LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES Managing Editor: Professor J.W.S. Cassels, Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, 16 Mill Lane, Cambridge CB2 1SB, England The titles below are available from booksellers, or, in case of difficulty, from Cambridge University Press. 34 Representation theory of Lie groups, M.F. ATIYAH et al 36 Homological group theory, C.T.C. WALL (ed) 39 Affine sets and affine groups, D.G. NORTHCOTT 46 p-adic analysis: a short course on recent work, N. KOBLITZ 50 Commutator calculus and groups of homotopy classes, H.J. BAUES 59 Applicable differential geometry, M. CRAMPIN & F.A.E. PIRANI 66 Several complex variables and complex manifolds II, M.J. FIELD 69 Representation theory, I.M. GELFAND et al 76 Spectral theory of linear differential operators and comparison algebras, H.O. CORDES 77 Isolated singular points on complete intersections, E.J.N. LOOIJENGA 83 Homogeneous structures on Riemannian manifolds, F. TRICERRI & L. VANHECKE 86 Topological topics, I.M. JAMES (ed) 87 Surveys in set theory, A.R.D. MATHIAS (ed) 88 FPF ring theory, C. FAITH & S. PAGE 89 An F-space sampler, N.J. KALTON, N.T. PECK & J.W. ROBERTS 90 Polytopes and symmetry, S.A. ROBERTSON 92 Representation of rings over skew fields, A.H. SCHOFIELD 93 Aspects of topology, I.M. JAMES & E.H. KRONHEIMER (eds) 94 Representations of general linear groups, G.D. JAMES 95 Low-dimensional topology 1982, R.A. FENN (ed) 96 Diophantine equations over function fields, R.C. MASON 97 Varieties of constructive mathematics, D.S. BRIDGES & F. RICHMAN 98 Localization in Noetherian rings, A.V. JATEGAONKAR 99 Methods of differential geometry in algebraic topology, M. KAROUBI & C. LERUSTE 100 Stopping time techniques for analysts and probabilists, L. EGGHE 101 Groups and geometry, ROGER C. LYNDON 103 Surveys in combinatorics 1985, I. ANDERSON (ed) 104 Elliptic structures on 3-manifolds, C.B. THOMAS 105 A local spectral theory for closed operators, I. ERDELYI & WANG SHENGWANG 106 Syzygies, E.G. EVANS & P. GRIFFITH 107 Compactification of Siegel moduli schemes, C-L. CHAI 108 Some topics in graph theory, H.P. YAP 109 Diophantine analysis, J. LOXTON & A. VAN DER POORTEN (eds) 110 An introduction to surreal numbers, H. GONSHOR 113 Lectures on the asymptotic theory of ideals, D. REES 114 Lectures on Bochner-Riesz means, K.M. DAVIS & Y-C. CHANG 115 An introduction to independence for analysts, H.G. DALES & W.H. WOODIN 116 Representations of algebras, PJ. WEBB (ed) 117 Homotopy theory, E. REES & J.D.S. JONES (eds) 118 Skew linear groups, M. SHIRVANI & B. WEHRFRITZ 119 Triangulated categories in the representation theory of finite-dimensional algebras, D. HAPPEL 121 Proceedings of Groups - St Andrews 1985, E. ROBERTSON & C. CAMPBELL (eds) 122 Non-classical continuum mechanics, R.J. KNOPS & A.A. LACEY (eds) 125 Commutator theory for congruence modular varieties, R. FREESE & R. MCKENZIE 126 Van der Corpus's method of exponential sums, S.W. GRAHAM & G. KOLESNIK 127 New directions in dynamical systems, T.J. BEDFORD & J.W. SWIFT (eds) 128 Descriptive set theory and the structure of sets of uniqueness, A.S. KECHRIS & A. LOUVEAU 129 The subgroup structure of the finite classical groups, P.B. KLEIDMAN & M.W.LIEBECK 130 Model theory and modules, M. PREST 131 Algebraic, extremal & metric combinatorics, M-M. DEZA, P. FRANKL & I.G. ROSENBERG (eds) 132 Whitehead groups of finite groups, ROBERT OLIVER 133 Linear algebraic monoids, MOHAN S. PUTCHA 134 Number theory and dynamical systems, M. DODSON & J. VICKERS (eds) 135 Operator algebras and applications, 1, D. EVANS & M. TAKESAKI (eds) 136 Operator algebras and applications, 2, D. EVANS & M. TAKESAKI (eds) 137 Analysis at Urbana, I, E. BERKSON, T. PECK, &J. UHL (eds) 138 Analysis at Urbana, II, E. BERKSON, T. PECK, & J. UHL (eds) 139 Advances in homotopy theory, S. SALAMON, B. STEER & W. SUTHERLAND (eds) 140 Geometric aspects of Banach spaces, E.M. PEINADOR & A. RODES (eds) 