Groups St Andrews 2001 in Oxford

This page intentionally left blank LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES Managing Editor: Professor N.J. Hi...

Author: C. M. Campbell | E. F. Robertson | G. C. Smith

16 downloads 529 Views 5MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!

Report copyright / DMCA form

DOWNLOAD PDF

This page intentionally left blank

LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES Managing Editor: Professor N.J. Hitchin, Mathematical Institute, University of Oxford, 24–29 St Giles, Oxford OX1 3LB, United Kingdom The titles below are available from booksellers, or, in case of diﬃculty, from Cambridge University Press at www.cambridge.org. 159 160 161 163 164 166 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 194 195 196 197 198 199 200 201 202 203 204 205 207 208 209 210 211 212 214 215 216 217 218 220 221 222 223 224 225 226 227 228 229

Groups St Andrews 1989 volume 1, C.M. CAMPBELL & E.F. ROBERTSON (eds) Groups St Andrews 1989 volume 2, C.M. CAMPBELL & E.F. ROBERTSON (eds) ¨ Lectures on block theory, BURKHARD KULSHAMMER Topics in varieties of group representations, S.M. VOVSI Quasi-symmetric designs, M.S. SHRIKANDE & S.S. SANE Surveys in combinatorics, 1991, A.D. KEEDWELL (ed) Representations of algebras, H. TACHIKAWA & S. BRENNER (eds) Boolean function complexity, M.S. PATERSON (ed) Manifolds with singularities and the Adams-Novikov spectral sequence, B. BOTVINNIK Squares, A.R. RAJWADE Algebraic varieties, GEORGE R. KEMPF Discrete groups and geometry, W.J. HARVEY & C. MACLACHLAN (eds) Lectures on mechanics, J.E. MARSDEN Adams memorial symposium on algebraic topology 1, N. RAY & G. WALKER (eds) Adams memorial symposium on algebraic topology 2, N. RAY & G. WALKER (eds) Applications of categories in computer science, M. FOURMAN, P. JOHNSTONE & A. PITTS (eds) Lower K-and L-theory, A. RANICKI Complex projective geometry, G. ELLINGSRUD et al Lectures on ergodic theory and Pesin theory on compact manifolds, M. POLLICOTT Geometric group theory I, G.A. NIBLO & M.A. ROLLER (eds) Geometric group theory II, G.A. NIBLO & M.A. ROLLER (eds) Shintani zeta functions, A. YUKIE Arithmetical functions, W. SCHWARZ & J. SPILKER Representations of solvable groups, O. MANZ & T.R. WOLF Complexity: knots, colourings and counting, D.J.A. WELSH Surveys in combinatorics, 1993, K. WALKER (ed) Local analysis for the odd order theorem, H. BENDER & G. GLAUBERMAN Locally presentable and accessible categories, J. ADAMEK & J. ROSICKY Polynomial invariants of ﬁnite groups, D.J. BENSON Finite geometry and combinatorics, F. DE CLERCK et al Symplectic geometry, D. SALAMON (ed) Independent random variables and rearrangement invariant spaces, M. BRAVERMAN Arithmetic of blowup algebras, WOLMER VASCONCELOS ¨ Microlocal analysis for diﬀerential operators, A. GRIGIS & J. SJOSTRAND Two-dimensional homotopy and combinatorial group theory, C. HOG-ANGELONI et al The algebraic characterization of geometric 4-manifolds, J.A. HILLMAN Invariant potential theory in the unit ball of Cn , MANFRED STOLL The Grothendieck theory of dessins d’enfant, L. SCHNEPS (ed) Singularities, JEAN-PAUL BRASSELET (ed) The technique of pseudodiﬀerential operators, H.O. CORDES Hochschild cohomology of von Neumann algebras, A. SINCLAIR & R. SMITH Combinatorial and geometric group theory, A.J. DUNCAN, N.D. GILBERT & J. HOWIE (eds) Ergodic theory and its connections with harmonic analysis, K. PETERSEN & I. SALAMA (eds) Groups of Lie type and their geometries, W.M. KANTOR & L. DI MARTINO (eds) Vector bundles in algebraic geometry, N.J. HITCHIN, P. NEWSTEAD & W.M. OXBURY (eds) Arithmetic of diagonal hypersurfaces over ﬁnite ﬁelds, F.Q. GOUVEA & N. YUI Hilbert C∗ -modules, E.C. LANCE Groups 93 Galway/St Andrews I, C.M. CAMPBELL et al (eds) Groups 93 Galway/St Andrews II, C.M. CAMPBELL et al (eds) Generalised Euler-Jacobi inversion formula and asymptotics beyond all orders, V. KOWALENKO et al Number theory 1992–93, S. DAVID (ed) Stochastic partial diﬀerential equation, A. ETHERIDGE (ed) Quadratic forms with applications to algebraic geometry and topology, A. PFISTER Surveys in combinatorics, 1995, PETER ROWLINSON (ed) Algebraic set theory, A. JOYAL & I. MOERDIJK Harmonic approximation, S.J. GARDINER Advances in linear logic, J.-Y. GIRARD, Y. LAFONT & L. REGNIER (eds) Analytic semigroups and semilinear initial boundary value problems, KAZUAKITAIRA Computability, enumerability, unsolvability, S.B. COOPER, T.A. SLAMAN & S.S. WAINER (eds) A mathematical introduction to string theory, S. ALBEVERIO, J. JOST, S. PAYCHA, S. SCARLATTI Novikov conjectures, index theorems and rigidity I, S. FERRY, A. RANICKI & J. ROSENBERG (eds) Novikov conjectures, index theorems and rigidity II, S. FERRY, A. RANICKI & J. ROSENBERG (eds) Ergodic theory of Zd actions, M. POLLICOTT & K. SCHMIDT (eds) Ergodicity for inﬁnite dimensional systems, G. DA PRATO & J. ZABCZYK

230 Prolegomena to a middlebrow arithmetic of curves of genus 2, J.W.S. CASSELS & E.V. FLYNN 231 Semigroup theory and its applications, K.H. HOFMANN & M.W. MISLOVE (eds) 232 The descriptive set theory of Polish group actions, H. BECKER & A.S. KECHRIS 233 Finite ﬁelds and applications, S. COHEN & H. NIEDERREITER (eds) 234 Introduction to subfactors, V. JONES & V.S. SUNDER 235 Number theory 1993–94, S. DAVID (ed) 236 The James forest, H. FETTER & B. GAMBOA DE BUEN 237 Sieve methods, exponential sums, and their applications in number theory, G.R.H. GREAVES et al 238 Representation theory and algebraic geometry, A. MARTSINKOVSKY & G. TODOROV (eds) 239 Cliﬀord algebras and spinors, P. LOUNESTO 240 Stable groups, FRANK. O. WAGNER 241 Surveys in combinatorics, 1997, R.A. BAILEY (ed) 242 Geometric Galois actions I, L. SCHNEPS & P. LOCHAK (eds) 243 Geometric Galois actions II, L. SCHNEPS & P. LOCHAK (eds) 244 Model theory of groups and automorphism groups, D. EVANS (ed) 245 Geometry, combinatorial designs and related structures, J.W.P. HIRSCHFELD et al 246 p-Automorphisms of ﬁnite p-groups, E.I. KHUKHRO 247 Analytic number theory, Y. MOTOHASHI (ed) 248 Tame topology and o-minimal structures, LOU VAN DEN DRIES 249 The atlas of ﬁnite groups: ten years on, ROBERT CURTIS & ROBERT WILSON (eds) 250 Characters and blocks of ﬁnite groups, G. NAVARRO 251 Gr¨ obner bases and applications, B.BUCHBERGER & F. WINKLER (eds) ¨ 252 Geometry and cohomology in group theory, P. KROPHOLLER, G. NIBLO, R. STOHR (eds) 253 The q-Schur algebra, S. DONKIN 254 Galois representations in arithmetic algebraic geometry, A.J. SCHOLL & R.L. TAYLOR (eds) 255 Symmetries and integrability of diﬀerence equations, P.A. CLARKSON & F.W. NIJHOFF (eds) ¨ 256 Aspects of Galois theory, HELMUT VOLKLEIN et al 257 An introduction to noncommutative diﬀerential geometry and its physical applications 2ed, J. MADORE 258 Sets and proofs, S.B. COOPER & J. TRUSS (eds) 259 Models and computability, S.B. COOPER & J. TRUSS (eds) 260 Groups St Andrews 1997 in Bath, I, C.M. CAMPBELL et al 261 Groups St Andrews 1997 in Bath, II, C.M. CAMPBELL et al 263 Singularity theory, BILL BRUCE & DAVID MOND (eds) 264 New trends in algebraic geometry, K. HULEK, F. CATANESE, C. PETERS & M. REID (eds) 265 Elliptic curves in cryptography, I. BLAKE, G. SEROUSSI & N. SMART 267 Surveys in combinatorics, 1999, J.D. LAMB & D.A. PREECE (eds) ¨ 268 Spectral asymptotics in the semi-classical limit, M. DIMASSI & J. SJOSTRAND 269 Ergodic theory and topological dynamics, M.B. BEKKA & M. MAYER 270 Analysis on Lie Groups, N.T. VAROPOULOS & S. MUSTAPHA 271 Singular perturbations of diﬀerential operators, S. ALBEVERIO & P. KURASOV 272 Character theory for the odd order function, T. PETERFALVI 273 Spectral theory and geometry, E.B, DAVIES & Y. SAFAROV (eds) 274 The Mandelbrot set, theme and variations, TAN LEI (ed) 275 Computational and geometric aspects of modern algebra, M.D. ATKINSON et al (eds) 276 Singularities of plane curves, E. CASAS-ALVERO 277 Descriptive set theory and dynamical systems, M.FOREMAN et al (eds) 278 Global attractors in abstract parabolic problems, J.W. CHOLEWA & T. DLOTKO 279 Topics in symbolic dynamics and applications, F. BLANCHARD, A. MAASS & A. NOGUEIRA (eds) 280 Characters and Automorphism Groups of Compact Riemann Surfaces, T. BREUER 281 Explicit birational geometry of 3-folds, ALESSIO CORTI & MILES REID (eds) 282 Auslander-Buchweitz approximations of equivariant modules, M. HASHIMOTO 283 Nonlinear elasticity, R. OGDEN & Y. FU (eds) 284 Foundations of computational mathematics, R. DEVORE, A. ISERLES & E. SULI (eds) 285 Rational points on curves over ﬁnite ﬁelds: Theory and Applications, H. NIEDERREITER & C. XING 286 Cliﬀord algebras and spinors 2nd edn, P. LOUNESTO 287 Topics on Riemann surfaces and Fuchsian groups, E. BUJALANCE, A.F. COSTA & E. MARTINEZ (eds) 288 Surveys in combinatorics, 2001, J.W.P. HIRSCHFELD (ed) 289 Aspects of Sobolev-type inequalities, L. SALOFF-COSTE 290 Quantum groups and Lie theory, A. PRESSLEY 291 Tits buildings and the model theory of groups, K. TENT 292 A quantum groups primer, S. MAJID 293 Second order partial diﬀerential equations in Hilbert spaces, G. DA PRATO & J. ZABCZYK 294 Introduction to operator space theory, G. PISIER 296 Lectures on invariant theory, I. DOLGACHEV 297 The homotopy category of simply connected 4-manifolds, H.-J. BAUES 298 Higher Operads, higher categories, T. LEINSTER 299 Kleinian groups and hyperbolic 3-manifolds, Y. KOMORI, V. MARKOVIC & C. SERIES (eds) 300 Introduction to M¨ obius diﬀerential geometry, U. HERTRICH-JEROMIN 301 Stable modules and the D(2)-problem, F. E. A. JOHNSON 302 Discrete and continous nonlinear Schr¨ odinger systems, M. ABLOWITZ, B. PRINARI & D. TRUBATCH 304 Groups St Andrews 2001 in Oxford v1, C. M. CAMPBELL, E. F. ROBERTSON & G. C. SMITH (eds) 305 Groups St Andrews 2001 in Oxford v2, C. M. CAMPBELL, E. F. ROBERTSON & G. C. SMITH (eds)

London Mathematical Society Lecture Note Series. 304

Groups St Andrews 2001 in Oxford Volume I

Edited by C.M. Campbell University of St Andrews E.F. Robertson University of St Andrews G.C. Smith University of Bath

   Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge  , United Kingdom Published in the United States by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521537391 © Cambridge University Press 2003 This book is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2003 ISBN-13 978-0-511-06560-6 eBook ISBN-10 0-511-06560-4 eBook ISBN-13 978-0-521-53739-1 paperback ISBN-10 0-521-53739-8 paperback

Cambridge University Press has no responsibility for the persistence or accuracy of s for external or third-party internet websites referred to in this book, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

Contents of Volume 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi Permutability and subnormality in ﬁnite groups M J Alejandre, A Ballester-Bolinches, R Esteban-Romero & M C Pedraza-Aguilera . . . . . . . . . . . . . . 1 (Pro)-ﬁnite and (topologically) locally ﬁnite groups with a CC-subgroup Z Arad & W Herfort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Table algebras generated by elements of small degrees Z Arad & M Muzychuk . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Subgroups which are a union of a given number of conjugacy classes A R Ashraﬁ & H Sahraei . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Some results on ﬁnite factorized groups A Ballester-Bolinches, J Cossey, X Guo & M C Pedraza-Aguilera . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 On nilpotent-like Fitting formations A Ballester-Bolinches, A Mart´ınez-Pastor, M C Pedraza-Aguilera & M D P´erez-Ramos . . . . . . . . . . . . . 31 Locally ﬁnite groups with min-p for all primes p A Ballester-Bolinches & T Pedraza . . . . . . . . . . . . . . . . . . . 39 Quasi-permutation representations of 2-groups of class 2 with cyclic centre H Behravesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 v

vi

CONTENTS: VOLUME 1

Groups acting on bordered Klein surfaces with maximal symmetry E Bujalance, F J Cirre & P Turbek . . . . . . . . . . . . . . . . . . . 50 Breaking points in subgroup lattices G Calugareanu & M Deaconescu . . . . . . . . . . . . . . . . . . . . . 59 Group actions on graphs, maps and surfaces with maximum symmetry M D E Conder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 On dual pronormal subgroups and Fitting classes A D’Aniello & M D P´erez-Ramos . . . . . . . . . . . . . . . . . . . . 92 (p, q, r)-generations of the sporadic group O’N M R Darafsheh, A R Ashraﬁ & G A Moghani . . . . . . . . . . . 101 Computations with almost-crystallographic groups K Dekimpe & B Eick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 Random walks on groups: characters and geometry P Diaconis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 On distances of 2-groups and 3-groups A Dr´ apal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Zeta functions of groups: the quest for order versus the ﬂight from ennui M P F du Sautoy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 Some factorizations involving hypercentrally embedded subgroups in ﬁnite soluble groups L M Ezquerro & X Soler-Escriv` a . . . . . . . . . . . . . . . . . . . . 190 Elementary theory of groups B Fine, A M Gaglione, A G Myasnikov, G Rosenberger & D Spellman . . . . . . . . . . . . . . . . . . . . . . . 197 Andrews-Curtis and Todd-Coxeter proof words G Havas & C Ramsay . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

CONTENTS: VOLUME 1

vii

Short balanced presentations of perfect groups G Havas & C Ramsay . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 Finite p-extensions of free pro-p groups W Herfort & P A Zalesskii . . . . . . . . . . . . . . . . . . . . . . . . 244 Elements and groups of ﬁnite length M Herzog, P Longobardi & M Maj . . . . . . . . . . . . . . . . . . . 249 Logged rewriting and identities among relators A Heyworth & C D Wensley . . . . . . . . . . . . . . . . . . . . . . . 256 A characterization of F4 (q) where q is an odd prime power A Iranmanesh & B Khosravi . . . . . . . . . . . . . . . . . . . . . . . 277 On associated groups of rings Y B Ishchuk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284

