcsanyi, Applications and adaptations of the low index subgroups procedure, Math. Comp. 74 (2005), 485-497. [10] Gene Cooperman and Victor Grinberg, Scalable parallel coset enumeration: bulk definition and the memory wall, J. Symbolic Comput. 33 (2002), 563-585. [11] F. Digne and Y. Gomi, Presentation of pure braid groups, J. Knot Theory Ramifications 10 (2001), 609-623. [12] Volkmar Felsch, Ludger Hippe and Joachim Neubiiser, GAP package ITC Interactive Todd-Coxeter, 2004; http://www.gap-system.org/Packages/itc.html

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[13] Greg Gamble, Alexander Hulpke, George Havas and Colin Ramsay, GAP package ACE; Advanced Coset Enumerator, 2006; http://www.gap-system.org/Packages/ace.html [14] The GAP Group, GAP - Groups, Algorithms, and Programming, Version 4.4, 2006; http://www.gap-system.org/ [15] Volker Gebhardt, Constructing a short denning set of relations for a finite group, J. Algebra 233 (2000), 526-542. [16] George Havas and Colin Ramsay, Proving a group trivial made easy: a case study in coset enumeration, Bull Austral Math. Soc. 62 (2000), 105-118. [17] George Havas and Colin Ramsay, Experiments in coset enumeration, in Groups and Computation III, Ohio State Univ. Math. Res. Inst. Publ. 8, de Gruyter (2001), 183-192. [18] George Havas and Colin Ramsay, Coset enumeration: ACE version 3.001, 2001; http://www.itee.uq.edu.au/~havas/ace3001.tar.gz [19] George Havas and Colin Ramsay, On proofs in finitely presented groups, in Groups St Andrews 2005, London Math. Soc. Lecture Note Ser., Cambridge Univ. Press (to appear). [20] George Havas and Edmund F. Robertson, Irreducible cyclic presentations of the trivial group, Experiment Math. 12 (2003), 487-490. [21] George Havas, Edmund F. Robertson and Dale C. Sutherland, The Fa'b'c conjecture is true, II, J. Algebra 300 (2006), 57-72. [22] Derek F. Holt, Bettina Eick and Eamonn A. O'Brien, Handbook of Computational Group Theory, CRC Press (2005). [23] Derek F. Holt and Sarah Rees, A graphics system for displaying finite quotients of finitely presented groups, in Groups and computation, DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 11 (1993), 113-126. [24] John Leech, Computer proof of relations in groups, in Topics in Group Theory and Computation, Academic Press (1977), 38-61. [25] John Leech, Coset enumeration, in Computational Group Theory, Academic Press (1984), 3-18. [26] Stephen A. Linton, Double coset enumeration, in Computational group theory, Part 2, J. Symbolic Comput. 12 (1991), 415-426. [27] S. A. Linton, On vector enumeration, in Computational linear algebra in algebraic and related problems, Linear Algebra Appl. 192 (1993), 235-248. [28] S. A. Linton, Generalisations of the Todd-Coxeter algorithm, in Computational algebra and number theory, Math. Appl, 325, Kluwer Acad. Publ., (1995), 29-51. [29] The Magma Development Team. Interactive Coset Enumeration, available via http://magma.maths.usyd.edu.au/magma/htmlhelp/MAGMA.htm [30] J. Neubiiser, An elementary introduction to coset table methods in computational group theory, in Groups St Andrews 1981, London Math. Soc. Lecture Note Ser. 71, Cambridge Univ. Press (1982), 1-45. [31] J. Neubiiser, An invitation to computational group theory, in Groups '93 Galway/St. Andrews, Vol. 2, London Math. Soc. Lecture Note Ser. 212, Cambridge Univ. Press (1995), 457-475. [32] E. A. O'Brien, Towards effective algorithms for linear groups, in Finite Geometries, Groups, and Computation, Walter de Gruyter (2006), 163-190. [33] N. Ruskuc, Presentations for subgroups of monoids, J. Algebra 220 (1999), 365-380. [34] Akos Seress, Permutation group algorithms, Cambridge Tracts in Mathematics 152, Cambridge Univ. Press (2003). [35] C. C. Sims, Computation with finitely presented groups, Cambridge Univ. Press, 1994.