141 Surveys in combinatorics 1989, J. SIEMONS (ed) 142 The geometry of jet bundles, D.J. SAUNDERS 143 The ergodic theory of discrete groups, PETER J. NICHOLLS 144 Introduction to uniform spaces, I.M. JAMES 145 Homological questions in local algebra, JAN R. STROOKER 146 Cohen-Macaulay modules over Cohen-Macaulay rings, Y. YOSHINO 147 Continuous and discrete modules, S.H. MOHAMED & B.J. MULLER 148 Helices and vector bundles, A.N. RUDAKOV et al 149 Solitons, nonlinear evolution equations and inverse scattering, M. ABLOWITZ & P. CLARKSON 150 Geometry of low-dimensional manifolds 1, S. DONALDSON & C.B. THOMAS (eds) 151 Geometry of low-dimensional manifolds 2, S. DONALDSON & C.B. THOMAS (eds) 152 Oligomorphic permutation groups, P. CAMERON 153 L-functions and arithmetic, J. COATES & M.J. TAYLOR (eds) 154 Number theory and cryptography, J. LOXTON (ed) 155 Classification theories of polarized varieties, TAKAO FUJITA 156 Twistors in mathematics and physics, T.N. BAILEY & R.J. BASTON (eds) 157 Analytic pro-p groups, J.D. DIXON, M.P.F. DU SAUTOY, A. MANN & D. SEGAL 158 Geometry of Banach spaces, P.F.X. MULLER & W. SCHACHERMAYER (eds) 159 Groups St Andrews 1989 volume 1, C.M. CAMPBELL & E.F. ROBERTSON (eds) 160 Groups St Andrews 1989 volume 2, C.M. CAMPBELL & E.F. ROBERTSON (eds) 161 Lectures on block theory, BURY-HARD KIOLSHAMMER. 162 Harmonic analysis and representation theory, A. FIGA-TALAMANCA & C. NEBBIA 163 Topics in varieties of group representations, S.M. VOVSI 164 Quasi-symmetric designs, M.S. SHRIKANDE & S.S. SANE 165 Groups, combinatorics & geometry, M.W. LIEBECK & J. SAXL (eds) 166 Surveys in combinatorics, 1991, A.D. KEEDWELL (ed) 167 Stochastic analysis, M.T. BARLOW & N.H. BINGHAM (eds) 168 Representations of algebras, H. TACHIKAWA & S. BRENNER (eds) 169 Boolean function complexity, M.S. PATERSON (ed) 170 Manifolds with singularities and the Adams-Novikov spectral sequence, B. BOTVINNIK 171 Squares, A.R. RAJWADE 172 Algebraic varieties, GEORGE R. KEMPF 173 Discrete groups and geometry, W.J. HARVEY & C. MACLACHLAN (eds) 174 Lectures on mechanics, J.E. MARSDEN 175 Adams memorial symposium on algebraic topology 1, N. RAY & G. WALKER (eds) 176 Adams memorial symposium on algebraic topology 2, N. RAY & G. WALKER (eds) 177 Applications of categories in computer science, M. FOURMAN, P. JOHNSTONE, & A. PITTS (eds) 178 Lower K- and L-theory, A. RANICKI 179 Complex projective geometry, G. ELLINGSRUD et al 180 Lectures on ergodic theory and Pesin theory on compact manifolds, M. POLLICOTT 181 Geometric group theory I, G.A. NIBLO & M.A. ROLLER (eds) 182 Geometric group theory H, G.A. NIBLO & M.A. ROLLER (eds) 183 Shintani zeta functions, A. YUKIE 184 Arithmetical functions, W. SCHWARZ & J. SPILKER 185 Representations of solvable groups, O. MANZ & T.R. WOLF 186 Complexity: knots, colourings and counting, D.J.A. WELSH 187 Surveys in combinatorics, 1993, K. WALKER (ed) 188 Local analysis for the odd order theorem, H. BENDER & G. GLAUBERMAN 189 Locally presentable and accessible categories, J. ADAMEK & J. ROSICKY 190 Polynomial invariants of finite groups, D.J. BENSON 191 Finite geometry and combinatorics, F. DE CLERCK et al 192 Symplectic geometry, D. SALAMON (ed) 193 Computer algebra and differential equations, E. TOURNIER (ed) 194 Independent random variables and rearrangement invariant spaces, M. BRAVERMAN 195 Arithmetic of blowup algebras, WOLMER VASCONCELOS 196 Microlocal analysis for differential operators, A. GRIGIS & J. SJOSTRAND 197 Two-dimensional homotopy and combinatorial group theory, C. HOG-ANGELONI, W. METZLER & A.J. SIERADSKI (eds) 198 The algebraic characterization of geometric 4-manifolds, J.A. HILLMAN 199 Invariant potential theory in the unit ball of C'n, MANFRED STOLL 200 The Grothendieck theory of dessins d'enfant, L. SCHNEPS (ed) 201 Singularities, JEAN-PAUL BRASSELET (ed) 202 The technique of pseudodifferential operators, H.O. CORDES 203 Hochschild cohomology of von Neumann algebras, A. SINCLAIR & R. SMITH 204 Combinatorial and geometric group theory, A.J. DUNCAN, N.D. GILBERT & J. HOWIE (eds) 207 Groups of Lie type and their geometries, W.M. KANTOR & L. DI MARTINO (eds) 208 Vector bundles in algebraic geometry, N.J. HITCHIN, P. NEWSTEAD & W.M. OXBURY (eds) London Mathematical Society Lecture Note Series. 