Contents of Volume 2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi Gracefulness, group sequencings and graph factorizations G Kaplan, A Lev & Y Roditty . . . . . . . . . . . . . . . . . . . . . . 295 Orbits in ﬁnite group actions T M Keller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 Groups with ﬁnitely generated integral homologies D H Kochloukova . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 Invariants of discrete groups, Lie algebras and pro-p groups D H Kochloukova . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344 Groups with all non-subnormal subgroups of ﬁnite rank L A Kurdachenko & P Soules . . . . . . . . . . . . . . . . . . . . . . 366 On some inﬁnite dimensional linear groups L A Kurdachenko & I Y Subbotin . . . . . . . . . . . . . . . . . . . . 377 Groups and semisymmetric graphs S Lipschutz & Ming-Yao Xu . . . . . . . . . . . . . . . . . . . . . . . 385 On the covers of ﬁnite groups M S Lucido . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 Groupland O Macedo´ nska . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400 On maximal nilpotent π-subgroups J Medina . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405

viii

CONTENTS: VOLUME 2

ix

Characters of p-groups and Sylow p-subgroups A Moret´ o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 On the relation between group theory and loop theory M Niemenmaa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422 Groups and lattices P P P´ alfy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428 Finite generalized tetrahedron groups with a cubic relator G Rosenberger, M Scheer & R M Thomas . . . . . . . . . . . . . . 455 Character degrees of the Sylow p-subgroups of classical groups J Sangroniz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487 Character correspondences and perfect isometries L Sanus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494 The characters of ﬁnite projective symplectic group P Sp(4, q) M A Shahabi & H Mohtadifar . . . . . . . . . . . . . . . . . . . . . . 496 Exponent of ﬁnite groups with automorphisms P Shumyatsky . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528 Classifying irreducible representations in characteristic zero A Turull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537 Lie methods in group theory M R Vaughan-Lee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547 Chevalley groups of type G2 as automorphism groups of loops P Vojtˇechovsk´y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586

INTRODUCTION Groups St Andrews 2001 in Oxford was another highly successful conference in the continuing series. The conference, held from 5 August 2001 to 18 August 2001, was attended by 230 mathematicians from 35 countries. The lectures and talks were given in the Mathematical Institute of the University of Oxford. We thank Lady Margaret Hall, St Anne’s College, The Queen’s College and Merton College for providing accommodation. We also acknowledge with gratitude the ﬁnancial support of the Edinburgh and London Mathematical Societies. The Mathematical Institute of the University of Oxford generously made lecture rooms available, and supported the conference in various other important ways. The organizing committee consisted of C M Campbell (University of St Andrews), D P Groves (Merton), R P Martineau (Wadham), P M Neumann (The Queen’s College), E F Robertson (University of St Andrews), G C Smith (University of Bath), W B Stewart (Exeter) and G A Stoy (LMH). Administrative support was provided by Jan Campbell and Maureen White. The main speakers, who were invited to give a series of talks, were Marston D E Conder (Auckland), Persi Diaconis (Stanford), Peter P P´ alfy (E¨ otv¨ os Lor´and, Budapest), Marcus du Sautoy (Cambridge), and Michael R Vaughan-Lee (Oxford). As has become the tradition, all these invited speakers have written substantial articles for these Proceedings. All papers have been subjected to a formal refereeing process comparable to that of a major international journal. Publishing constraints have forced the editors to exclude some very worthwhile papers, and this is of course a matter of regret. The Theory of Groups continues to move forward on many fronts, and twenty years on from the announcement of the classiﬁcation of the ﬁnite simple groups, it prospers perhaps surprisingly well (rather like Mark Twain). It is a measure of the success of this conference series and this subject that mathematical libraries around the world are collecting the series of St Andrews Conference Proceedings. As LATEX has become the almost universal typesetting language for mathematics, the nuts and bolts of the editorial job are perhaps getting a little easier. We hope that in future even more contributors will stick faithfully to the typesetting standard prescribed by the editors (so that these hacks will have more time to do research in group theory, lie on beaches and so on). Fran Burstall of the University of Bath provided typesetting counselling. It is hoped that the next conference in this series will be held in 2005 and will be Groups St Andrews in St Andrews (revisiting the scene of the original crime), and it is also hoped that in 2009 we will return once more to Bath. As these proceedings go to press, news has just arrived that one of the grandfathers of the subject, B H Neumann, has died. He attended the ﬁrst Groups St Andrews conference in 1981 and some later ones in the series. Many of our community knew him with great aﬀection – his warm smile and shoestring tie will linger in the memory almost as long as his important contributions to mathematics. CMC, EFR, GCS

xi

PERMUTABILITY AND SUBNORMALITY IN FINITE GROUPS MANUEL J. ALEJANDRE ∗ , A. BALLESTER-BOLINCHES † , R. ESTEBANROMERO ‡ and M. C. PEDRAZA-AGUILERA ‡1 ∗

Centro de Investigaci´ on Operativa, Universidad Miguel Hern´ andez, Avda. del Ferrocarril s/n, 03202 Elche, Spain † ` Departament d’Algebra, Universitat de Val`encia, C/ Doctor Moliner 50, 46100 Burjassot (Val`encia), Spain ‡ Departamento de Matem´ atica Aplicada, E.U.I., Universidad Polit´ecnica de Valencia, Camino de Vera, s/n, 46071 Valencia, Spain

All groups considered in this report will be ﬁnite.

1

Notation and terminology

A group G is said to be a T -group if every subnormal subgroup of G is normal in G, that is, if normality is a transitive relation. These groups have been widely studied (see [10], [11], or [14]). A subgroup H of a group G is said to be permutable (or quasinormal) in G if HK = KH for all subgroups K of G. Permutability can be considered thus as a weak form of normality. The study of groups G in which permutability is transitive, that is, H permutable in K and K permutable in G always imply that H is permutable in G, has been a successful ﬁeld of research in recent years. Such groups are called P T -groups. According to a theorem of Kegel [12, Satz 1], every permutable subgroup of G is subnormal in G. Consequently, P T -groups are exactly those groups in which subnormality and permutability coincide; that is, those groups in which every subnormal subgroup permutes with every other subgroup. Therefore, every T -group is clearly a P T -group. One could wonder what would happen if we did not require that every subnormal subgroup of a group G permutes with any other subgroup of G, but only with a certain family of its subgroups. In this direction, those groups in which every subnormal subgroup of G permutes with every Sylow p-subgroup of G for each prime p have sometimes been called T ∗ -groups (see [3]) or also (π − q)-groups (see [1]). Nevertheless, in recent years the expression P ST -groups has become more popular for them. It was ﬁrst used in [15]. Again, by a result of Kegel, each subgroup of a group G permuting with every Sylow subgroup of G is subnormal. Therefore, P ST -groups are exactly those groups in which permutability with Sylow subgroups is a transitive relation; that is, G is a P ST -group when given H ≤ K ≤ G such that H permutes with every Sylow subgroup of K and K permutes with every Sylow subgroup of G, then H permutes with every Sylow subgroup of G. It is clear that {T -groups} ⊂ {P T -groups} ⊂ {P ST -groups} 1 The work of the second and last authors is supported by Proyecto PB97-0674-C02-02 of DGICYT, MEC, Spain

2

ALEJANDRE et al.

and the above three classes are distinct. The study of these classes of groups has undoubtedly constituted a fruitful topic of research in group theory, due to the eﬀorts of many leading mathematicians. There are in essence three diﬀerent ways of approaching the question of characterizing T -groups, P T -groups and P ST -groups. The aim of this survey is to provide a general perspective of these three lines. Note that every simple group belongs to these classes. Hence most of the papers on this topic frame their work in the soluble universe. In fact, as we shall see below, in some cases the solubility of the group appears in a spontaneous way.

2

Characterizations based on the normal structure

This is possibly the most classical way of studying these groups. The structure of soluble T -groups was determined by Gasch¨ utz ([11]) in 1957. They are exactly the soluble groups G with an abelian normal Hall subgroup L of odd order such that G/L is a Dedekind group and such that the elements of G induce power automorphisms in L. Some years later, in 1964, Zacher gave in [16] the corresponding theorem for soluble P T -groups: one just has to replace “Dedekind” by “nilpotent modular” in Gasch¨ utz’s theorem. Finally, the structure of soluble P ST -groups was obtained by Agrawal (see [1]), in a way similar to the Gasch¨ utz and Zacher characterizations: G is a soluble P ST group if and only if G has an abelian normal Hall subgroup L of odd order such that G/L is nilpotent and the elements of G induce power automorphisms in L. As straightforward consequences of these theorems, one can state that the classes of soluble T -groups, soluble P T -groups and soluble P ST -groups are subgroupclosed. Recently, the second and third authors provided in [6, Theorem A] local versions of the above theorems.

3

Characterizations based on the Sylow structure

Several papers have explored the inﬂuence of the Sylow structure of a group on the condition of being a P ST -group, P T -group or T -group. The natural outcome of these investigations is the fact that in the soluble universe, the diﬀerence between these three classes is quite simply their Sylow structure. Therefore several unifying points of view for these classes have been obtained in the soluble universe. These ideas were ﬁrstly used by Bryce and Cossey in [10]. With p a prime, they deﬁned the class Up∗ of p-supersoluble groups G such that all the p-chief factors of G form a single isomorphism class of G-modules. They proved that a soluble group G is a T -group if and only if G satisﬁes Up∗ for every prime p and all its Sylow subgroups are T -groups. The ﬁrst, second and fourth authors introduced in [2] the use of the class Up∗ for a single prime p and the class ∩p∈P Up∗ to describe the classes of soluble P T -groups and P ST -groups. Many other relevant local deﬁnitions were introduced in [10] and in [2].

PERMUTABILITY AND SUBNORMALITY IN FINITE GROUPS

3

This procedure of deﬁning local versions in order to simplify the study of the global properties has shown itself to be very useful. The following three deﬁnitions are the main keys which allow us to describe the three classes we are working with in a natural way. Definition 1 Let G be a group and p a prime. We say that G: 1. Enjoys property Cp (see [14]) if each subgroup of a Sylow p-subgroup P of G is normal in the normalizer NG (P ). 2. Satisﬁes property Xp (as in [7]) if each subgroup of a Sylow p-subgroup P of G is permutable in the normalizer NG (P ). 3. Enjoys property Yp (see [5]) if for all p-subgroups H and S of G such that H ≤ S, H permutes with every Sylow subgroup of NG (S). Note that the property Cp is inherited by subgroups. This fact follows from the abnormality of the Sylow normalizers in the group. By its deﬁnition, the class of Yp -groups is subgroup-closed as well. Nevertheless, proving that Xp is inherited by subgroups is of extreme diﬃculty. This fact was ﬁrst observed by Beidleman, Brewster and Robinson in [7], but only as a consequence of a much stronger theorem. The following theorem, obtained by the second and third authors in [5], summarizes the relationship between the three properties listed above and is crucial to having a global knowledge of their behaviour. Theorem 1 [5, Theorem 3] A group G satisfies Xp (respectively, Cp ) if and only if G satisfies Yp and its Sylow p-subgroups are modular (respectively, Dedekind). This theorem is a consequence of the following p-nilpotence criterion, which was proved in the same paper: Theorem 2 [5, Theorem 1] Let p be a prime and let G be a group with a modular Sylow p-subgroup P . Then G is p-nilpotent if and only if NG (P ) is p-nilpotent. As a natural consequence of the above statements, the property Xp is inherited by subgroups. Not surprisingly, this fact, whose direct proof has been obtained only very recently, simpliﬁes in a dramatic way the proofs of many of the existing theorems. Robinson had proved in [14] that a group G is a soluble T -group if and only if it satisﬁes property Cp for all primes p. Years later, Beidleman et al. proved in [7] that a group G is a soluble P T -group if and only if it enjoys property Xp for every prime p. Finally, the second and third authors proved in [5] that a group G is a soluble P T -group if and only if it satisﬁes Yp for every prime p. One can see that in the theorem by Beidleman et al., a lot of eﬀort is put into proving that every group satisfying Xp for every prime p must be soluble. However, this assertion follows directly from the fact that Xp is closed under subgroups and factor groups. Therefore, one can travel easily among the classes of soluble P ST -groups, soluble P T -groups and soluble T -groups (or among their local versions) just by changing the requirements on their Sylow p-subgroups.

ALEJANDRE et al.

4

Moreover, any hope of creating a similar landscape which could be valid also outside the soluble universe is soon dispelled. For instance, as soon as we have a local property Jp which is subgroup-closed and such that a ﬁnite group G is a P ST -group if and only if G satisﬁes Jp for every prime p, then it is possible to prove that ∩p∈P Jp is contained in the class of soluble groups. Hence we had better desist from our expectations. Finally, the situation would be excellent if we could describe in a precise way the class of Yp -groups for a prime p, which is the central point which our description relies on. This wish was achieved in [5]: Theorem 3 [5] A group G is a Yp -group if and only if G is either p-nilpotent or G has abelian Sylow p-subgroups and G satisfies Cp .

4

Characterizations based on embedding properties

A strong connection between the classes described above and some embedding properties has recently been shown in several papers. Peng proved in [13] that a soluble group G is a T -group if and only if every p-subgroup of G, for every prime p, is pronormal in G. In addition, Bianchi et al. presented in [9] the following embedding property: A subgroup H of a group G is said to be an H-subgroup of G if for all g ∈ G, NG (H) ∩ H g ≤ H. They proved that a group G is a soluble T -group if and only if every subgroup of G is an H-subgroup of G. With a similar philosophy in mind, the second and third authors proved the following theorem. Theorem 4 [4] Let G be a group. The following statements are equivalent: 1. G is a soluble T -group. 2. Every subgroup of G is weakly normal in G. 3. Every subgroup of G satisfies the subnormalizer condition. A subgroup H of G is said to be weakly normal in G if H g ≤ NG (H) implies that g ∈ NG (H) and is said to satisfy the subnormalizer condition if for every subgroup K of G such that H is normal in K, it follows that NG (K) ≤ NG (H). It is said that a subgroup H of G is hypercentrally embedded in G if H/CoreG (H) ≤ Z∞ (G/CoreG (H)). Beidleman and Heineken [8] proved that a soluble group G is a P ST -group if and only if every subnormal subgroup permutes with every Carter subgroup of G and the subnormal subgroups of G are hypercentrally embedded in G. The second and third authors showed in [6] that permutability with Carter subgroups could be removed, that is, a soluble group G is a P ST -group if and only if every subnormal subgroup of G is hypercentrally embedded in G.

PERMUTABILITY AND SUBNORMALITY IN FINITE GROUPS

5

References [1] R.K. Agrawal “Finite groups whose subnormal subgroups permute with all Sylow subgroups” Proc. Amer. Math. Soc. 47 (1975), 77-83 [2] M. J. Alejandre, A. Ballester-Bolinches, M.C. Pedraza-Aguilera “Finite soluble groups with permutable subnormal subgroups” J. Algebra, 240 (2001), 705-721 [3] M. Asaad, P. Cs¨org¨o “On T ∗ -groups” Acta Math. Hungar. 213 (1997), 235-243 [4] A. Ballester-Bolinches, R. Esteban-Romero “On finite T -groups” Preprint [5] A. Ballester-Bolinches, R. Esteban-Romero “Sylow permutable subnormal subgroups of finite groups” Preprint [6] A. Ballester-Bolinches, R. Esteban-Romero “Sylow permutable subnormal subgroups of finite groups II” To appear in Bull. Austral. Math. Soc. [7] J.C. Beidleman, B. Brewster, D.J.S. Robinson “Criteria for permutability to be transitive in finite groups” J. Algebra, 222 (1999), 400-412 [8] J.C. Beidleman, H. Heineken “Finite soluble groups whose subnormal subgroups permute with certain classes of subgroups” Preprint [9] M. Bianchi, A. Gillio Berta Mauri, M. Herzog, L. Verardi “On finite solvable groups in which normality is a transitive relation” To appear in J. of Group Theory. [10] R.A. Bryce, J. Cossey “The Wielandt subgroup of a finite soluble group” J. London Math. Soc. 40 (1989), 244-256 [11] W. Gasch¨ utz “Gruppen in deinen das Normalteilersein transitiv ist” J. reine angew. Math. 198 (1957), 87-92 [12] O.H. Kegel “Sylow-Gruppen und Subnormalteiler endlicher Gruppen” Math. Z. 78 (1962), 205-221 [13] T.A. Peng “Finite groups with pro-normal subgroups” Proc. Amer. Math. Soc. 20 (1969), 232-234 [14] D.J.S. Robinson “A note on finite groups in which normality is transitive” Proc. Amer. Math. Soc. 19 (1968), 933-937 [15] D.J.S. Robinson “The structure of finite groups in which permutability is a transitive relation”, J. Aust. Math. Soc. 70 - 2 (2001), 143-149 [16] G. Zacher “I gruppi risolubili finiti in cui i sottogruppi di composizione coincidono con i sottogruppi quasi-normali” Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 37 (1964), 150-154

(PRO)-FINITE AND (TOPOLOGICALLY) LOCALLY FINITE GROUPS WITH A CC-SUBGROUP Z. ARAD∗ and W. HERFORT†1 ∗

Department of Mathematics, Bar–Ilan University, Ramat–Gan, Israel and Department of Computer Science and Mathematics, Netanya Academic College, Netanya, Israel † Institut f¨ ur Angewandte und Numerische Mathematik, Technische Universit¨at Vienna, Austria

Abstract A proper subgroup H of a group G is called a CC-subgroup of G if the centralizer CG (h) of h ∈ H # = H \ {1} is contained in H. Such finite groups were partially classified by G. Frobenius , W. Feit , K. W. Gruenberg and O. H. Kegel , J.S. Williams , A. S. Kondrat’iev , N. Iiyori and H. Yamaki , M. Suzuki , M. Herzog , Z. Arad , D. Chillag , Ch. Praeger and others. In this report2 , using the classification of finite simple groups, we give a complete list of all finite groups containing a CC-subgroup. As a corollary we classify infinite profinite groups, locally finite groups and certain classes of topological groups containing a CC-subgroup under certain conditions.