APPLICATIONS OF COHOMOLOGY TO THE THEORY OF GROUPS — A POSTSCRIPT DEREK J. S. ROBINSON

In 1981 the importance of homological algebra as a tool in group theory was beginning to be recognised. After the pioneering work in the 1940's by S. Eilenberg, S. MacLane and B. Eckmann on the homology and cohomology of groups, twenty years elapsed before really convincing applications appeared: the prime example was Gaschiitz's famous theorem on the existence of outer automorphisms of finite p-groups. The well known sets of notes by K. W. Gruenberg [5] and U. Stammbach [16], which were published in the 1970's, had proved to be a stimulus to research, and already a body of work had appeared in the literature. It seemed timely to write a survey for Groups St Andrews 1981. The twenty five years which have elapsed since that critical conference have witnessed a continuation of the trend in group theory to introduce techniques from homological algebra, as well as other areas of mathematics. Today many group theorists are conversant with a variety of homological methods, including spectral sequences. Our aim here is to survey some of the achievements during this period. Until about 1980 group theoretic interpretations of the cohomology groups Hn(G,M) had only been found only for n < 3; these arise of course from the classical theory of group extensions. The problem of finding group theoretic interpretations of Hn(G,M) for arbitrary n was solved by D. F. Holt [6] and J. Huebschmann [8]. They showed how to associate to an element of Hn(G,M) an equivalence class of crossed (n — l)-fold extensions, exact sequences of groups and modules beginning with M and ending in G. However, despite the elegance of this construction, its consequences for group theory remain unexplored at this date. One of the commonest uses of cohomology in group theory has been to establish splitting theorems by showing that a second cohomology group vanishes. In some cases where splitting does not occur, it is possible to establish near splitting, i.e., splitting up to finite index and finite intersection, which typically happens when the second cohomology group has finite exponent. This technique has been particularly useful in the study of soluble groups with finite rank. It has also played a significant role in the theory of just non-polycyclic groups (see Robinson and Wilson [15]). The first cohomology group has also found applications, since by showing that it vanishes, one can conclude that the complements of a normal subgroup in a semidirect product are conjugate. This method has been widely used in the study of soluble groups which are products of subgroups with prescribed properties. For a detailed account of this area see the book on products by Amberg, Franciosi and de Giovanni [1]. Indeed it is the natural occurrence of derivations in a product of groups which underlies this application. A high point in this line of research is to be found in the work of J. S. Wilson ([17], [18], [19]), and we quote one of his results: a soluble group which is a product of two minimax subgroups is a

mimimax group. Another less expected application has been to show that there is

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an algorithm which is able to decide if a given subgroup of a polycyclic-by-finite group is permutable (Robinson [14]). Cohomology has continued to be popular in the study of automorphisms of composite groups, where the Wells exact sequence can be used. We mention in particular work on the group of central automorphisms of a group — see for example, Pranciosi and de Giovanni [4], and Curzio et al. [3]. A very active area of research has been the study of groups, especially soluble groups, with finite homological or cohomological dimension. As a result a link between these homological invariants and the rank of a soluble group has been established. Initial results were found by Bieri, Gruenberg and Stammbach. It was shown that a soluble group has finite homological or cohomogical dimension if and only if it is torsion-free with finite total rank. Also the cohomological dimension exceeds the homological dimension in such a group by at most 1. There remained the very difficult problem of identifying the soluble groups G for which hd(G) = cd(G) < oo. This was finally solved by Kropholler [11], who showed that these are exactly the torsion-free constructible groups. (For an account of this theory see Bieri [2] and also Chapter 11 of [12]). One of the most astonishing recent applications of cohomology to group theory has been Kropholler's theorem on soluble minimax groups. A finitely generated soluble group is a minimax group if and only if it has no sections of the type Zp wr Z for any prime p. The proof involves a delicate analysis of the interplay between cohomology and direct limits of modules. It may also be mentioned that the relation between direct limits of groups and cohomology has been fruitful, thanks to work of C. Jensen [9], who describes a spectral sequence converging to the cohomology of a group which is expressed as a direct limit. Using this machinery the following theorem of Holt [7] can be proved: Let G be a locally finite group with \G\ = Km and m finite. Let M be an abelian torsion group such that G and M do not have elements of the same prime order, and suppose that M is a G-module. Then Hn(G,M) = 0 for n > m + 2. The spectral sequence has also been used to obtain vanishing theorems for the cohomology of locally nilpotent and locally supersoluble groups (see Robinson [13]). For further information about recent homological applications in group theory the reader is referred to the surveys [2] and [13], and also to Chapter 11 of [12]. As for the future, it seems that whenever methods from one area of mathematics are brought to bear on the problems of another area, a rich theory is likely to evolve. This trend seems set to continue in the case of homological algebra and group theory.