211 Groups '93 Galway / St Andrews Galway 1993 Volume 1 Edited by C.M. Campbell University of St Andrews T.C. Hurley University College, Galway E.F. Robertson University of St Andrews S.J. Tobin University College, Galway J.J. Ward University College, Galway AMBRIDGE UNIVERSITY PRESS CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 2RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521477499 © Cambridge University Press 1995 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 1995 A catalogue record for this publication is available from the British Library ISBN-13 978-0-521-47749-9 paperback ISBN-10 0-521-47749-2 paperback Transferred to digital printing 2005 CONTENTS Volume I Preface .................................................................ix Introduction ........................................................... xi Geometry, Steinberg representations and complexity J L Alperin & G Mason ................................................. 1 The structure of metabelian finite groups Z Arad, E Fisman & M Muzychuk ...................................... 6 Table algebras of extended Gagola-type and applications to finite group theory Z Arad, E Fisman & C M Scoppola .................................... 13 On the saturation of formations of finite groups A Ballester-Bolinches & L M Ezquerro ................................. 23 Locally constructed formations of finite groups ............................... 32 A Ballester-Bolinches & C Juan-Martinez Reflections on virtually one-relator groups K Bencsath & B Fine ......................... 37 Rickard equivalences and block theory M Broue ............................................................... 58 Computing the conjugacy classes of elements of a finite group G Butler ............................................................... 80 Quotient categories of modules over group algebras J F Carlson ........................................................... 113 Weak chain conditions for non-almost normal subgroups G Cutolo & L A Kurdachenko ... ....................................120 Computation of the character table of affine groups using Fischer matrices M R Darafsheh & A Iranmanesh ....................................... 131 The lattice of compact representations of an infinite group D Dikranjan ...........................................................138 Automorphisms of nilpotent and related groups M R Dixon ............................................................ 156 Generation of orthogonal groups over finite fields A G Earnest .......................................................... 172 The structure of certain Coxeter groups ................... 177 V Felsch, D L Johnson, J Neubuser & S V Tsaranov n-free groups and questions about universally free groups .................. 191 B Fine, A M Gaglione, G Rosenberger & D Spellman Classification of all generating pairs of two generator Fuchsian groups B Fine & G Rosenberger .............................................. 205 Parametric words and models of the elementary theory of non-abelian free groups A M Gaglione & D Spellman .......................................... 233 The groups G(n, 1) as fundamental groups of Seifert fibered homology spheres L Grasselli ............................................................ 244 Lifting automorphisms : a survey C K Gupta & V Shpilrain .............................................249 (MI)-groups acting uniserially on a normal subgroup P Z Hermann ......................................................... 264 Revisiting a theorem of Higman E Jespers, M M Parmenter & P F Smith ............................... 