1

Introduction

Let G denote a finite group. According to M. Herzog [18] a subgroup M ≤ G is a CC-subgroup (”centralizers contained“), if CG (m) ≤ M for every m ∈ M \ {1}. The example with smallest cardinality is G := S3 with either M := (123) or M := (12) being a CC-subgroup. More generally, by the well known result of G. Frobenius , every Frobenius group has CC-subgroups either the kernel or any complement. 1.1

Sketching the thread

One finds the concept of a CC-subgroup (without calling it that) in work of W. Feit describing doubly transitive groups which fix 3 letters (e.g. in [13]). He considered the situation of a group containing a CC-subgroup of order divisible by 3 and conjectured that G is either Frobenius with kernel M or G = PSL(2, 3n ) with n ≥ 1, provided certain extra conditions hold. In [25] M. Suzuki classified all groups containing a CC-subgroup of even order and in the proof of his Theorem 1 in [25] he said a group G with a CC-subgroup to satisfying condition (c). M. Suzuki found that such a group is either Frobenius, 1 The second author would like to thank for greatful hospitality at the Bar-Ilan University, the Netanya Academic College and the Tel-Aviv University in February 2000 2 The results of this report and their complete proofs are contained in [10]

FINITE AND LOCALLY FINITE GROUPS WITH A CC-SUBGROUP

7

PSL(2, q) for q a power of 2, or in today’s terminology, G is a Suzuki group over a field of characteristic 2. In [14] W. Feit and J. G. Thompson showed that if M ∼ = C3 then G is either an extension of a nilpotent group by either A3 or S3 , or G is an extension of a 2-group by A5 or, third possibility, G ∼ = PSL(2, 7) and in [1] Z. Arad generalized their result to classifying all groups containing a CC Sylow 3subgroup. M. Herzog in [18] proved the aforementioned conjecture of Feit under additional assumptions. Z. Arad in [2] and Z. Arad together with M. Herzog in [7, 8], using results of W. B. Stewart [26], succeeded in determining a complete list of finite groups containing a CC-subgroup of order divisible by 3. Apparently, in an unpublished note of K. W. Gruenberg and O. H. Kegel and later by K. W. Gruenberg and K. W. Roggenkamp in [17] the notion of the prime graph of a finite group G has been introduced to have vertices the primes dividing |G| and edges (p, q) whenever exist commuting elements x, y ∈ G of respective orders p = q. Its connected components are denoted by sets of primes such that 2 ∈ π1 provided G has even order. J.S. Williams ’s Theorem 3 in [27] shows that to every odd component πi (only odd primes, i > 1) there exists a πi -Hall subgroup M ≤ G, which is a an odd order CC-subgroup. The existence of such M together with results in [17] show that the prime graph is disconnected if and only if the augmentation ideal decomposes as a G-module. The same authors introduce the notion of a 2-Frobenius group for a group G containing a normal Frobenius group H = KL, K G with G/K again Frobenius group. In Williams’s Theorem 3 each such M turns out to be nilpotent. Therefore it is desirable to give a complete description of groups containing a CC-subgroup including structural information on the group M as well. Z. Arad and D. Chillag in [4, 5, 6] continued classifying groups with a CC-subgroup. As a final result we present Theorem A below. In [21] O. H. Kegel and B. A. F. Wehrfritz describe the situation of locally finite Frobenius groups and, answering a question originally posed by O. H. Kegel, D. Gildenhuys , L. Ribes and W. Herfort provide a description of profinite Frobenius groups. Since in the profinite completion of the infinite dihedral group C2 ∗ C2 each of it’s factors is a CC-subgroup and no normal complement to any of them exists, the extra condition of M being a Hall-subgroup, has been included in the definition (see section 4.6 on profinite Frobenius groups in [24]). In [15] ˇakov gives a unifying result on algebraic and finite Frobenius groups. Yu. M. Gorc In [23] Yu. N. Mukhin deals with certain topological Frobenius groups. We present results for profinite groups (Theorem B), locally finite groups (Theorem C) and topologically locally finite groups (every compact subset is contained in a compact subgroup – we denote this fact by G ∈ [LF]− ) in Theorem D.

2

Announcement of the results

Following P.Hall [19] we say that G satisfies Eπ if G has a Hall π-subgroup denoted by Gπ . G satisfies Cπ if it satisfies Eπ and any two Sπ -subgroups of G are conjugate. G satisfies Dπ if it satisfies Cπ and every π-subgroup of G is contained in some Sπ -subgroup of G.

8

ARAD, HERFORT

The following result contains a complete classification of finite groups containing a CC-subgroup. Theorem A Let G be a finite group containing a CC-subgroup M . Let π := π(M ). Then G satisfies Dπ . Furthermore we have one of the following four cases: (1) M is non-nilpotent and of even order and one of the following holds: (a) G is a Frobenius group with complement M ; ∼ PSL(2, 2n ), n ≥ 2 and M is solvable; (b) G = (c) G ∼ = Sz (q), q = 22n+1 , n ≥ 1 and M is solvable. (2) M is nilpotent of even order and one of the following holds: (a) G is a solvable Frobenius group with complement M ; (b) G is a solvable Frobenius group with kernel M ; ∼ PSL(2, 2n ), n ≥ 2 and M is a 2-Sylow subgroup; (c) G = (d) G ∼ = Sz (q), q = 22n+1 , n ≥ 1 and M is a 2-Sylow subgroup. (3) M is non-nilpotent of odd order and one of the following holds (a) G is a solvable Frobenius group with complement M ; ∼ PSL(2, q), q ≡ 3 (mod 4) and M is solvable of odd order |M | = (b) G = q q−1 2 ; (4) M is nilpotent of odd order and one of the following holds: (a) G is a Frobenius group with M either kernel or complement; (b) G is simple non-abelian and G and M are classified in [27] and [28] ([29]) (see as well A. S. Kondrat’iev in [22]); (c) G is not simple. With H := (M )G and S := H/F (H), one finds S to be simple containing the CC-subgroup M F (H)/F (H) ∼ = M (S and M are classified in (4)(b)), and F (H) and G/H are π1 –groups. (d) G is a 2-Frobenius group, i.e., exists F G such that F M G is a Frobenius group with kernel F and cyclic complement M , and exists cyclic R ≤ G with M R Frobenius group (having kernel M and cyclic complement R) and G = F M R. Proof If M ≤ G is a CC-subgroup, then certainly the prime graph is disconnected. If G is nilpotent then, using the classification of finite simple groups, the aforementioned result of Gruenberg and Kegel yields part of (b) and (d). If M has even order then Suzuki’s result yields (a) and the other part of (b). From now on M is supposed to be of odd order and not nilpotent. Then NG (M ) = M (else NG (M ) is Frobenius and hence M nilpotent) and [6] shows that M = KL is a π := π(M ) Hall subgroup, and at the same time is Frobenius with kernel K and cyclic kernel

FINITE AND LOCALLY FINITE GROUPS WITH A CC-SUBGROUP

9

L. By the same paper G satisfies Dπ . From all this conclude that the prime graph of G has at least 3 components and that π contains at least two primes p, q with p dividing q − 1. The list in (c) is found combining information on the order of G from [12] and checking the tables in [27] and [28] for divisibility. Sporadic and alternating groups are excluded by using the fact that Hall subgroups involving precisely two primes p, q (with p < q ≤ n if G = An ) do not exist in neither An [11] or any sporadic group [19]. ✷ Theorem B Let a profinite group G contain a CC-Hall subgroup M . One of the following holds. (i) G is finite and contained in the list of Theorem A; (ii) G is infinite, M is finite and one of the following holds: (a) G is a profinite Frobenius group with M the Frobenius complement; (b) With H := (M )G the quotient S := H/F (H) is a finite simple group, M F (H)/F (H) ∼ = M is a CC-subgroup of S, F (H) is a nilpotent π1 group, and G/H is a finite π1 -group; (c) G is a profinite 2-Frobenius group, i.e., exists F G open, such that F M G is a profinite Frobenius group with kernel F and finite cyclic complement M and exists a finite R ≤ G with M R a finite Frobenius group (kernel M , finite cyclic complement R) and G = F M R; in (b) an isomorphic copy of M is an odd order CC-subgroup of S; the group S is as in Theorem A (4)(c); (iii) G and M are both infinite and G is a profinite Frobenius group with M the Frobenius kernel. Theorem C Let a locally finite group G contain a CC-Hall subgroup M . One of the following holds. (i) G is finite and contained in the list of Theorem A; (ii) G is infinite, M either contains an involution or is not locally nilpotent and one of the following holds: (a) G is a Frobenius group with M either kernel or complement; (b) G ∼ = PSL(2, F ), with F a locally finite field; (c) G ∼ = Sz (F ), F a locally finite field of even characteristics and M locally solvable; (iii) G is infinite, M does not contain any involution and is locally nilpotent and one of the following holds: (a) G is a Frobenius group with M either kernel or complement;

10

ARAD, HERFORT (b) G is a locally finite 2-Frobenius group, i.e., exists F G with F M G a locally finite Frobenius group (having kernel F and locally cyclic complement M ) and exists R ≤ G with M R a locally finite Frobenius group (kernel H and finite cyclic complement R) and G = F M R; (c) For H := (M )G the quotient S := H/F (H) is a locally finite simple group with M F (H)/F (H) a CC-subgroup of S, F (H) a nilpotent π1 normal subgroup of G, and G/H is a locally finite π1 group. S is a direct limit of groups each of them an extensions of a π1 -group by a simple group from (4)(b) in Theorem A.

With the help of Theorems B and C we establish a classification result for the class [LF]− of topologically locally finite groups (as introduced in [20]). Theorem D Let G ∈[LF]− be totally disconnected and neither be locally finite, nor compact. One of the following holds: (i) G is a topologically locally finite group. M is locally finite, iso Frobenius lated and complement and F := G \ g∈G H g ∪ {1} is a CC-normal subgroup of G (the kernel); (ii) G is a topologically locally finite 2-Frobenius group, i.e., exists F G open such that F M G is topological Frobenius group (F the kernel and M finite complement) and exists finite cyclic R ≤ G such that M R is a finite Frobenius group with kernel M and complement R (G is topologically 2-Frobenius) and G = F M R; (iii) G is a topological Frobenius group with M G an open CC-normal subgroup (the kernel) and it possesses a locally finite Frobenius-complement H, which is an isolated Hall subgroup of G; (iv) M is locally finite; for H := (M )G the quotient S := H/F (H) is a locally finite simple group with M F (H)/F (H) ∼ = M a CC-subgroup (as classified in Theorem C), F (H) is a nilpotent normal π1 subgroup of G, and G/H is a locally finite π1 -group. It should be desirable to extend our classification theorems to classes of groups ˇakov and containing algebraic groups as well as ours. The results of Yu. M. Gorc Yu. N. Mukhin indicate such a possibility. References [1] Z. Arad , A classification of 3CC groups and applications to Glauberman-Scmidt theorem, J. Alg. 42, 176–180, (1976) [2] Z. Arad , A classification of groups with a centralizer condition, Bull. Austral. Math. Soc. 15, 81–85, (1976) [3] Z. Arad & D. Chillag , On Finite Groups Containing a CC-Subgroup, Arch. d. Math. 24, 225–234, (1977) [4] Z. Arad & D. Chillag , On Finite Groups Containing a CC-Subgroup, Arch. d. Math. 35, 401–405, (1980)

FINITE AND LOCALLY FINITE GROUPS WITH A CC-SUBGROUP

11

[5] Z. Arad & D. Chillag , Finite Groups Containing a Nilpotent Hall Subgroup of Even Order, Houston J. of Math. 7(1), 23–32, (1981) [6] Z. Arad & D. Chillag , A Criterion for the Existence of Normal π-complements in Finite Groups, J. Alg. 87, 472–482, (1984) [7] Z. Arad & M. Herzog , A classification of groups with a centralizer condition II: Corrigendum and addendum, Bull. Austral. Math. Soc. 16, 55–60, (1977) [8] Z. Arad & M. Herzog , A classification of groups with a centralizer condition II: Corrigendum and addendum, Bull. Austral. Math. Soc. 17, 157–160, (1977) [9] Z. Arad & W. Herfort , A classification of locally finite and profinite groups with a centralizer condition, Communications in Algebra 10, 1749–1764, (1982) [10] Z. Arad & W. Herfort , Classification of Finite Groups with a CC-subgroup and Applications to Infinite and Topological Groups, (submitted for publication) [11] Z. Arad & M. B. Ward , New Criteria for the Solvability of Finite Groups, J. Alg. 77, 234–246, (1982) [12] J. H. Conway , R. T. Curtis , S. P. Norton , R. A. Parker , R. A. Wilson , Atlas of finite groups. Maximal subgroups and ordinary characters for simple groups. With comput. assist. from J. G. Thackray. Oxford: Clarendon Press, (1985). [13] W. Feit On a class of doubly transitive permutation groups. Illinois J. Math.4,1960, 170–186. [14] W. Feit & J. G. Thompson , Finite groups which contain a self-centralizing subgroup of order 3. Nagoya Math. J. 21, 1962, 185–197 ˇakov , On infinite Frobenius groups, Dokl.Akad.Nauk.SSSR (AMS [15] Yu. M. Gorc Transl. (1963) 11, 1397-1399) [16] K. W. Gruenberg & O. H. Kegel , Unpublished manuscript, (1957) [17] K. W. Gruenberg & K. W. Roggenkamp , Decomposition of the augmentation ideal and of the relation modules of a finite group. Proc. London Math. Soc. 31, 149–166, (1975) [18] M. Herzog , On finite groups which contain a Frobenius group, J. Alg. 6, 192–221, (1967) [19] P. Hall , Theorem like Sylow’s, Proc. London Math. Soc. 6, 286–304, (1956) [20] W. Herfort & L. M. Manevitz , Topological Frobenius groups, J. Alg. 92, 16–32, (1985) [21] O. H. Kegel & B. A. F. Wehrfritz , Locally Finite Groups, North Holland, Mathematical Library, 1973 [22] A. S. Kondrat’iev , Prime graph components of finite simple groups, (Russian, English) Math. USSR, Sb. 67, No.1, 235-247 (1990); translation from Mat. Sb. 180, No.6, 787-797 (1989) [23] Yu. N. Mukhin , On Frobenius topological groups, Investigations in group theory, Collect. Artic., Sverdlovsk 1984, 120-130 (1984) [24] L. Ribes & P. A. Zalesskii , Profinite Groups, Springer 2000 [25] M. Suzuki , On a class of doubly transitive groups, Ann. Math. II 75, 105–145, (1962) [26] W. B. Stewart , Groups having strongly self-centralizing 3-centralizers, Proc. London Math. Soc. 26, 653–680, (1973) [27] J.S. Williams , Prime Graph Components of Finite Groups, J. Alg. 69, 487–513, (1981) [28] N. Iiyori & H. Yamaki , Prime Graph Components of the Simple Groups of Lie Type over the Field of Even Characteristic, J. Alg. 155, 335–343, (1993) [29] N. Iiyori & H. Yamaki , Corrigendum to: Prime Graph Components of the Simple Groups of Lie Type over the Field of Even Characteristic, J. Alg. 181, 659, (1996)

TABLE ALGEBRAS GENERATED BY ELEMENTS OF SMALL DEGREES ZVI ARAD∗† and MIKHAIL MUZYCHUK

†1

∗

Department of Mathematics, Bar-Ilan University , Ramat-Gan, 52900, Israel Department of Computer Sciences and Mathematics Netanya Academic College, Netanya,42365, Israel †

Abstract The pair (A, B) is called a table algebra if A is a commutative associative C-algebra of dimension k with a distinguished basis B = {1 = b1 , b2 , . . . , bk } which satisﬁes the following conditions: k λijm bm where λijm ∈ R+ ∪ {0} (nonnegative reals). I) For all i, j bi bj = m=1

¯=a II) There is an algebra automorphism − : A → A such that B = B and a for all a ∈ A. (If bi = bi = bi , then bi is called real). III) For all i, j λij1 = 0 iﬀ j = i. Arad and Blau proved that there exists a unique algebra homomorphism f : A → C such that f (bi ) = f (bi ) ∈ R+ for all i. The numbers {f (bi ) bi ∈ B} are called the degrees of (A, B). The goal of our paper is to give a survey of current research on algebras generated by a basis element of small degree n where n ∈ {2, 3, 4, 5}.