References [1] B. Amberg, S. Franciosi and F. de Giovanni, Products of Groups, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1992. [2] R. Bieri, Homological Dimension of Discrete Groups, 2nd ed., Mathematics Notes, Queen Mary College, London, 1981.

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[3] M. Curzio, D. J. S. Robinson, H. Smith and J. Wiegold, Some remarks on central automorphisms of hypercentral groups, Arch. Math. (Basel) 53 (1989), no. 4, 327331. [4] S. Franciosi and F. de Giovanni, On central automorphisms of finite-by-nilpotent groups, Proc. Edinburgh Math. Soc. (2) 33 (1990), no. 2, 191-201. [5] K. W. Gruenberg, Cohomological Topics in Group Theory, Lecture Notes in Mathematics 143, Springer-Verlag, Berlin-New York, 1970. [6] D. F. Holt, An interpretation of the cohomology groups Hn(G, M), J. Algebra 60 (1979), no. 2, 307-320. [7] D. F. Holt, On the cohomology of locally finite groups, Quart. J. Math. (2) 32 (1981), 165-172. [8] J. Huebschmann, Crossed n-fold extensions of groups and cohomology, Comment. Math. Helv. 55 (1980), no. 2, 302-313. [9] C. U. Jensen, Les foncteurs derives de lira et leurs applications en theorie des modules, Lecture Notes in Mathematics 254, Springer-Verlag, Berlin-New York, 1972. [10] P. H. Kropholler, On finitely generated soluble groups with no large wreath product sections, Proc. London Math. Soc. (3) 49 (1984), no. 1, 155-169. [11] P. H. Kropholler, Cohomological dimension of soluble groups, J. Pure Appl. Algebra 43 (1986), no. 3, 281-287. [12] J. C. Lennox and D. J. S. Robinson, The Theory of Infinite Soluble Groups, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2004. [13] D. J. S. Robinson, Cohomology in infinite group theory, in Group theory (Singapore, 1987), 29-53, de Gruyter, Berlin, 1989. [14] D. J. S. Robinson, Derivations and the permutability of subgroups in polycyclic-byfinite groups, Proc. Amer. Math. Soc. 130 (2002), no. 12, 3461-3464. [15] D. J. S. Robinson and J. S. Wilson, Soluble groups with many polycyclic quotients, Proc. London Math. Soc. (3) 48 (1984), no. 2, 193-229. [16] U. Stammbach, Homology in Group Theory, Lecture Notes in Mathematics 359, Springer-Verlag, Berlin-New York, 1973. [17] J. S. Wilson, Soluble groups which are products of minimax groups, Arch. Math. (Basel) 50 (1988), no. 3, 193-198. [18] J. S. Wilson, Soluble products of minimax groups and nearly surjective derivations, J. Pure Appl. Algebra 53 (1988), no. 3, 297-318. [19] J. S. Wilson, Soluble groups which are products of groups of finite rank, J. London Math. Soc. (2) 40 (1989), no. 3, 405-419.