269 Cohomological finiteness conditions P H Kropholler ....................................................... 274 vi Contents of Volume II Preface .................................................................ix Introduction............................................................ xi An army of cohomology against residual finiteness P R Hewitt ........................................................... 305 On some questions concerning subnormally monomial groups ............................................................ 314 E Horvath A conjecture concerning the evaluation of products of class-sums of the symmetric group J Katriel .............................................................. 322 Automorphisms of Burnside rings W Kimmerle & K W Roggenkamp ..................................... 333 On finite generation of unit groups for group rings J Krempa ............................................................. 352 Counting finite index subgroups A Lubotzky........................................................... 368 The quantum double of a finite group and its role in conformal field theory G Mason.............................................................. 405 Closure properties of supersoluble Fitting classes M Menth ..............................................................418 Groups acting on locally finite graphs - a survey of the infinitely ended case R G Moller ............................................................426 An invitation to computational group theory J Neubuser ............................................................ 457 On subgroups, transversals and commutators M Niemenmaa & A Vesanen .......................................... 476 Intervals in subgroup lattices of finite groups P P Palfy ............................................................. 482 Amalgams of minimal local subgroups and sporadic simple groups C W Parker & P Rowley .............................................. 495 Vanishing orbit sums in group algebras of p-groups M A Rao & R Sandling ............................................... 507 From stable equivalences to Rickard equivalences for blocks with cyclic defect R Rouquier ........................................................... 512 Factorizations in which the factors have relatively prime orders A D Sands ............................................................ 524 Some problems and results in the theory of pro-p groups A Shalev .............................................................. 528 On equations in finite groups and invariants of subgroups S P Strunkov ..........................................................543 Group presentations where the relators are proper powers R M Thomas .......................................................... 549 A condensing theorem N B Tinberg .......................................................... 561 Lie methods in group theory E I Zel'manov ......................................................... 567 Some new results on arithmetical problems in the theory of finite groups J P Zhang .............................................................586 Groups that admit partial power automorphisms J Zimmerman ......................................................... 594 Problems .............................................................. 602 viii PREFACE This is the first of two volumes of the Proceedings of Groups 1993 Galway / St Andrews. There is a full contents of both volumes and papers written by authors with the name of the first author lying in the range A-K. Contained in this part are the papers of the main speakers J L Alperin, M Broue and P H Kropholler, and we would like especially to thank them for their contributions both to the conference and to this volume. There is some perturbation in strict alphabetical order at the join between the volumes and the second volume contains papers written by authors with the name of the first author lying in the range H-Z.