1

Introduction

First we give all deﬁnitions which are necessary in order to formulate our results. A table algebra (A, B) is called integral if λijm ∈ N ∪ {0} and f (bi ) ∈ N for all i, j, m; homogeneous of degree λ if f (b) = λ for all b ∈ B\ {1}; standard if f (bi ) = λii1 for all i; normalized if λii1 = 1 for all i. k λi bi ∈ A we deﬁne the support of a by Let (A, B) be a table algebra. For a = i=1 the following formula: suppB (a) = {bi ∈ B λi = 0}. There is a scalar product ( , ) associated with a table algebra (A, B): xb b, yb b = xb yb λb¯b1 b∈B

b∈B

b∈B

A subset C ⊆ B is called a table subset of B iﬀ suppB (bi bj ) ⊆ C for all bi , bj ∈ C. ∞ supp(cn ) called the For each c ∈ B we deﬁne a table subset Bc by Bc = table subset generated by c. 1

n=1

The second author was supported by the Israeli Ministry of Absorption.

INTEGRAL TABLE ALGEBRAS

13

A table algebra (A, B) is called simple iﬀ the only table subsets of B are {1} and B. An element c ∈ B is called faithful iﬀ Bc = B; linear iﬀ suppB (cn ) = {1} for some n > 0. The set of all linear elements in B is denoted by L(B). A table algebra is called abelian iﬀ all elements b ∈ B are linear. Two table algebras (A, B) and (A , B ) are called isomorphic, notation B B , if there exists an algebra isomorphism ψ : A → A such that ψ(B) is a rescaling of B ; and the algebras are called exactly isomorphic, notation B x B , when ψ(B) = B (in other words, B and B have the same structure constants). 1.1

Examples

There are two natural examples of table algebras related to ﬁnite groups. Let G be a ﬁnite group, then the centre of Z(C[G]) of the group algebra C[G] is an integralstandard table algebra a distinguished basis of which is formed by sums Cˆ := c∈C c ∈ Z(C[G]) where C runs within the set of G-conjugacy classes. The algebra of complex valued class functions on G with a distinguished basis of irreducible characters B = Irr(G) is an integral normalized table algebra. If G is ﬁnite simple group, then both algebras are simple. Another example arises from algebraic combinatorics. Let (X; R = {Ri }di=0 ) be an association scheme on a ﬁnite set X (we refer the reader to [7] for the deﬁnition of an association scheme). Then a C-vector space A spanned by the adjacency matrices Ai of the relations Ri , i = 0, ..., d is a complex matrix algebra which is called Bose-Mesner algebra of the association scheme. It is a standard integral table algebra with a distinguished basis A0 , ..., Ad . If the automorphism group of the association scheme (X, R) contains a regular subgroup H, then the association scheme is equivalent to so-called Schur ring. Let H be a ﬁnite group. An element x = h∈H xh h ∈ C[H] is called a simple quantity if xh ∈ {0, 1} for each h ∈ H. It is easy to see that there exists a one-to-one correspondence between subsets of H and simple quantities. We write T , T ⊆ H for the formal sum t∈T t. A subalgebra A ⊆ C[H] is called a Schur ring (brieﬂy an S-ring) over the group H if there exists a basis B = {T 0 = {1}, ..., T d } of A consisting of simple quantities which satisﬁes the following conditions [15]: S1. H = T0 ∪ ...Td is a partition of H; := S2. For each i ∈ {0, 1, ..., d} there exists i ∈ {0, ..., d} such that Ti ∈ T } = Ti . The sets Ti , i = 0, ..., d are called the basic sets of A. If A is commutative (for example, if H is abelain), then (A, B) becomes an integral standard table algebra. We say that a table algebra is group-like if it is exactly isomorphic to an S-ring over some group. Let Φ ≤ Aut(H) and let T0 , ..., Td be the complete set of Φ-orbits of the Φ-action on H. Then the vector space spanned by the simple quantities T i , i = 0, ..., d is a Schur ring over H. We shall write O(Φ, H) for the set {T 0 , ..., T d }. A particular case of this construction gives us the center of the group algebra C[H] if we take Φ = Inn(H). (−1)

{t−1 | t

14 1.2

ARAD, MUZYCHUK Wreath, tensor and fibred products

Let (A, B), (C, D) be two table algebras. Then one can deﬁne their tensor product (A ⊗C C, B ⊗ D) where the distinguished basis B ⊗ D is the set of tensors b ⊗ d, b ∈ B, d ∈ D. If both (A, B) and (C, D) are standard then so is their tensor product. If (C, D) is standard, then (A ⊗C C, B ⊗ D) contains a subalgebra (A, B) (C, D) spanned, as a vector space, by the following basis: B D = {1 ⊗ d | d ∈ D} ∪ {b ⊗ D+ | b ∈ B, b = 1}. A direct check shows that a subspace spanned by the above basis is a subalgebra of (A ⊗C C, B ⊗ D) that satisﬁes all the axioms. In what follows we shall denote it by (A, B) (C, D) and call it the wreath product of (C, D) by (A, B). The dimension of (A C, B D) is always equal to dim(A) + dim(C) − 1. A tensor and wreath product of table algebras are particular cases of a more general operation: ﬁbred product. Consider two table algebras (A, B) and (S, T). A family U = {Ub }b∈B of subsets of S will be called a B-graded set of fibres if it satisﬁes the following conditions 1. ∀a∈B the set Ua is linearly independent. 2. ∀a∈B ∀b∈B ∀c∈B (λabc = 0 =⇒ CUa CUb ⊆ CUc ) . The second property is equivalent to an existence of the elements µuvw abc ∈ C such that for each triple a, b, c ∈ B with λabc = 0 it holds that ∀u∈Ua ∀v∈Ub uv = µuvw (1.1) abc w w∈Uc

µuvw abc

are reals and µuvw In what follows we always assume that abc ≥ 0. If the table algebras (A, B), (C, D) integral we shall always require that µuvw abc ∈ N. The following claim follows directly from the deﬁnition of the B-graded set of ﬁbres. Proposition 1.1 The subspace spanned by the set b∈B {b ⊗ u | u ∈ Ub } is an subalgebra of A ⊗ S. In what follows we shall denote the set b∈B {b ⊗ u | u ∈ Ub } as B ⊗ U. Proposition 1.2 Let (A, B) and (S, T) be two standard table algebras. Assume that a B-graded set of ﬁbres U has the following properties: 3. 1 ∈ U1 ; 4. ∀a∈B Ua = Ua ; 5. ∀a∈B ∀u,v∈Ua µuv1 aa1 = δu,v f (u) . Then the C-algebra C(B ⊗ U) with a distinguished basis B ⊗ U is a standard table algebra. Its structure constants are computed by the following formula: λa⊗u b⊗v c⊗w = λabc µuvw abc ,

(1.2)

INTEGRAL TABLE ALGEBRAS

2

15

Classification results about standard integral table algebras

H.Blau and B.Xu proved in [13] that each integral table algebra may be rescaled into a homogeneous integral table algebra. This makes this class of algebras of a special interest. A homogeneous table algebra of degree 1 is just a group algebra of a ﬁnite abelian group. Table algebras generated by an element of degree 2 were completely classiﬁed by H.Blau in [9] under the assumption that either a generating element is real or the algebra does not contain linear elements of degree a power of 2. The homogeneous table algebras of degree 3 under certain conditions were classiﬁed in [11]. The following result may be considered as the ﬁrst step towards the classiﬁcation of homogeneous table algebras of degree 4. Theorem 1 [Arad-Erez-Muzychuk],[5] Let (A, B) be a standard integral homogeneous table algebra of degree 4 which contains a faithful element. Then it is exactly σ is an automorphism of Zn × Zn isomorphic to O(σ, Zn × Zn ),n is odd, where

0 1 . of order 4 deﬁned by the matrix −1 0 Theorem 2 [Arad-Fisman-Muzychuk],[6] Let (A, B) be a standard integral table algebra with L(B) = 1 generated by basis elements of degree 2. Then there exists a ﬁnite group G = EH, where E is an elementary abelian 2-group, H is abelian, NG (h) CG (h) for all h ∈ H and CH (E) = 1G such that B is exactly isomorphic to {cl G (h) h ∈ H}. Moreover, the degrees of the basis elements of B are the same as the set of orders of subgroups of E. Also H is a group of odd order. Theorem 3 [Arad-Arisha-Fisman-Muzychuk],[1] Let (A, B) be a standard integral table algebra with a faithful nonreal element a ∈ B of degree 3. Assume that L(B) = {1}. Then one of the following holds: (a) aa = 3·1+3b where f (b) = 2 and B is exactly isomorphic to a wreath product Zm V where V = {1, v}, v 2 = 2 · 1 + v. (b) aa = 3 · 1 + b, b ∈ B and f (b) = 6 (such algebras were classiﬁed by Blau; see Theorem 4.) (c) B is a homogeneous standard integral table algebra of degree 3. Algebras of type (c) were completely classiﬁed in [11]. Algebras of type (b) were classiﬁed by H.Blau in [10]. Theorem 4 [Harvey I. Blau],[10] Assume the conditions of Theorem 3 of type (b), then B is exactly isomorphic to either Gn for some n ≥ 3 or O(S3 , Zm × Zm ) for some m ≥ 3. We refer the reader to [10] for exact deﬁnition of algebras Gn . The distinguished basis of the table algebra O(S3 , Zm × Zm ) is formed by the orbits of the symmetric group S3 acting in a natural way on Zm × Zm .

16

ARAD, MUZYCHUK

Remark 1: Theorem 3 is not true if b is not required to be non-real. H.Blau constructed an inﬁnite family of integral standard table algebras with a faithful real element of degree 3. Thousands of such algebras were constructed by Z.Arad and his students using computer. Theorem 5 [Arad–Muzychuk-Arisha-Fisman],[2] Let (A, B) be a standard integral table algebra with a faithful nonreal element b ∈ B of degree 4. Assume that f (c) ≥ 3 for all c ∈ B \ {1}. Then one of the following holds: (a) bb = 4b1 + 4c where c ∈ B is of degree 3 and (A, B) is exactly isomorphic to a wreath product Zn {1, c} where {1, c} is uniquely deﬁned by sup(bb) = {1, c}. (b) bb = 4b1 + c4 + d4 + e4 where f (c4 ) = f (d4 ) = f (e4 ), c4 , d4 , e4 ∈ B and (A, B) is exactly isomorphic to Fm . 2 (c) bb = 4b1 + c4 + d8 where f (c4 ) = 4, f (c8 ) = 8, c4 , d8 ∈ B and (A, B) is exactly . 3 isomorphic to Fm (d) bb = 4b1 + b12 where b12 ∈ B, f (b12 ) = 12. Case (d) is the most diﬃcult one. We have made a certain progress towards resolving this case [2]. Up to now we know only two series of examples of standard integral table algebras of this type: group-like algebras and the algebras An (sgn) described in Subsection 2.2. Group-like algebras are of type O(H, Z3m ), m ≥ 2 where H is a subgroup of S4 which contains a 4-cycle. Algebras with a faithful non-real element of degree ﬁve were studied in a series of papers [4],[12]. We give here one of the classiﬁcation results obtained there. Theorem 6 If (A, B) is a standard integral table algebra with L(B) = {1} and f (b) ≥ 4 for all b ∈ B \ {1} which contains a nonreal faithful basis element b of degree 5 such that (bb, bb) ≥ 65, then one of the following cases holds: (i) bb = 5 + 4a, where f (a) = 4; and B is exactly isomorphic to Zm V , where V = {1, a} and a2 = 4 + 3a. (ii) bb = 5 + 2(b + b), and B is exactly isomorphic to Tm (5). (iii) bb = 5 + 2(c + c) for a nonreal c ∈ B\{b, b}, and B is exactly isomorphic to Y5 or Tm (5). Here Tm (5) is a table algebra deﬁned in [13] and Y5 = {1, b, b, b2 , b2 } is a table algebra with the following multiplication table. bb = 5 + 2(b2 + b2 ) bb2 = 2(b + b2 ) + b2 b2 b2 = 5 + b + b + b2 + b2 2 3

The algebras Fm are defined in the next subsection The algebras Fm are defined in the next subsection

b2 = 3b + 2b2 bb2 = 2(b + b) + b2 b22 = 2(b + b2 ) + b2

INTEGRAL TABLE ALGEBRAS

17

The algebras Fm , Fm .

2.1

We shall build a series of algebras of dimension 4m + 2, m ≥ 1. We start with the following 6-dimensional algebra (H, E) where E = {1, h, c1 , c2 , v, u, } the multiplication table of which looks as follows h 4·1 + h + c1 + c2

h c1 c2 u v

h+v h+v v 3c1 + 3c2 +4u + v

c1 h+v

c2 h+v

u v

v 3c1 + 3c2 +4u + v 4c2 4·1 + 4u 3c1 3h + 3v 4·1 + 4u 4c1 3c2 3h + 3v 3c1 3c2 3·1 + 2u 3h + 2v 3h + 3v 3h + 3v 3h + 2v 12 · 1 + 9c1 + 9c2 +3h + 8u + 2v

(2.1)

A routine check shows that all the axioms of integral standard table algebras hold in this case. Deﬁne a subset U ⊂ H as follows U = {h, v, s = 2 · 1 + 12 (c1 + c2 ), t = −1 + u + 12 (c1 + c2 )}. By direct computations we obtain the following multiplication tables h v s t h h + 2s 2s + 4t + v 3h + v 2v v 2s + 4t + v 10s + 8t + 3h + 2v 5v + 3h 4v + 6h s 3h + v 5v + 3h 4s + 2t 2s + 4t t 2v 4v + 6h 2s + 4t 4s + 2t

h v 4·1 + c1 3c1 + 3c2 + +c2 + h 4u + v v 3c1 + 3c2 + 12·1 + 9c1 + 9c2 4u + v +8u + 3h + 2v s 3h + v 5v + 3h

h

t

2v

4v + 6h

s 3h + v

t 2v

5v + 3h

4v + 6h

(2.2)

(2.3) 6 · 1 + 3c1 4u + 3c1 + 3c2 +3c2 + 2u 4u + 3c1 + 3c2 6 · 1 + 3c1 +3c2 + 2u

h v s t h h + 2s 2s + 4t + v 3h + v 2v c1 h+v 3h + 3v 2s + 2t 2s + 2t c2 h+v 3h + 3v 2s + 2t 2s + 2t u v 2v + 3h 2t + s t + 2s v v + 4t + 2s 3h + 2v 3h + 5v 4v + 6h +8t + 10s

(2.4)

18

ARAD, MUZYCHUK

Let Cm = g be a cyclic group of order m written multiplicatively and let (C[Cm ]; Cm ) be the corresponding table algebra. For each a ∈ Cm we set: Ua =

E, a = 1; U, a = 1.

It follows from (2.2)-(2.4) that U = {Ua }a∈Cm is a set of Cm -graded ﬁbres. Therefore the set Fm := Cm ⊗ U is a distinguished basis of an integral standard table algebra (cf. Proposition 1.2). Since g ⊗ h is a faithful non-real element, the algebra Fm is an example of algebras which satisfy part (b) Theorem 5. The dimension of this algebra is 4m + 2. A direct check shows that the set Fm := (Fm \ {c1 , c2 }) ∪ {c1 + c2 } is a basis of a table subalgebra of Fm (of dimension 4m + 1). 2.2

The algebras An (sgn).

We start with the following two-parameter series of table algebras which we denote by H(m, n), n ∈ N, m ∈ R>0 . It is deﬁned by the following data: H(m, n) = {c0 , ..., cn−1 } ∪ {o0 , ...on−1 } ∪ {e0 , ..., en−1 } cα cβ cα+β eβ eα+β oβ oα+β

eα eα+β (m − 1)cα+β + (m − 2)eα+β (m − 1)oα+β

oα oα+β (m − 1)oα+β mcα+β+1 + meα+β+1

where the addition of indices is done modulo n. A direct check shows that H(m, n) is a standard table algebra which is integral if m ∈ N. Now let C2n be a cyclic group generated by g ∈ C2n . In a group algebra C[C2n ] we chose the following basis xi = sgn(i)g i where sgn : Z2n → {−1, 1} is an arbitrary function which satisﬁes sgn(0) = 1, sgn(−x) = sgn(x). We have xα xβ = S(α, β)xα+β where S(α, β) = sgn(α)sgn(β)/sgn(α + β). Since sgn is symmetric S(α, −α) = 1. Now we consider the direct sum H(4, n)⊕C[C2n ]. Clearly this is an associative commutative algebra. A direct check shows that the following elements form a basis of H(4, n) ⊕ C[C2n ]: − o+ α = oα + 2x2α+1 , oα = 3oα − 2x2α+1 ,

2eα + 2x2α , α = 0; − eα := 4e0 , α = 0 4cα , α = 0; − := := c+ , c α α c0 + x0 , α = 0

e+ α :=

2eα − 2x2α , α = 0; 0, α = 0 0, α = 0; 3c0 − x0 , α = 0

where α runs through the set {0, ..., n−1} ⊂ Z2n . We denote this basis by An (sgn). Clearly, |An (sgn)| = 5n. The following equalities show that the structure constants in the above basis are nonnegative integers.