GROUPS WITH EXPONENT FOUR SEAN TOBIN In the quarter of a century which has elapsed since the St Andrews meeting in 1981 a lot of information has been gathered about Burnside groups, whether free or restricted. We will use RBP generally as shorthand for the restricted Burnside problem and in this case the work of Kostrikin [52] and Zelmanov [66] and [67] has established the local finiteness property. Their success in obtaining an affirmative answer to the RBP for all prime-power exponents and so, by the theorem of Hall & Higman [26], for all finite exponents has naturally directed attention to the structure of such groups — in particular, questions about order, class and derived length. Much of the work — including further studies of B(r, 4) — has depended on (a) Computer-aided calculations, using Todd-Coxeter coset enumeration at first, and more recently employing nilpotent quotient algorithms which produce presentations of groups through power-commutator relations. Pioneers in this work were John Leech [31] and I. D. Macdonald [33] and [54]; the procedures involved are described in Appendix B of [61]. (b) Linearising problems about group commutators by studying the associated Lie algebras, yielding connections between commutator identities in certain groups and identities in associated Lie rings. Kostrikin in his book [52] refers to classical papers of Magnus, Griin, Zassenhaus and Baer in the early nineteen-forties, followed by contributions of Lazard and Higman in the nineteen-fifties, which drew attention to the connection between the RBP for a prime exponent p and local nilpotency of an associated Lie algebra of characteristic p. The most convenient way to explore developments in the decade after the 1981 Conference is to read the book [61] by Michael Vaughan-Lee. He gives a very good description of the main results, with proofs, often with the introduction of useful notations and worthwhile simplifications. The book includes "... a relatively short proof of Kostrikin's Theorem" (on Lie algebras with an Engel condition), also "... a treatment of Razmyslov's theorem about non-solvable groups of prime-power exponent, and a treatment of groups of exponent four." Furthermore, the book in its second edition includes a new chapter of forty pages on Zelmanov's solution of the RBP for prime-power exponents. In the original proof Zelmanov used Jordan algebras, but in his book Vaughan-Lee gives a proof depending only on Lie algebras (a theme to which he returns in [62] where he asserts that "the key to the solution of RBP for groups of prime-power exponent lies in Lie algebras"). The importance of groups of exponent four, not only because of their intrinsic interest, but also as a test-bed so to speak for application of multilinear identities, is reflected in the fact that over a fifth of the book is devoted to "Chapter 6: Groups of exponent four". Before proceeding further I wish to correct an error in section 2 B) supra where

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I wrongly transcribed the presentation for B(2,4) given in Coxeter & Moser [9] by omitting the relation (a 2 6 2 ) 4 = 1 and referring to the subgroup (a 2 , b) rather than (a, b2). In fact I had inadvertently substituted the presentation proposed in [25] — which could not be correct since Macdonald [54] had shown that B(2,4) cannot be presented by fewer than nine relations. In 1981 I was able to include in my survey the result of A. J. S. Mann on the orders of groups of exponent four; see Theorem 5.10 supra. The precise result proved is: T h e o r e m ( M a n n [56]) If d > 1 then 4d < 21og2 |S(d,4)| < (4 + 2>/2)d. Mann's paper starts with the relation [ai,<22,... ,a/~]2 e 7&+2G when G has exponent four (see Theorem 2.1 supra) and uses identities deduced from Lie ring relations as well as from computed power-commutator presentations for 5(2,4). An outline of his work is given already in the comments before Theorem 5.10. In her thesis [55] Anne-Marie McManus has reduced the upper bound shown here, by further careful examination of the Lie ring identities. Her work naturally relies heavily on manipulation of these identities, using an improved nilpotent quotient algorithm adapted for the study of rings. It may be appropriate at this point to mention that the precise order of 5(5,4) is 2 2728 (Newman & O'Brien [57], but see also remarks at the beginning of Chapter 6 in [61]). The search for bounds to the orders of finite images of B(d,n) has been taken forward by Vaughan-Lee and Zelmanov in [63], [64] and [65] where, apart from the technical work, they make some interesting remarks on the problem of establishing upper bounds for the orders and suggest parallels with aspects of Ramsey theory. These researches culminated in the paper [49] by Groves and Vaughan-Lee. In all cases the upper bounds are very large, given by a kind of exponential "stairs" with a huge number of steps. The best result is as follows: T h e o r e m (Groves & Vaughan-Lee [49]) Let G be a finite d-generator group of exponent n. Define the function T: N x N - ^ N inductively: T(d, l) = d

and, for n>l,

T(d, n + 1) =

dT^n).

Then This improves on the result in [65] by giving 4 n as the exponent of n in the expression for the bound, rather than the previous nn. We next consider three papers written by Martyn Quick about 2$4, the Burnside variety of all groups with exponent four. In [58] he establishes the structure of the last two nontrivial lower central factors of B(r, 4), namely 737—3/73r-2 and 737—25 where r > 3. Each is an elementary abelian 2-group; the ranks were already determined in Chapter 6 of [61] and the structure is given in terms of GL(r, 2)-modules

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over the field of two elements. The basic idea here is that, writing G = B(r, 4), the action of Aut G on G/G2 gives rise to a homomorphism Aut G —• GL(r, 2). Using results from [45] yields the following: Theorem C (Quick, 1996) Letr>3 be an integer, G = B(r,4) be the Burnside group of exponent four on free generators a\, <22,..., ar and w(xi,X2, • • •, xr) be a word in r variables such that the value w(a\, <22,..., ar) is a nontrivial element in the penultimate nontrivial term 737.-3 of the lower central series of G. Then any subvariety of the Burnside variety $$4 in which w defines a law is soluble. In [59] and [60] Quick derives further information about subvarieties of V&4. As background we may note that $$4 is itself a subvariety of the 5-Engel variety £5 (see Theorem 5.9 supra); it is insoluble (otherwise termed non-solvable), but not minimally so since (see [46]) it contains the insoluble group F2 which is 4-Engel, where F represents the relatively free group of countably infinite rank in $$4. Furthermore, in [21] it is shown that the variety V&4 D (83 is soluble. This lends particular interest to Quick's discussion of the variety V&4 D £4, concerning questions of solubility and finite bases for subvarieties. In [60] relying on power-commutator presentations computed for several Burnside groups of exponent 4 he proves (Theorem 2) that the 4-Engel verbal subgroup of F is a minimal non-trivial verbal subgroup; this gives: Theorem A The variety V&4 D (84 of all 4-Engel groups of exponent 4 is a maximal subvariety of the Burnside variety 534. A consequence (Corollary 3) is that if V&4 contains a subvariety which is not finitely based then so also does 2$4 D €4. Using results from [44] and [53] he derives: Theorem C Any variety of soluble groups of exponent 4 is finitely based. He introduces a law £: [y, x, x, x, z] [y, x, x, y, z, x] [y, x, x, z, x, z, z] [z2, x, y2, x,x] = l

such that the subvariety £ of V&4 determined by £ is 4-Engel, is a proper subvariety of 2$4 D £4 and contains the insoluble group F2. This gives: Theorem D The variety of 4-Engel groups of exponent 4 is not minimally insoluble. In [59] Quick determines (i) all the subvarieties of V&4 D £4 which contain £, (ii) all other subvarieties of V&4 which contain £.

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The varieties in (ii) form a single ascending chain from £ to ^ 4 ; the varieties in (i) form a chain from £ to 2$40(84, in 1-to-l correspondence with the members of (ii); each of (i) being a subvariety of the corresponding member in (ii). It is not known whether these chains are finite or countably infinite. All these subvarieties, since they contain £, are insoluble, and all are finitely based. Finally, the existence of these chains reduces the finite basis problem for 2$4 to the problem for £: Theorem (Quick, 1997) If there exists a subvariety of 2$4 which is not finitely based then there is some subvariety of £ which is not finitely based. In conclusion, one can find a fairly brisk, but readable and comprehensive, survey of the Burnside problem and known finiteness results in [50] by Narain Gupta. He points out that the techniques developed to study Burnside's problem have revealed some deep facts in group theory; and, for example, explains the construction and properties of the Gupta-Sidki infinite 2-generator p-group (which is periodic but the periods are not bounded). Another excellent survey on the general Burnside problem appears more recently as a Report by M. F. Newman, entitled A Still Undecided Question, in the CMA Research Reports series of the ANU, Canberra, dated March 2000. It is available on the Internet at wwwmaths. aim. edu. au/research. reports/mrr/00/007/. Each of these authors has made a huge contribution to the study of Burnside groups, and each in his survey quotes a remark by Magnus & Chandler comparing Burnside's Problem to Fermat's Last Theorem as a catalyst for research. I quote Gupta's concluding sentence as a suitable one with which to end this article: "The fascination exerted by a problem with an extremely simple formulation which then turns out to be extremely difficult has something irresistible about it to the mind of the mathematician."