INTEGRAL TABLE ALGEBRAS

19

1. When s = S(2α + 1, 2β + 1): + o+ α oβ

=

+ − c+ α+β+1 + (1 + s)eα+β+1 + (1 − s)eα+β+1 , α + β + 1 = 0; + + 4c0 + e0 , α + β + 1 = 0

2. When s = S(2α + 1, 2β + 1): − o+ α oβ =

+ − 3c+ α+β+1 + (3 − s)eα+β+1 + (3 + s)eα+β+1 , α + β + 1 = 0 + + 4c0 + 3e0 , α + β + 1 = 0

3. When s = S(2α + 1, 2β + 1): + o+ α eβ =

3+3s + 3−3s − 2 oα+β + 2 oα+β , β − + 3oα + 3oα,β , β = 0

= 0

4. When s = S(2α + 1, 2β): − o+ α eβ =

3−3s + 2 oα+β

+

0, β = 0

5.

+ o+ α cβ

=

3+s − 2 oα+β , β

= 0

− o+ α+β + oα+β , β = 0 + oα , β = 0

− − 6. o+ α c0 = oα . 7. When s = S(2α + 1, 2β + 1): − o− α oβ

=

+ − 9c+ α+β+1 + (9 − s)eα+β+1 + (9 + s)eα+β+1 , α + β + 1 = 0 + − + 12c0 + 8c0 + 9e0 , α + β + 1 = 0

8. When s = S(2α + 1, 2β): + o− α eβ =

9. − o− α eβ

=

10. + o− α cβ =

9−3s + 9+s − 2 oα+β + 2 oα+β , β + − 3oα + 3oα , β = 0

= 0

9−s − 2 oα+β , β

= 0

9+3s + 2 oα+β

0, β = 0.

+

− 3o+ α+β + 3oα+β , β = 0 + − 3oα + 2oα , β = 0

− + − 11. o− α c0 = 3oα + 2oα . 12. When s = S(2α, 2β):

 + + −  6cα + 4(eα + eα ), β = 0 + + − + + (2 + s)e 0, α + β = 0 3c e = e+ α β α+β + (2 − s)eα+β , α = 0, β =  α+β + − + 6c0 + 2c0 + 2e0 , α + β = 0

20

ARAD, MUZYCHUK

13. When s = S(2α, 2β):  0, β = 0    6(c+ + c− ) + 2e+ , α = 0 + − 0 0 0 eα eβ = + + − + (2 − s)e 3c  α+β α+β + (2 + s)eα+β , α + β = 0   − + 4c0 + 2e0 , α + β = 0 14.

 + eα , β = 0    4(e+ + e− ), α = 0 + + β β eα cβ = + , α = 0, β = 0, α + β = 0 e   α+β  , α = 0, β = 0, α + β = 0 2e+ 0

15. − e+ α c0

=

− e+ α + eα , α = 0 + 3e0 , α = 0.

16. When s = S(2α, 2β).   0, α = 0 or β = 0 − − 3c+ + (2 + s)e+ e = e− α β α+β + (2 − s)eα+β , α = 0, β = 0, α + β = 0  α+β + − + 6c0 + 2c0 + 2e0 , α = 0, β = 0, α + β = 0 17.

 − eα , β = 0    0, α = 0 + e− α cβ = e+ , α = 0, β = 0, α + β = 0    α+β 2e+ 0 , α = 0, β = 0, α + β = 0

− + − 18. e− α c0 = 2eα + eα , α = 0 19.

+ c+ α cβ =

4c+ α+β , α + β = 0; − 4(c+ 0 + c0 ), α + β = 0

20. − c+ α c0 =

21.

3

3c+ α , α = 0; c− 0 ,α = 0

− + − c− 0 c0 = 3c0 + 2c0 .

Normalized table algebras generated by a nonreal element of degree 3

Examples of normalized integral table algebras appear in the representation theory of ﬁnite groups and fusion rule algebras in conformal ﬁeld theory. If such an algebra contains a faithful element of degree two and does not contain linear elements of degree 2m , then its structure is known by [9]. Concerning normalized table algebras with a faithful element of degree 3, nothing was known. A.Arad and C.Guiyun made a certain progress towards a classiﬁcation of these algebras [14]. They found 4 examples of normalized integral table algebras with L(B) = {1}

INTEGRAL TABLE ALGEBRAS

21

generated by a non-real element of degree 3. Two of them are exactly isomorphic to (Ch(P SL(2, 7)), Irr(P SL(2, 7))) and (Ch(3·A6 ), Irr(3·A6 )) where 3·A6 is a triple cover of A6 . Two other algebras are not induced from the algebra of characters of any group. Their dimensions are: 32 and 22. References [1] Z. Arad, H. Arisha, E. Fisman and M. Muzychuk, Integral tables algebras with a faithful nonreal element of degree 3, Journal of Algebra, 231 (2000), 473-483 [2] Z. Arad, H. Arisha, E. Fisman and M. Muzychuk, Integral tables algebras with a faithful nonreal element of degree 4, In: Standard Integral Table Algebras Generated by a Nonreal element of Small Degree, Z.Arad and M.Muzychuk eds, to appear in the Lecture Notes in Mathematics, Springer-Verlag. [3] Z. Arad and H. Blau, On table algebras and applications to ﬁnite group theory, J. of Algebra 138 (1991), 137-185. [4] Z. Arad, F. B¨ unger, E. Fisman and M. Muzychuk, On standard integral table algebras with a faithful nonreal element of degree ﬁve, In: Standard Integral Table Algebras Generated by a Nonreal element of Small Degree, Z.Arad and M.Muzychuk eds, to appear in the Lecture Notes in Mathematics, Springer-Verlag. [5] Z.Arad, Y. Erez and M.Muzychuk, Classiﬁcation of standard integral homogeneous table algebras of degree 4 (manuscript). [6] Z. Arad, E. Fisman, M. Muzychuk, Standard integral table algebras generated by elements of degree two, (submitted to J. of Algebra). [7] E.Bannai and T.Ito, Algebraic Combinatorics I: Association Schemes, Benjamin /Cummings, Menlo Park, 1984. [8] H.I. Blau, Integral table algebras, aﬃne diagrams and the analysis of degree two, J. of Algebra 178 (1995), 872-918. [9] H.I. Blau, Homogeneous integral table algebras of degree two, Algebra Colloq. 4 (1997), 872-918. [10] H.I. Blau, Integral Table Algebras and Bose-Mesner Algebras with a Faithful Nonreal Element of Degree Three J. of Algebra 231 (2000), 484-545. [11] H.I.Blau, B.Xu, Z,Arad, E,Fisman, V. Miloslavsky and M.Muzychuk Homogeneous Integral Table Algebras of Degree Three: A Trilogy, Memoirs of the AMS 144, N 684 (2000). [12] F. B¨ unger Standard integral table algebras with a faithful real element of degree ﬁve and width three, In: Standard Integral Table Algebras Generated by a Nonreal element of Small Degree, Z.Arad and M.Muzychuk eds, to appear in the Lecture Notes in Mathematics, Springer-Verlag. [13] H.I.Blau and B.Xu, On homogeneous table algebras, J. of Algebra 199(1998), 142-168. [14] C. Guiyun and Z. Arad, On normalized table algebras generated by a faithful non-real element of degree 3, (submitted). [15] H. Wielandt, Finite Permutation Groups, Academic Press, 1964, Berlin.

SUBGROUPS WHICH ARE A UNION OF A GIVEN NUMBER OF CONJUGACY CLASSES ALI REZA ASHRAFI and HEYDAR SAHRAEI Department of Mathematics, University of Kashan, Kashan, Iran E-mail: [email protected]

Abstract In [11] and [12] Shahryari and Shahabi investigated the structure of ﬁnite groups containing a normal subgroup which is a union of two or three conjugacy classes. In [13] Riese and Shahabi investigated the similar problem for normal subgroups which are a union of four conjugacy classes. In [2], we investigated the structure of ﬁnite non-perfect groups in which every non-trivial proper normal subgroup is a union of n conjugacy classes, for a given integer n. In this survey paper we report these results and investigate some new problems. 2000 Mathematics Subject Classification: 20E34, 20D10. Keywords and phrases: Conjugacy class, normal subgroup

1

Introduction

Let G be a ﬁnite group and h be a non-central element of G. Following Shahryari and Shahabi [11], we say that a normal subgroup H of the group G is a small subgroup if H = 1 ∪ ClG (h), in which ClG (h) denotes the G-conjugacy class containing h. It is easy to see that H ≤ G and |H|(|H| − 1) ||G|. Moreover, H is an elementary abelian normal subgroup of G. In [11], Shahryari and Shahabi studied the structure of ﬁnite centerless groups in which G , the derived subgroup of G, is a small subgroup. They proved that: Theorem 1.1 (Shahryari and Shahabi [11]) Let G be a finite centerless group and G be a small subgroup of G. Then: (a) G is a Frobenius group with kernel G and its kernel is abelian. (b) G has exactly one irreducible non-linear character χ. (c) χ(1) = |G : G | and χ(h) = −1. (d) CG (h) = G and |G| = |G |(|G | − 1). In this connection one might ask about the structure of G, if G has a normal subgroup which is a union of three conjugacy classes. In [12], Shahryari and Shahabi studied the structure of a ﬁnite group G with a normal subgroup H which is a union of three conjugacy classes. They proved that:

SUBGROUPS WHICH ARE A UNION

23

Theorem 1.2 (Shahryari and Shahabi [12]) Let G be a finite group and H = 1 ∪ ClG (h) ∪ ClG (g) be a normal subgroup of G. Then one of the following holds: (a) h−1 ∈ ClG (g) and H is an elementary abelian p-group of odd order. Also, |H|−1 2 |χ(1), for any irreducible character χ of G with the property [χ, 1H ] = 0. (b) h−1 ∈ ClG (h), (o(h), o(g)) = 1, and H is a metabelian p-group.

(c) h−1 ∈ ClG (h), (o(h), o(g)) = 1 and H is a Frobenius group with the elementary abelian kernel H . Also, in the mentioned paper, authors proved that H is a Frobenius group of order pa q b , where p, q are primes and a, b are positive integers. There is an error in Corollary 7 of this paper. In the following lemma, we correct this error. Lemma 1.3 Let H = 1 ∪ ClG (g) ∪ ClG (h), h−1 ∈ ClG (h) and (o(g), o(h)) = 1. Then H is a Frobenius group of order 2n p, where p = 2n − 1 is prime. Proof Without loss of generality we can assume that gh, hg ∈ ClG (h). By Lemma 3 of [12], H is a Frobenius group of order pm q n , for some distinct primes p and q, and some positive integers n and m. By Lemma 5 of [12], H is a small subgroup of H, and, by Lemma 4 of [12], Z(H) = 1. So by Theorem 2.1 of [11], |H| = |H |(|H |−1). On the other hand, by Lemma 6 of [12], |H| = pq n . This shows that p = q n − 1 and so q = 2. This concludes the proof of lemma. Now, under certain conditions, we can improve the Theorem 1.1, to the case that G is a union of three conjugacy classes. Theorem 1.4 (Ashraﬁ and Sahraei [2]) Let G be a finite centerless group, G = 1 ∪ ClG (g) ∪ ClG (h), g, h are non-conjugate and non-central elements of G and h−1 ∈ ClG (g). Then the following assertions holds: (i) G is solvable and G is the unique minimal normal subgroup of G, (ii) G is a Frobenius group with kernel G and its complement is cyclic, (iii) G has exactly two irreducible non-linear character χ and ψ with χ(1) = ψ(1) = |G : G |, (iv) |G| = 12 pa (pa − 1), in which pa = |G |. The most important development concerning our problem has been achieved by Udo Riese and Mohammad Ali Shahabi. They have characterized [13] the structure of a ﬁnite group G containing a normal subgroup H, which is a union of four conjugacy classes. So, presently the smallest value of n for which no characterization of G has yet been found is ﬁve. In the end of this section, we state the result of Riese and Shahabi [13].

24

ASHRAFI, SAHRAEI

Theorem 1.5 Let G be a finite group and H be a normal subgroup of G which is a union of four conjugacy classes in G. Then the number of characteristic subgroups of H is at most four, and one of the following holds: (a) H is a p-group and H = 1. (b) H ∼ = A5 , and CGG(H) ∼ = S5 . (c) H is a solvable group of order pa q b , where p and q are distinct primes and a, b are positive integers. Also, one of the following holds: (i) r = b and H is a direct product of its elementary abelian Sylow p- and q-subgroups. (ii) r = b and H is a Frobenius group with kernel N , where N is the union of two conjugacy classes in G. Furthermore H N is a cyclic group of order G 2 p, p2 or the quaternion group of order 8. If | H N | = p , then CG (N )

is isomorphic to a subgroup of ΓL1 (q b ) and, in particular, solvable. If H ∼ N = Q8 is a quaternion group of order 8, then N is elementary abelian of order 9 and CGG(N ) is isomorphic to a subgroup of GL2 (3) containing Q8 .

(iii) r ≤ b and H is a Frobenius group with kernel M , where M is a Sylow H is q-subgroup of H, which is a union of three conjugacy classes and M cyclic of order p. Throughout this paper all groups considered are assumed to be ﬁnite groups. Our notation is standard and taken mainly from [3], [5] and [7].

2

On n-decomposable finite groups

In this section G is a ﬁnite non-perfect group. We investigate the structure of the group G with the condition that every non-trivial proper normal subgroup of G is a union of n distinct conjugacy classes, for a given n. In [2], we called such a group, n-decomposable. We denote the set of all such positive integers by Λ. It is easy to see that if G is a ﬁnite abelian n-decomposable group, then n is a prime number and G has order n2 . This shows that p ∈ Λ, p is prime. Using a non-abelian group of order pq, where p, q are primes and p < q, we can see that 1 + p−1 q ∈ Λ. Hence it is natural to ask: Problem 2.1 Is it true that Λ = N − {1}? In the following theorem, we investigate the structure of a ﬁnite solvable group G with the condition that every normal subgroup of G is a union of n conjugacy classes of G. We have: Theorem 2.2 Suppose G is a non-abelian solvable group and every non-trivial proper normal subgroup of G is a union of n conjugacy classes. Then G is a

SUBGROUPS WHICH ARE A UNION

25

Frobenius group with kernel G and its complement is a cyclic group of prime order. Moreover, G has order pr q, in which p and q are primes, r is a positive integer and pr − 1 = (n − 1)q. Proof. It is obvious that G is a maximal subgroup of G and so |G : G | = q, q is prime. Since G is a minimal normal subgroup of G, G is an elementary abelian subgroup of order, say pr . Thus, |G| = pr q. Since G is not abelian, q = p. If 1 = x ∈ G then it is an easy fact that CG (x) = G . Therefore, by [Theorem 1.2, p. 1136] of [8], G is a Frobenius group with kernel G . Since G is abelian, by [Theorem 5.1, p.1160] of [8], n − 1 = |G q|−1 . Therefore, pr − 1 = (n − 1)q, proving the theorem. In the following theorems, we determine the structure of n-decomposable nonperfect ﬁnite groups, for n = 2, 3, 4. Theorem 2.3 Suppose every proper non-trivial normal subgroup of G is small. Then one of the following holds: (a) G is an abelian group of order 4, (b) G is isomorphic to S3 , the symmetric group on three symbols, (c) G is isomorphic to the semidirect product Zp ∝ E(2n ), in which p = 2n − 1 is prime, and, for given positive integer n and prime number p such that p = 2n − 1, there exists at most one such a group. Proof We can assume that G is not abelian. According to Theorem 2.1 of [11], G is the unique non-trivial proper normal subgroup of G and is an elementary abelian subgroup of G. Therefore, G is solvable and by Theorem 2.2, G is a semidirect product of an elementary abelian subgroup of order q n by a cyclic group of order p, p is prime, and that p = q n − 1. Therefore, q = 2 or q = p + 1. If q = p + 1, then p = 2, q = 3 and G is isomorphic to S3 . Suppose q = 2 then G is isomorphic to the semidirect product Zp ∝ E(2n ), in which p = 2n − 1. It is well known that Aut(G ) ∼ = GL(2, n) and that |GL(2, n)| = pm, where (p, m) = 1. Suppose f : Zp −→ GL(2, n) is a group homomorphism, then o(f (1)) = 1 or p. If o(f (1)) = 1 then G is abelian, a contradiction. Thus, o(f (1)) = p and the image of Zp is a Sylow subgroup of GL(2, n), proving the theorem. We will omit the proof of the following theorem, which determines the structure of 3-decomposable non-perfect ﬁnite groups. Theorem 2.4 (Ashraﬁ and Sahraei [2]) Suppose that G is a 3-decomposable group. Then one of the following holds: (a) G is an abelian group of order 9, (b) G is a group of order pq, p and q are primes and q =

p−1 2 .

26

ASHRAFI, SAHRAEI n

(c) G is isomorphic to the semidirect product Zq ∝ E(3n ), in which q = 3 2−1 is prime, and, for a given positive integer n and prime number q such that n q = 3 2−1 , there exists at most one such a group. Finally, using similar argument as in Theorem 2.3, we can determine the structure of 4-decomposable ﬁnite groups. We have: Theorem 2.5 (Ashraﬁ and Sahraei [2]) Suppose that every proper non-trivial normal subgroup of G is a union of four conjugacy classes of G. Then one of the following holds: (a) G ∼ = S5 , the symmetric group on five letters, p−1 3 . n E(2 ), in

(b) G is a group of order pq, p and q are primes and q =

n

which q = 2 3−1 (c) G is isomorphic to the semidirect product Zq ∝ is prime, and, for a given positive integer n and prime number q such that n q = 2 3−1 , there exists at most one such a group. Now, we consider the semidirect product Zq ∝ E(2n ), in which q = prime. We end this paper with the following problem:

2n −1 3

is

Problem 2.6 Which pairs (n, q) can be represented a n-decomposable ﬁnite group? References [1] J. L. Alperin, Groups and Representations, Springer-Verlag, New York, 1995. [2] A. R. Ashraﬁ and H. Sahraei, On Finite Groups Whose Normal Subgroups Have the Same Number of Conjugacy Classes, To appear. [3] D. Gorenstein, Finite Groups, New York, 1968. [4] Jr. Marshall Hall, The Theory of Groups, Chelsea Publishing Company, New York, 1976. [5] B. Huppert, Endliche Gruppen, Springer-Verlag, Berlin, 1967. [6] I. M. Isaacs, Character theory of ﬁnite groups, Academic Press, 1978. [7] G. James and M. Liebeck, Representations and Characters of Groups, Cambridge University Press, 1993. [8] G. Karpilovsky, Group Representations, Volume 1, North-Holland Mathematical Studies, Vol. 175, Amesterdam, 1992. [9] Derek J. S. Robinson, A Course in the Theory of Groups, 2nd ed., Graduate Text in Mathematics ; 80, Springer-Verlag, New York, 1996. [10] H. Sahraei, Subgroups which is the Union of Conjugacy Classes, M.Sc. thesis, University of Kashan, 2000. [11] M. Shahryari and M. A. Shahabi, Subgroups which are the union of two conjugacy classes, Bull. Iranian Math. Soc. Vol. 25, No. 1 (1999), 59-71. [12] M. Shahryari and M. A. Shahabi, Subgroups which are the union of three conjugate classes, J. Algebra 207, (1998), 326-332. [13] Udo Riese and M. A. Shahabi, Subgroups which are the union of four conjugacy classes, To appear in Pure Mathematics and Its Applications

SOME RESULTS ON FINITE FACTORIZED GROUPS A. BALLESTER-BOLINCHES ∗ , JOHN COSSEY † , XIUYUN GUO M. C. PEDRAZA-AGUILERA ∗∗1

‡

and

∗

` Departament d’Algebra, Universitat de Val`encia, C/ Doctor Moliner 50, 46100 Burjassot (Val`encia), Spain † Mathematics Department, School of Mathematical Sciences, The Australian National University, Canberra, 0200, Australia ‡ Department of Mathematics, Shanxi University, Taiyuan, Shanxi 030006, People’s Republic of China ∗∗ Departamento de Matem´atica Aplicada, E.U.I., Universidad Polit´ecnica de Valencia, Camino de Vera, s/n, 46071 Valencia, Spain

The well-known fact that a product of two normal supersoluble subgroups of a group is not necessarily supersoluble shows that the saturated formation of supersoluble groups need not be closed under the product of normal subgroups. This makes interesting the study of factorized groups whose subgroup factors are connected by certain permutability properties. Baer (see [2]) proved that if a group G is the product of two normal supersoluble subgroups, then G is supersoluble if and only if the commutator subgroup of G is nilpotent. This result has been generalized by Asaad and Shaalan ([1]) in the following sense: If G is the product of two subgroups H and K such that H permutes with every subgroup of K and K permutes with every subgroup of H, that is, G is the mutually permutable product of H and K, and G , the commutator subgroup of G is nilpotent, then G is supersoluble. Moreover they prove that in the case G = HK such that every subgroup of H permutes with every subgroup of K, that is, G is the totally permutable product of H and K, then if the factors H and K are supersoluble the group G is also supersoluble. Further studies have been done by several authors within the framework of formation theory. More precisely, Maier ([14]) proves that the above property on totally permutable products is not only true for the class of all supersoluble groups but also for all saturated formations F containing U, the class of supersoluble groups. Two diﬀerent extensions to this result are made in [7] and [11]. In [7] BallesterBolinches and P´erez-Ramos extend the above result to non-saturated formations containing U. On the other hand, a generalization of Maier’s result to an arbitrary ﬁnite number of factors is given by Carocca in [11]. Furthermore, it turns out that the residual, projectors and normalizers associated to these kind of formations behave well in totally permutable products (see [3] and [6]). Mutually permutable products are studied in [4] and [5]. The results contained in those papers should be seen in the light of the fact that structural information about permutable products is notoriously hard. They have a good behaviour only in some special cases, for instance when the conmutator subgroup is nilpotent. A complete account of the 1 The work of the first and last authors is supported by Proyecto PB97-0674-C02-02 of DGICYT, MEC, Spain

28

BALLESTER-BOLINCHES et al.

majority of those results can be found in [10] and [12]. In the last years some weaker versions of the concepts of totally and mutually permutable products have been studied. Permutability of each factor only with some concrete families of subgroups of the other one is analyzed. In this context, our main objective in [8] and [9] is to obtain relevant properties of m-permutable products. A group G is said to be the m- permutable product of the subgroups H and K if G = HK and H permutes with every maximal subgroup of K and K permutes with every maximal subgroup of H. It is clear that totally permutable products and mutually permutable products are m-permutable. However, there exists m-permutable products which are not mutually permutable. Our ﬁrst result proves that saturated formations containing the class U are closed under m-permutable products with nilpotent conmutator subgroup. Notice that in this case, the groups in question are soluble. Theorem 1 Let G = G1 G2 . . . Gr be the pairwise m-permutable product of the subgroups G1 , G2 , . . . , Gr . Let F be a saturated formation such that U ⊆ F. If Gi is an F-group for each i and G is nilpotent, then G belongs to F. On the other hand, it is well-known that formations composed of nilpotent groups are subgroup closed ([13, IV, 1.16]). This motivates the following question: If G = AB and G is nilpotent, are A and B in the formation generated by G?. The answer to this question is negative in general as the following example shows. Let F be the class of soluble groups whose Carter subgroups are 2-groups and take for G the symmetric group of degree 3. By [13, IV, 1.1], F is a formation. Since the Carter subgroups of G are 2-groups it follows that the formation generated by G is contained in F. Now G = AB, where A is a Sylow 3-subgroup of G and B is a Sylow 2-subgroup of G, and G is nilpotent. Clearly A is not in F. However, our next result shows that the answer to the above question is aﬃrmative in the case of saturated formations. Theorem 2 Let G = G1 G2 . . . Gr be the pairwise permutable product of the subgroups G1 , G2 , . . . , Gr and suppose that G is nilpotent. Assume that F is a saturated formation such that G belongs to F. Then each subgroup Gi belongs to F. Combining Theorems 1 and 2, we have the following result. Corollary 1 Let G = G1 G2 . . . Gr be the pairwise m-permutable product of the subgroups G1 , G2 , . . . , Gr . Assume that G is nilpotent. If F is a saturated formation containing U, then G ∈ F if and only if Gi ∈ F for each i. The celebrated theorem of Kegel and Wielandt states the solubility of every group which is the product of two nilpotent subgroups. The nilpotent condition

SOME RESULTS ON FINITE FACTORIZED GROUPS

29

on the factors can not be relaxed to supersolubility. The group P SL(2, 7) is a non-soluble group which is the product of a nilpotent group and a supersoluble group. In the following we present some theorems of Kegel-Wielandt type for products connected by certain permutability properties. Theorem 3 Let the group G = HK be the product of the subgroups H and K. If H is supersoluble, K is nilpotent and K permutes with every maximal subgroup of H, then G is soluble. Theorem 4 Let the group G = HK be the m-permutable product of the subgroups H and K. If H and K are supersoluble, then G is soluble. As we have seen before, the m-permutable product G of two supersoluble groups is supersoluble if and only if G is nilpotent. The following theorem analyzes the case K nilpotent and it is an application of the above results. Theorem 5 Let the group G = HK be the m-permutable product of the subgroups H and K. Assume that H is supersoluble and K is nilpotent. If K permutes with every Sylow subgroup of H, then G is supersoluble. We ﬁnish by showing that the hypotheses on K in the above theorem are necessary in order to get supersolubility. Remarks (a) Permutability of H with every maximal subgroup of K is essential in order to get supersolubility in Theorem 5. Let G be the symmetric group of degree four. Then G = HK where K is the Klein four group and H is isomorphic to the symmetric group of degree 3. Since K is normal in G, we have that K permutes with every subgroup of H and H is supersoluble. However, G is not supersoluble. Notice that H does not permute with the maximal subgroups of K. (b) Theorem 5 is not true if K does not permute with every Sylow subgroup of H as the following example shows. Example Let X be the symmetric group of degree 3 and consider S = C7 ∼ X the natural wreath product of X with a cyclic group of degree 7. Let {v1 , v2 , v3 } be a basis of B, the base group of S, on which X acts naturally by permuting the suﬃces. Then it is clear that V7 =< v3 − v1 , v3 − v2 > is an irreducible Xsubmodule. Denote by G = [V7 ]X the corresponding semidirect product. Notice that the subgroup H = [V7 ]C3 of G is supersoluble since 3 divides 7 − 1. Moreover if C2 =< (12) >, then V7|C2 = W ⊕ W1 where W is isomorphic to the trivial C2 module. Hence K = W × C2 is a nilpotent subgroup of G. Notice that G = HK. Moreover, G is the m-permutable product of H and K. However G is not a supersoluble group. Notice that K does not permute with every Sylow subgroup of H.

30

BALLESTER-BOLINCHES et al.

References [1] M. Asaad and A. Shaalan, On the supersolvability of ﬁnite groups. Arch. Math., Vol. 53, 318-326 (1989). [2] R. Baer, Classes of ﬁnite groups and their properties. Illinois J. Math., Vol. 1, 115-187 (1957). [3] A. Ballester-Bolinches and M. C. Pedraza-Aguilera and M. D. P´erez-Ramos, On ﬁnite products of totally permutable groups. Bull. Austral. Math. Soc, Vol. 53, 441-445 (1996). [4] A. Ballester-Bolinches, M. C. Pedraza-Aguilera and M. D. P´erez- Ramos, Mutually permutable products of ﬁnite groups. J. Algebra, Vol. 213, 369-377 (1999). [5] A. Ballester-Bolinches and M.C. Pedraza-Aguilera, Mutually permutable products of ﬁnite groups II. J. Algebra, Vol 218, 563-572, (1999). [6] A. Ballester-Bolinches and M.C. Pedraza-Aguilera and M. D. P´erez-Ramos, Finite groups which are products of pairwise totally permutable subgroups. Proc. Edinburgh Math. Soc., Vol 41, 567-572, (1998). [7] A. Ballester-Bolinches and M. D. P´erez-Ramos, A Question of R. Maier concerning formations. J. Algebra, Vol. 182, 738-747, (1996). [8] A. Ballester-Bolinches Xiuyun Guo and M. C. Pedraza-Aguilera, A note on m-permutable products of ﬁnite groups. J. Group Theory, Vol. 3, 381- 384 (2000). [9] A. Ballester-Bolinches, John Cossey and M. C. Pedraza-Aguilera, On products of ﬁnite supersoluble groups. Accepted in Comm. in Algebra. [10] A. Ballester-Bolinches, M. D. P´erez-Ramos and M. C. Pedraza- Aguilera, Totally and Mutually permutable products of ﬁnite groups, Proc. Group Bath/St. Andrews(1997). London Math. Soc. Lecture Notes Series 260, Vol. I, 65- 68 (1999). [11] A. Carocca, A note on the product of F-subgroups in a ﬁnite group, Proc. Edinburgh Math. Soc., Vol. 39, 37-42 (1996). [12] A. Carocca and R. Maier, Theorems of Kegel-Wielandt type, Proc. Group Bath/St. Andrews(1997). London Math. Soc. Lecture Notes Series 260, Vol. I, 195-201 (1999). [13] K. Doerk and T. Hawkes, Finite Soluble Groups, Walter De Gruyter, Berlin / New York, (1992). [14] R. Maier, A completeness property of certain formations, Bull. London Math. Soc., Vol. 24, 540-544 (1992).

ON NILPOTENT-LIKE FITTING FORMATIONS A. BALLESTER-BOLINCHES∗ , A. MART´INEZ-PASTOR† , M.C. PEDRAZA∗ 1 ´ AGUILERA† and M.D. PEREZ-RAMOS ∗

` Departament d’Algebra, Universitat de Val`encia, C/ Doctor Moliner 50, 46100 Burjassot (Val`encia), Spain † Escuela Universitaria de Inform´ atica, Departamento de Matem´ atica Aplicada, Universidad Polit´ecnica de Valencia, Camino de Vera, s/n, 46071 Valencia, Spain

1

Introduction

Only ﬁnite groups will be considered here. Although theories of formations and Fitting classes are quite independent generalizations of the classical theory of Sylow and Hall, many of their results have been motivated by the good behaviour of the Fitting formation of nilpotent groups as a class of groups. In the last years some diﬀerent extensions of nilpotent groups have been studied mainly within the framework of formation theory but also from the point of view of Fitting classes. This paper is a survey article containing a detailed account of recent works concerning these classes of nilpotent type. Let us start by looking at the class of nilpotent groups. The following characterizations of a ﬁnite nilpotent group are well-known: G is a nilpotent group if and only if G satisﬁes one of the following conditions: (1) G is the direct product of its Sylow subgroups. (2) G has a normal p-complement, for every prime p. (3) (Frobenius p-nilpotence criterion) For every prime p and every p-subgroup P of G one of the following equivalent conditions holds: (i) NG (P )/CG (P ) is a p-group. (ii) NG (P ) is p-nilpotent. We are going to see how these characterizations can be generalized in diﬀerent ways to obtain classes F extending the class of nilpotent groups. These classes have a common property: They are all subgroup-closed saturated formations which are locally deﬁned by Sπ(p) , the class of all π(p)-groups for a set of primes π(p), for each prime p and p ∈ π(p). Diﬀerent restrictions on the sets of primes π(p) will provide several extensions of the class of nilpotent groups which arise from a wide variety of interesting questions in the theory of classes of groups. We will consider only ﬁnite soluble groups, although some of the mentioned results can be extended to ﬁnite groups in general. We refer to Doerk and Hawkes’ book [12] for the basic notation, terminology and results about Fitting classes and formations. 1 This research has been supported by Proyecto PB 97-0674-C02-02 of DGICYT, Ministerio de Educaci´ on y Ciencia, Spain.

BALLESTER-BOLINCHES et al.