Additional References [49] D. P. Groves and M. R. Vaughan-Lee, Finite groups of bounded exponent, Bull. London Math. Soc. 35 (2003), 37-40. [50] N. Gupta, Groups in which every element has finite order, Amer. Math. Monthly 96 (1989), 297-308. [51] P. Hall, A contribution to the theory of groups of prime-power order, Proc. London Math. Soc. (2) 36 (1934), 29-95. [52] A. I. Kostrikin, Around Burnside, Moscow, Nauka (1986); English translation by James Wiegold, Springer (1990). [53] A. N. Krasil'nikov, The identities of a group with nilpotent commutator subgroup are finitely based, Math. USSR Izv. 37 (1991), 539-553. [54] I. D. Macdonald, A computer application to finite p-groups, J. Austral. Math. Soc. 17 (1974), 102-112. [55] Anne-Marie McManus, Computing in Lie Rings, Ph.D. Thesis, A.N.U., Canberra, 1996. [56] A. J. S. Mann, On the orders of groups of exponent four, J. London Math. Soc. 26 (1982), 64-76.

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[57] M. F. Newman and E. A. O'Brien, Applications of computers to questions like those of Burnside II, Internat J. Algebra Comput. 6 (1996), 593-605. [58] M. Quick, The module structure of some lower central factors of Burnside groups of exponent four, Quart J. Math. 47 (1996), 469-492. [59] M. Quick, A classification of some insoluble varieties of groups of exponent four, J. Algebra 197 (1997), 342-371. [60] M. Quick, Varieties of groups of exponent four, J. London Math. Soc. (2) 60 (1999), 747-756. [61] M. R. Vaughan-Lee, The Restricted Burnside Problem (2nd Edition), London Math. Soc. Monographs (N.S.) 5, Oxford University Press (1993). [62] M. R. Vaughan-Lee, On Zelmanov's Theorem, J. Group Theory 1 (1998), 65-94. [63] M. R. Vaughan-Lee and E. I. Zelmanov, Upper bounds in the restricted Burnside problem, J. Algebra 162 (1993), 107-145. [64] M. R. Vaughan-Lee and E. I. Zelmanov, Upper bounds in the restricted Burnside problem II, Internat. J. Algebra Comput. 6 (1996), 735-744. [65] M. R. Vaughan-Lee and E. I. Zelmanov, Bounds in the restricted Burnside problem, J. Austral. Math. Soc. (Ser. A) 67 (1999), 261-271. [66] E. I. Zelmanov, The solution of the restricted Burnside problem for odd exponent, Izv. Math. USSR 36 (1991), 41-60. [67] E. I. Zelmanov, The solution of the restricted Burnside problem for 2-groups, Mat. Sb. 182 (1991), 568-592.