32

2

Lattice formations

Our ﬁrst extension comes from the study of the lattice properties of some special subgroups of a group related to saturated formations. It is well known that the set of all subnormal subgroups of a ﬁnite group G is a lattice. On the other hand, it is abundantly clear that in the ﬁnite soluble universe, and within the framework of formation theory, the natural extension of the subnormal embedding is the Fsubnormal embedding, F a saturated formation. Therefore, the natural question arising from the above fact is the following: Which are the subgroup-closed saturated formations F for which the set of all F-subnormal subgroups is a lattice in every group G? We recall here the concept of F-subnormal subgroup. For a saturated formation F, a maximal subgroup M of a group G is said to be F-normal in G if G/CoreG (M ) ∈ F. A subgroup H of G is called F-subnormal in G if either H = G, or there exists a chain H = H0 ≤ H1 ≤ · · · ≤ Hn = G such that Hi is an F-normal maximal subgroup of Hi+1 for 0 ≤ i < n. It is clear that if F = N , the saturated formation of nilpotent groups, then the F- subnormal subgroups of G are exactly the subnormal subgroups of G. The answer to the above question was obtained in 1992 by Ballester-Bolinches, Doerk and P´erez-Ramos [4]: Theorem 1 Let F be a subgroup-closed saturated formation containing N . Then the following statements are pairwise equivalent: (1) The set of all F-subnormal subgroups is a lattice for every group G. (2) Let F be the full and integrated local formation function defining F. Then there exists a partition {πi }i∈I of the set of all prime numbers P such that F (p) = Sπi for every prime number p ∈ πi and for every i ∈ I. (3) There exists a partition {πi }i∈I of the set of all prime numbers P such that a group G is an F-group if and only if G is the direct product of its Hall πi subgroups, for i ∈ I. We refer to these classes as lattice formations. Notice that they are extensions of the class of nilpotent groups linked to the ﬁrst characterization mentioned. These formations can be also characterized through the behaviour of the F-residual with respect to F-subnormal subgroups [9]: Theorem 2 Let F be a subgroup-closed saturated formation containing N . Then the following statements are equivalent: (1) F is a lattice formation. (2) If H and K are two F-subnormal subgroups of a group G, then H, K F = H F , K F . Related results to Theorem 1 and Theorem 2, even in the ﬁnite universe, were obtained by Vasilev, Kamornikov and Semenchuk [22] and by Kamornikov [15], respectively.

ON NILPOTENT-LIKE FITTING FORMATIONS

33

Lattice formations are also Fitting classes. In fact, these classes were also studied by Lockett [17] from this point of view. He proved that they are dominant Fitting classes and obtained the exact description of the associated injectors. So, in this framework, it is natural to ask about the behaviour of the F-radical and the Finjectors with respect to the F-subnormal subgroups of a group. This idea provides also interesting characterizations of the lattice formations in terms of Fitting type properties [8]: Theorem 3 Let F be an sn -closed saturated formation containing N . Then the following statements are pairwise equivalent: (1) F is a lattice formation. (2) F is a Fitting class satisfying that if G is a group, V is an F-injector of G and H is an F-subnormal subgroup of G, then V ∩ H is an F-injector of H. (3) F is a Fitting class such that the F-radical GF of a group G can be described in the following way: GF = X ∈ F | X is an F-subnormal subgroup in G . (4) If H and K are two F-subnormal F-subgroups of a group G, then H, K ∈ F. (5) F is a Fitting class and if H is an F-subnormal F-subgroup of a group G, then H, H g ∈ F for every g ∈ G. Lattice formations have been also involved recently in the study of F-normality associated to saturated formations. Originally the notion of F-normality was restricted to maximal subgroups. A ﬁrst attempt to give a deﬁnition valid for arbitrary subgroups was made in 1995 by Ballester-Bolinches, Doerk and P´erez-Ramos [5]. Although this deﬁnition has a quite good behaviour in many aspects, the results regarding the lattice properties of F-normal subgroups diﬀer from the corresponding ones for F-subnormal subgroups. So, later on, an alternative deﬁnition of F-subnormality suggested by K. Doerk and so called F-Dnormality has been studied by Arroyo-Jord´ a and P´erez-Ramos in [2]. This new deﬁnition satisﬁes again all the desired properties. Moreover, in this case, lattice formations turn out to be again the subgroup-closed saturated formations for which the set of all FDnormal subgroups is a lattice i! n every group G. Besides, the analogous results to Theorem 1 and Theorem 2 remain valid for F-Dnormal subgroups in place of F-subnormal ones. Finally, we deal with lattice formations in the context of factorized groups, that is, groups of the form G = AB where A and B are subgroups of G. In the study of such groups it is interesting to know relevant subgroups which inherit the factorization, in the following sense: If G = AB is the product of two subgroups A and B, a subgroup S is factorized if S = (A ∩ S)(B ∩ S) and A ∩ B ≤ S. In this respect, it is well-known that the Fitting subgroup of a group which is the product of two nilpotent subgroups is a factorized group (see [1]). The natural question is now for which subgroup-closed Fitting formations F the F-radical of a

34

BALLESTER-BOLINCHES et al.

group G which is the product of two F-groups inherits the factorization. The next result which appears in [8] shows that this property also characterizes the aforesaid subgroup-closed Fitting formations. Theorem 4 Let F be a subgroup-closed Fitting formation containing N . Then the following statements are equivalent: (1) F is a lattice-formation. (2) F satisfies the property: If G = AB and A and B are F-groups, then the F-radical GF of G is a factorized subgroup, that is, GF = (GF ∩ A)(GF ∩ B) and A ∩ B ≤ GF .

3

Formations with the Shemetkov property

Now we will look at other questions in the framework of formation theory. First of all, notice that, in view of property (4) of Theorem 3, it is natural to wonder which are the subgroup-closed saturated formations F closed under taking products of F-subnormal subgroups. Of course, lattice formations satisfy this property, but so does the class of p-nilpotent groups, for every prime p. On the other hand, in ﬁnite groups it is very usual to argue by minimal counterexample. Therefore it is of interest to know the structure of F-critical groups with respect to some class F. For a class of groups F, a group G is said to be minimal non-F-group or simply F-critical, if G is not in F but all proper subgroups of G are in F. In particular, when F = N is the class of all nilpotent groups, N -minimal groups are the well known Schmidt groups. In the Kourovka Notebook [16] Shemetkov proposed the following question: Which are the saturated formations F satisfying that every F-critical group is either a Schmidt group or a cyclic group of prime order? We say that a saturated formation F has the Shemetkov property if every F-critical group is either a Schmidt group or a cyclic group of prime order. The above questions provide a diﬀerent extension of the class of nilpotent groups. The groups in this extension can be caracterized through an analogue to the Frobenius p-nilpotence criterion. More concretely we have: Theorem 5 Let F be a subgroup-closed saturated formation containing N . Then the following statements are pairwise equivalent: (1) F satisfies: If H and K are two F-subnormal F-subgroups of a group G and G = HK, then G ∈ F. (2)F has the Shemetkov property. (3) For each prime number p, there exists a set of primes π(p), with p ∈ π(p), such that F is locally defined by the formation function f (p) = Sπ(p) . (4) If F is the integrated and full formation function defining F, F (p) = Sπ(p) ∩F where π(p) = charF (p). (5) G is an F-group if and only if for each p-subgroup P of G and each prime p one of the following equivalent conditions holds:

ON NILPOTENT-LIKE FITTING FORMATIONS

35

(i) NG (P )/CG (P ) ∈ Sπ(p) . (ii) NG (P ) ∈ Sp Sπ(p) . The equivalence between (1), (3) and (4) has been obtained in [3]. On the other hand, we want to point out that the equivalence between (2), (3) and (5) have been also analyzed in the ﬁnite universe and without the condition N ⊆ F in [10] and [11]. Moreover, some authors of the Gomel School have also studied related problems (see [18], [19] and [14]). We are going to see how this family of saturated formations appears again in the context of factorized groups. When studying factorized groups, one often has to consider triply factorized groups, that is, groups of the form G = AB = AC = BC, where A, B and C are subgroups of G. In this respect, one of the most signiﬁcant results about ﬁnite factorized groups is Kegel’s Theorem [13] which states that if G is a triply factorized group G = AB = AC = BC, then G is nilpotent (supersoluble) whenever A, B and C are nilpotent (supersoluble). So the following question can be formulated: Which are the subnormal subgroup closed saturated formations containing N which are closed under triply factorized groups? A particular case of this question is when C is an F-subnormal subgroup of G. The answer to both questions leads to the family of saturated formations of nilpotent type into consideration (see [6]). Theorem 6 Let F be an sn -closed saturated formation containing N . Then the following statements are equivalent: (1) F satisfies the property: If G is a group of the form G = AB = AC = BC, where A, B and C are F-subgroups of G, then G is an F-group. (2) F satisfies the property: If G is a group of the form G = AB = AC = BC, where A and B are Fsubgroups of G and C is an F-subnormal F-subgroup of G, then G is an F-group. (3) F is a subgroup-closed saturated formation such that if F is the integrated and full formation function defining F, then for each prime number p ∈ P, F (p) = Sπ(p) ∩ F where π(p) = charF (p). We remark that Vasilev also studied related problems in [20] and [21]. In the study of factorized groups, the case of a triply factorized group G = AB = AC = BC where C is a normal subgroup of G plays a special role. For instance, the factorizer of a normal subgroup of a factorized group always has this form. So, the following characterization of the above formations is also of interest [6]. Theorem 7 Let F be an sn -closed saturated formation containing N . Then the following statements are equivalent: (1) F satisfies the property: If G is a group of the form G = AB = AC = BC, where A and B are Fsubgroups of G and C is normal subgroup of G, then GF = C F . (2) F is a subgroup-closed saturated formation such that, if F is the integrated and full formation function defining F, then for each prime number p ∈ P, F (p) = Sπ(p) ∩ F where π(p) = charF (p).

BALLESTER-BOLINCHES et al.

36

4

Extensions of p-nilpotent groups

In this section we meet extensions of the class of nilpotent groups through the existence of normal complements. Our starting point is the following question which is related both to property (2) of Theorem 2 and property (1) of Theorem 5: Which are the saturated formations F satisfying: If H and K are two F-subnormal subgroups of a group G and G = HK, then GF = H F K F ? Theorem 8 Let F be a subgroup-closed saturated formation containing N . The following statements are pairwise equivalent: (1) F satisfies the property: If H and K are two F-subnormal subgroups of a group G and G = HK, then GF = H F K F . (2) For each prime number p, there exists a set of primes π(p) with p ∈ π(p) such that F is locally defined by the formation function f given by f (p) = Sπ(p) . These sets of primes satisfy the following property: If q ∈ π(p), then π(q) ⊆ π(p) for every pair of prime numbers p, q. (3) A group G is an F-group if and only if G has a normal π(p)-complement for every prime p, where π(p) is a set of primes with p ∈ π(p). Clearly, the saturated formations described in this theorem are extensions of the formations of all nilpotent and p-nilpotent groups (p a prime). Again, this family of classes appears when considering some interesting questions in the framework of factorized and trifactorized groups. In particular, we consider an extension of the properties studied in Theorem 6 and 7. More concretely, in [6] it is proved: Theorem 9 Let F be an sn -closed saturated formation containing N . Then the following statements are equivalent: (1) F satisfies the property: If G is a group of the form G = AB = AC = BC, where A and B are Fsubgroups of G and C is an F-subnormal subgroup of G, then GF = C F . (2) For each prime number p, there exists a set of primes π(p) with p ∈ π(p) such that F is locally defined by the formation function f given by f (p) = Sπ(p) . These sets of primes satisfy the following property: If q ∈ π(p), then π(q) ⊆ π(p) for every pair of prime numbers p, q.

5

Dominant Fitting classes

The common property enjoyed by all the formations considered in this paper is that they are saturated formations F = LF (f ) locally deﬁned by f (p) = Sπ(p) , for a set of primes π(p) such that p ∈ π(p), for every prime p. As we have seen, diﬀerent restrictions on the sets of primes π(p) give diﬀerent extensions of the class N of nilpotent groups. Now we want to draw attention to the fact that they are also Fitting classes. From this point of view, one would hope them to be dominant, like the class of nilpotent groups. Recall that a Fitting class F is dominant if in each group G

ON NILPOTENT-LIKE FITTING FORMATIONS

37

the F-maximal subgroups containing the F-radical GF of G are conjugate. In this case, the F-injectors of G are exactly the F-maximal subgroups of G containing GF . However, if F is a class as above, F is not in general a dominant Fitting class (even if we consider classes as in section 4). So the next step is to look for characterization theorems. In [7] we provide a characterization for these classes to be dominant. Again, this characterization rest on the appropriate restrictions for the sets of primes π(p). This leads to a nice connection between a property of Fitting type like dominance and the description of the canonical local deﬁnition as saturated formations. Theorem 10 Let F be a saturated formation containing N which is locally defined by the formation function f given by f (p) = Sπ(p) , where π(p) is a set of primes with p ∈ π(p), for every prime p. The following statements are equivalent: (i) F is a dominant Fitting class, (ii) If q ∈ π(p), then either π(q) = π(p) or π(q) = P or π(p) = P, for each prime p. In fact, to prove this theorem, the exact description of the F-maximal subgroups of a group G containing the F-radical GF is obtained for these classes. This construction rests strongly on a well-known result of Lockett [17] about the permutability of normally embedded subgroups of a group into which a given Hall system of the group reduces. Construction. For each prime s, consider the partition of π: π = {pi | i ∈ I} ∪ {pj |j ∈ J} with s ∈ ∪i∈I π(pi ) but s ∈ ∩j∈J π(pj ). For a group G, denote CG (Oπ r (G)/Oπ (G))(if I = ∅, Zs = G). Zs : = r∈{pi |i∈I}

Take Σ a Hall system of the group G. For every prime s, consider Hs = Gs ∩ Zs , for Gs ∈ Syls (G), Gs ∈ Σ. Consider: Hs HΣ := s∈π(G)

Theorem 11 Let F be as in Theorem 9. The set of F-maximal subgroups of G containing GF is {HΣ | Σ is a Hall system of G} and this is a conjugacy class of subgroups of G. It is interesting to point out that classical constructions of nilpotent injectors, Lockett injectors, as well as p-nilpotent injectors (p a prime), appear as particular cases of the above construction.

BALLESTER-BOLINCHES et al.

38 References

[1] B. Amberg, S. Franciosi and F. de Giovanni, Products of Groups, Clarendon PressOxford, (1992). [2] M. Arroyo-Jord´ a and M. D. P´erez-Ramos, On the lattice of F-Dnormal subgroups in ﬁnite soluble groups, to appear in J. Algebra. [3] A. Ballester-Bolinches, A note on saturated formations, Arch. Math. 58 (1992), 110-113. [4] A.Ballester-Bolinches, K. Doerk and M.D. P´erez-Ramos, On the lattice of Fsubnormal subgroups, J. Algebra 148 (1992), 42-52. [5] A. Ballester-Bolinches, K. Doerk and M.D. P´erez-Ramos, On F-normal subgroups of ﬁnite soluble groups, J. Algebra 171 (1995), 189-203. [6] A. Ballester-Bolinches, A. Mart´ınez-Pastor and M.C. Pedraza-Aguilera, Finite trifactorized groups and formations, J. Algebra, 226 (2000), 990-1000. [7] A. Ballester-Bolinches, A. Mart´ınez-Pastor and M. D. P´erez-Ramos, A family of dominant Fitting classes of ﬁnite soluble groups, J. Austral. Math. Soc. (A) 64 (1998), 33-43. [8] A. Ballester-Bolinches, A. Mart´ınez-Pastor and M. D. P´erez-Ramos, Nilpotent-like Fitting formations of ﬁnite soluble groups, Bull. Austr. Math. Soc. 62 (2000), 427-433. [9] A. Ballester-Bolinches, M.C. Pedraza-Aguilera and M.D. P´erez-Ramos, On F- subnormal subgroups and F-residuals of ﬁnite soluble groups, J. Algebra. 186 (1996), 314-322. [10] A. Ballester-Bolinches and M.D. P´erez-Ramos, On F-critical groups, J. Algebra 174 (1995), 948–958. [11] A. Ballester-Bolinches and M.D. P´erez-Ramos, Two questions on L.A. Shemetkov on critical groups, J. Algebra 179 (1996), 905–917. [12] K. Doerk and T.Hawkes, Finite soluble groups, Walter De Gruyter, Berlin-New York, (1992). [13] O. H. Kegel, Zur Struktur mehrfach faktorisierter endlicher Gruppen, Math. Z. 87 (1965), 42-48. [14] S.F. Kamornikov , On two problems of L.A. Shemtekov, Siberian Math. J. 35, no. 4, (1994), 713-721. [15] S.F. Kamornikov, Permutability of subgroups and F-subnormality, Siberian Math. J. 37 (1996), 936–949. [16] “The Korouvka Notebook”, Unsolved problems in Group Theory, Novosibirsk, (1992). [17] F.P. Lockett, “On the Theory of Fitting Classes of Finite Soluble groups”, Ph.D. Thesis, University of Warwick, 1971. [18] L.A. Shemetkov and A.N. Skiba “Formations of algebraic systems” Moscow, 1989. [19] A. N. Skiba, On a class of local formations of ﬁnite groups, Dokl. Akad. Nauk BSSR 34 , no. 11, (1990), 982–985. [20] A.F. Vasilev, On the problem of the enumeration of local formations with a given property, in “Problems in algebra”, no. 3, 3–11, Univ. Press, Minsk, 1987. [21] A.F. Vasilev, The maximal hereditary subformation of a local formation, in “Problems in algebra”, no. 5, 39–45 Univ. Press, Minsk, 1990. [22] A.F. Vasilev, S.F. Kamornikov and V.N. Semenchuk, On lattices of subgroups of ﬁnite groups, in “Inﬁnite groups and related algebraic structures”, 27–54, Kiev, 1993.