ADDENDUM TO "THE SCHUR MULTIPLIER: AN ELEMENTARY APPROACH" JAMES WIEGOLD

Firstly, the book [A7] is devoted to the Schur multiplier, with much wider scope than in these notes. Secondly, progress has been made on some of the problems posed. All groups are assumed finite. Additional references are in the form [Ax]. 1. Problem 5.3. Here we were looking at the deficiency def(Gn) of the n-th direct power Gn of a perfect group G with trivial multiplier. The question as to whether def(Gn) tends to infinity with n is extremely difficult. The best known so far [Al] is that the deficiency of *SZ(2,5)2 is zero. See also [A2], [A3], [A4] where the deficiencies of low powers of PSL(2,p) are investigated. 2. Problem 6.1. This asks: when exactly is the multiplier of a finite p-group trivial? Another hard problem. I mentioned that Andozhskii [1] had made infinitely many strictly 3-generator p-groups with trivial multiplier for p odd. The analogous result for p = 2 can be deduced from Jamali's article [A6]. 3. In [21], David Johnson showed that a non-cyclic p-group generated by elements of order p has non-trivial multiplier. This is generalized in [A12], where it is shown that such a group G sits at the bottom of an infinite properly ascending chain G = Gi < G2 < ... Gn < ... of p-groups in which G^+i) is a Schur cover for Gi for each i. 4. Are all p-groups with trivial multiplier tricyclic? Tricyclic in the sense of product of three cyclic subgroups. This verbally communicated problem of Wamsley is still open. Personally, I doubt it, even though there is no evidence to support this belief. Notice that if true, it would imply that groups with trivial multiplier would have large exponent compared to the order — something like the cube root. Maybe one should tackle this exponent problem first. M. F. Newman has recently done a computer check [A10] of all 2-generator 2-groups of orders up to 210, and found no exception to the tricyclicity hypothesis. 5. Problem 6.4. This has been answered positively. Lubotzky and Mann [A9] used the theory of powerful p-groups to show that the rank of the multiplier of a p-group of rank r is indeed bounded in terms of r. 6. Problems of a type not mentioned in the original article. 6.1. Is the solubility length of a finite group of deficiency zero bounded? Kenne [A8] has made a 2-generator 2-relator group of solubility length 6. In [A5] Havas, Holt, Kenne and Rees give a soluble group of derived length 7. Thus any bound must be at least 7.

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6.2. Is the nilpotency class of a non-met acyclic p- group of zero deficiency bounded, p > 3? There are [All], [A 13] examples of 2-generator 2relator non-metacyclic 2-groups of arbitrarily large nilpotency class; and similarly for p = 3 [A 13]. Finally, there are two serious misprints in the Exercise at the top of Page 145 of the original; it is obvious that two occurrences of "G" should be "M(G)". Much worse than that, the claims made in the Exercise about which groups have multipliers of maximal exponent compared to the order are false. Edmund Robertson has found a number of counterexamples. Warm thanks go to Colin Campbell, Edmund Robertson and Mike Newman for very helpful assistance in producing this Addendum. Additional References [Al] C. M. Campbell, E. F. Robertson, T. Kawamata, I. Miyamoto and P. D. Williams, Deficiency zero presentations for certain perfect groups, Proc. Roy. Soc. Edinburgh Sect. A 103 (1986), 63-71. [A2] C. M. Campbell, E. F. Robertson and P. D. Williams, Efficient presentations for the groups PSL(2,p) x PSL(2,p), p prime, J. London Math. Soc. (2) 41 (1990), 69-77. [A3] C. M. Campbell, E. F. Robertson and P. D. Williams, On the efficiency of some direct powers of groups, in Groups—Canberra 1989, Lecture Notes in Math. 1456, Springer, Berlin (1990), 106-113. [A4] C. M. Campbell, I. Miyamoto, E. F. Robertson and P. D. Williams, The efficiency of PSL(2,p)s and other direct powers of groups, Glasgow Math. J. 39 (1997), 259-268. [A5] George Havas, Derek F. Holt, P. E. Kenne and Sarah Rees, Some challenging group presentations, J. Austral. Math. Soc. (Ser. A) 67 (1999), 206-213. [A6] Ali-Reza Jamali, A further class of 3-generator 3-relator finite groups, Comm. Algebra 29 (2001), no. 2, 879-883. [A7] G. Karpilovsky, The Schur Multiplier, London Math. Soc. Monographs, New Series, Oxford 1987. [A8] P. E. Kenne, Some new efficient soluble groups, Comm. Algebra 18 (1990), no. 8, 2747-2753. [A9] A. Lubotzky and A. Mann, Powerful p-groups I. Finite groups, J. Algebra 105 (1987), 484-505. [A 10] M. F. Newman, unpublished. [All] E. F. Robertson, A comment on finite nilpotent groups of deficiency zero, Canad. Math. Bull. 23 (1980), no. 3, 313-316. [A12] James Wiegold, On a result of Johnson about Schur multipliers, Glasgow Math. J. 34 (1991), 347. [A13] James Wiegold, On some groups with trivial multipliers, Bull. Austral. Math. Soc. 40 (1989), 331-332.