LOCALLY FINITE GROUPS WITH MIN-p FOR ALL PRIMES p A. BALLESTER-BOLINCHES

∗

and TATIANA PEDRAZA

∗ 1

∗

` Departament d’Algebra, Universitat de Val`encia, C/ Doctor Moliner 50, 46100 Burjassot (Val`encia), Spain

It is well-known that there are numerous properties of ﬁnite groups which are equivalent to nilpotence. For instance, subnormality of each subgroup, normality of all Sylow subgroups, centrality of every chief factor and normality of all maximal subgroups. If attention is restricted to locally ﬁnite-soluble groups, the ﬁrst three properties are suﬃcient to ensure local nilpotence and the last three properties are enjoyed by each locally nilpotent group. It is also well-known that, for ﬁnite groups G, the conditions G ≤ φ(G) and G/φ(G) nilpotent are both equivalent to nilpotence. Taking into account that the Frattini subgroup φ(G) of a group G is deﬁned as the intersection of G with all its maximal subgroups, it is rather clear that the condition G ≤ φ(G) is for inﬁnite groups, even for locally ﬁnite groups, a weak property because an inﬁnite group can have insuﬃcient maximal subgroups or even none at all. In 1975, Tomkinson introduced a characteristic subgroup µ(G) with properties similar to those of the Frattini subgroup φ(G) of a ﬁnite group. Our ﬁrst objective in this work is to use this Frattini-like subgroup µ(G) to obtain a complete characterisation of a class of generalised nilpotent groups in the universe cL¯ of all radical locally ﬁnite groups with min-p for all primes p. We recall the deﬁnition of µ(G) (see [19]). If U is a proper subgroup of a group G then m(U ) is the least upper bound of the types of all properly ascending chains from U to G. Clearly m(U ) = 1 if and only if U is a maximal subgroup of G. A proper subgroup M of G is called a major subgroup of G if m(U ) = m(M ) whenever M ≤ U < G. The subgroup µ(G) is then the intersection of all major subgroups of G. In his paper, Tomkinson shows that every proper subgroup of a group G is contained in a major subgroup of G and then µ(G) is always a proper subgroup of ¯ G. Let G be a cL-group and consider H and K two normal subgroups of G such that K is contained in H. Then H/K is said to be a δ-chief factor of G if H/K is either a minimal normal subgroup of G/K or a divisibly irreducible ZG-module, that is, H/K has no proper inﬁniteG-invariant subgroups. ¯ Let B be the class of all cL-groups in which every proper subgroup has a proper normal closure. This is a class of generalised nilpotent groups in the universe ¯ cL¯ because every nilpotent cL-group is in B and every ﬁniteB-group is nilpotent. ¯ Moreover, this class contains the class of all cL-groups for which every subgroup is descendant. Now we can establish the ﬁrst of our main results. It shows that in ¯ B-groups are to inﬁnite groups as nilpotent groups to ﬁnite groups. the class cL, Moreover, we obtain a complete characterisation of the B-groups G, through the Frattini-like subgroup µ(G), analogously to the ﬁnite one for nilpotent groups and the Frattini subgroup. ¯ The following Theorem 1 [5, Theorem 1] Let G be a group in the class cL. 1

This work is supported by Proyecto PB97-0674-C02-02 of DGICYT, MEC, Spain

40

BALLESTER-BOLINCHES, PEDRAZA

statements are pairwise equivalent: (i) G is a B-group. (ii) G/µ(G) is a B-group. (iii) G ≤ µ(G). (iv) Every major subgroup of G is a normal subgroup of G. (v) G is a direct product of nilpotent Sylow subgroups. (vi) G is locally nilpotent and the radicable part of G is central. (vii) Every δ-chief factor of G is central. Let G be the split extension of a quasicyclic 2-group by its involution. Then G is a locally nilpotent group, in fact it is hypercentral, but it is not a B-group. Consider the set {pi }i1 of all prime numbers in their natural order. Let Gi be the split extension of the cyclic group xi of order pii by its automorphism yi of order pi which maps xi toxipi +1 . Then Gi is nilpotent of class i. Let G = Dr∞ i=1 Gi , then G is a B-group which is not nilpotent (see [19]). These examples show that the class ¯ B is intermediate between the classes of nilpotent cL-groups and locally nilpotent ¯ cL-groups. It is known that the product of two normal nilpotent subgroups is nilpotent - this is the Fitting’s theorem [16, Theorem 5.2.8]. The corresponding statement holds for locally nilpotent groups and is of great importance. Moreover in any group G there is a unique maximal normal locally nilpotent subgroup (called the Hirsch-Plotkin radical) containing all normal locally nilpotent subgroups of G (see[16, (12.1.3)]). We obtain analogous results for the class B by deﬁning the corresponding radical subgroup associated to this class. Theorem 2 [5, Theorem 5] Every group G ∈ cL¯ has a unique largest normal B-subgroup, denoted by δ(G). In fact δ(G) = F (G), the Fitting subgroup of G. It is known that in general the Fitting subgroup in an inﬁnite group gives little information about the structure of the group. However in this case it plays an important role as it inherits the properties of the B-radical. A well-known result of ﬁnite groups is that the Fitting subgroup of a group G is the intersection of the centralisers of all chief factors of G (see[11, (A.13.8)]). This result is also true for the Hirsch-Plotkin radical of a periodic locally soluble group (see [10, (1.3.5)],[10, (6.2.4)]). As one might expect, there is an important connection between δ(G) and the centralisers of theδ-chief factors as the following theorem shows. ¯ Theorem 3 [5, Theorem 7] Suppose that G is a cL-group. Then F (G) is the intersection of the centralisers of all δ-chief factors of G.

LOCALLY FINITE GROUPS WITH MIN-p

41

A famous theorem of Kegel and Wielandt ([1, (2.4.3)]) states the solubility of every ﬁnite group G = AB which is the product of two nilpotent subgroups A and B. Such groups have been widely studied over the past 50 years by several authors (see[1, Ch.2]). On the other hand, very little is known about the structure of a locally ﬁnite product of two nilpotent groups. However, some extensions of these results from ﬁnite to locally ﬁnite groups have been obtained in [12], replacing nilpotence by local nilpotence. Our second purpose in this work is to investigate the structure of a radical locally ﬁnite group with min-p for all p, G = AB, factorised by two B-subgroups A and B. We extend some well-known results of products of ﬁnite nilpotent groups. A group G is said to be metanilpotent, or G ∈ N2 , if there exists a nilpotent normal subgroup N of G such that the factor group G/N is nilpotent. There exist examples, in the ﬁnite universe, showing that a product of two nilpotent groups is not metanilpotent in general. However, in [15, Theorem 1], Maier shows that if the factor subgroups are modular ﬁnite groups, the result holds. Analogously, we say that a group G is locally nilpotent-by-locally nilpotent, or G ∈ LN2 , if there exists a locally nilpotent normal subgroup N of G such that G/N is locally nilpotent. Using this concept, we obtain a natural generalisation of Maier’s theorem in our universe. ¯ Theorem 4 [7, Theorem 1] Let G = AB be a cL-group, where A and B are 2 modular locally nilpotent groups. Then G ∈ LN . Some authors have been interested in the study of factorised groups whose subgroup factors are connected by certain permutability properties. In fact, the following question can be formulated. Let the group G = AB be the product of subgroups A and B which lie in a class of groups F. What is a relationship between the factors A andB -weaker than their elementwise permutability in the case of a direct productwhich guarantee G ∈ F?. In this direction, Asaad and Shaalan have obtained signiﬁcative results on the products of ﬁnite supersoluble groups which are linked by the following permutability requirements ([2]): A group G is the totally permutable product of the subgroups A and B if G = AB and every subgroup of A permutes with every subgroup of B. A group G is the mutually permutable product of the subgroups A and B if G = AB and A permutes with every subgroup of B and B permutes with every subgroup of A. Some weaker versions of these permutable products due to Carocca ([9]) and to Beidleman, Heineken, Galoppo and Manfredino ([8]) have been fruitful in studying the structure of a product of two soluble groups. Following this line of thought and concerning radical locally ﬁnite groups with min-p for all p, we obtain a structural result, analogous to Theorem 4, using the following permutability concept: A group G = AB is the mutually der-permutable product of the subgroups A and B if A permutes with every term of the derived series of B and B permutes with every term of the derived series of A. We say that a group G ∈ B2 if there exists a normal B-subgroup N of G such that G/N is a B-group.

42

BALLESTER-BOLINCHES, PEDRAZA

¯ Theorem 5 [7, Theorem 2] Let G be a cL-group. If G = AB is the mutually der-permutable product of the B-subgroups A and B. Then G ∈ B2 . A classical (and considerably hard) topic in the study of a product of two nilpotent ﬁnite groups has been trying to obtain bounds for their derived length. In 1973, Pennington shows that there are strong restrictions on the (c + d)th term of the derived series of a ﬁnite group G = AB where c and d are the classes of the nilpotent subgroups A and B, respectively (see[1, (2.5.3)]). This result has been generalised by Franciosi, DeGiovanni and Sysak to periodic radical groups ([12, Theorem D]). Now we obtain a result of this kind, using the derived lengths of the subgroups A and B instead of their classes, under some restrictions on the permutability of their derived series. Let us denote by dG the derived length of a soluble group G. ¯ Theorem 6 [7, Theorem 3] Let the cL-group G = AB be the mutually der-permutable product of two soluble B-subgroups A and B with derived lengths dA and dB , respectively. Then the (dA + dB )th term G(dA +dB ) of the derived series of G is a π-group in the class B, whereπ = π(A) ∩ π(B). References [1] B. Amberg, S. Franciosi and F. de Giovanni, “Products of Groups” (Clarendon Press, Oxford, 1992). [2] M. Asaad and A. Shaalan, “On the supersolubility of finite groups”, Arch. Math., 53 (1989), no.4, 318-326. [3] A. Ballester-Bolinches and S. Camp-Mora, “A Gasch¨ utz-Lubeseder Type Theorem in a Class of Locally Finite Groups”, J. Algebra, 221 (1999), 562-569. [4] A. Ballester-Bolinches and S. Camp-Mora, “A Bryce-Cossey type Theorem in a class of locally finite groups”, Bull. Austral. Math. Soc., 63 (2001),459-466. [5] A. Ballester-Bolinches and Tatiana Pedraza, “On a class of generalised nilpotent groups”, to appear in J. Algebra. [6] A. Ballester-Bolinches and Tatiana Pedraza, “The Fitting subgroup and some injectors of radical locally finite groups with min-p for all p”. (Preprint) [7] A. Ballester-Bolinches and Tatiana Pedraza, “On products of generalised nilpotent groups”. (Preprint) [8] J. C. Beidleman, A. Galoppo, H. Heineken and M. Manfredino, “On certain products of soluble groups”, Forum. Math., 13 (2001), no.4, 569-580. [9] A. Carocca, “On factorized finitegroups in which certain subgroups of the factors permute”, Arch. Math., 71 (1998), no. 4, 257-261. [10] M.R. Dixon, “Sylow Theory, Formations and Fitting classes in Locally Finite Groups”, World Scientiﬁc (Series in Algebravol 2), Singapore-New Jersey-London-Hong Kong, 1994. [11] K.Doerk and T. O. Hawkes, “Finite Soluble Groups”, Walter De Gruyter, Berlin-New York, 1992. [12] S. Franciosi,F. de Giovanni and Ya. P. Sysak, “On locally finite groupsfactorized by locally nilpotent subgroups”, J. Pure Appl.Algebra, 106 (1996), 45-56. [13] Burkhard H¨ oﬂing,“Subgroups of locally finite products of locally nilpotent groups”, Glasgow Math. J., 41 (1999), no. 3,323-343. [14] O. H. Kegel, “Zur Struktur mehrfachfaktorisierter endlicher Gruppen”, Math. Z., 87(1965), 42-48.

LOCALLY FINITE GROUPS WITH MIN-p

43

[15] R. Maier, “Endliche metanilpotenteGruppen”, Arch. Math., 23 (1972), 139-144. [16] D. J. S. Robinson, “A course in the theory of groups”, Springer-Verlag, 1982. [17] D. J. S. Robinson, “Finiteness conditions and generalized soluble groups” (vol 1 and vol 2), Springer-Verlag, 1972. [18] R. Schmidt, “Subgroup Lattices of Groups”, De Gruyter Expositions in Mathematics,14. Berlin, 1996. [19] M. J. Tomkinson, “A Frattini-like subgroup”, Math. Proc. Cambridge Phil. Soc.,77 (1975), 247-257. [20] M. J. Tomkinson, “Finiteness conditions and a Frattini-like subgroup”, Supp. Rend. Mat. Palermo, serie II, 23 (1990), 321-335. [21] M. J. Tomkinson, “Schunck classes and Projectors in a class of locally finite groups”, Proc. Edinburgh Math.Soc., 38 (1995), 511-522.

QUASI-PERMUTATION REPRESENTATIONS OF 2-GROUPS OF CLASS 2 WITH CYCLIC CENTRE HOUSHANG BEHRAVESH1 Department of Mathematics, University of Urmia, Urmia, Iran Email: [email protected]

Dedicated to the memory of Brian Hartley

Abstract In [HB1] we deﬁned c(G), q(G) and p(G) for ﬁnite groups. In this paper we will calculate these quantities for a p-group of class 2 with cyclic centre, where p is either an odd prime or 2.

1

Introduction

By a quasi-permutation matrix we mean a square matrix over the complex ﬁeld C with non-negative integral trace. Thus every permutation matrix over C is a quasi-permutation matrix. For a given ﬁnite group G, let p(G) denote the minimal degree of a faithful permutation representation of G (or of a faithful representation of G by permutation matrices), let q(G) denote the minimal degree of a faithful representation of G by quasi-permutation matrices over the rational ﬁeld Q, and let c(G) be the minimal degree of a faithful representation of G by complex quasipermutation matrices. See [BGHS]. It is easy to see that, when G is a ﬁnite group, then c(G) ≤ q(G) ≤ p(G). Now I would like to state a problem from Prof. Brian Hartley (1992-94). Problem : Let G be a ﬁnite p-group. Find G such that c(G) = q(G) = p(G). In fact it easy to prove that, when p is an odd prime, then c(G) = q(G). So in this case a good question to be asked is: For p be an odd prime, when is q(G) = p(G)? Now let p = 2. In [HB2] we showed that, when G is a generalized quaternion group then 2c(G) = q(G) = p(G). So in this case a good question to be asked is: 1 Institute for Studies in Theoretical Physics and Mathematics, P.O.Box 19395-5746, Tehran, Iran

QUASI-PERMUTATION REPRESENTATIONS OF 2-GROUPS OF CLASS 2 45 For p = 2, when is c(G) < q(G) < p(G)? In this paper we will show that when G is a p-group of class 2 with cyclic centre then c(G) = |Z(G)| |G : Z(G)|1/2 . Moreover

q(G) = p(G) =

2

c(G) 2c(G)

G has no Q8 section otherwise.

Calculating p(G), c(G) and q(G)

Lemma 2.1 Let G be a ﬁnite p-group of nilpotency class 2. Let Z(G) be cyclic. Then there exists a faithful irreducible character χ, and for all such χ we have χ(1)2 = |G : Z(G)| and χ(g) = 0 for all g ∈ G\Z(G). Furthermore |G| = p2r+s , where |Z(G)| = ps and χ(1) = pr . Proof : See [HB1], Lemma 4.3]. Note : Let χ be an irreducible character of G. Then by mQ (χ) we mean the Schur index of χ over the Q. Theorem 2.2 Let G be a p-group and χ ∈ Irr(G) such that mQ (χ) = 2. Then G contains a generalized quaternion section. More speciﬁcally there is an irreducible character ξ on a subgroup K of G for which ξ G = χ, Q(ξ) = Q(χ) is a subﬁeld of the real ﬁeld, and the image of K under a representation aﬀording ξ is a generalized quaternion group. Proof : See [F] Theorem 2. Remark (1): Let G be a 2-group of class 2. Let G have a generalized quaternion section, that is, let H K ≤ G and K/H ∼ = Q2n+1 , where n ≥ 2 and n

Q2n+1 =< a, b : a2 = 1, a2

n−1

= b2 , bab−1 = a−1 > .

Then it is easy to prove that n = 2. In other words G only can have Q8 as a section. So by Theorem 2.2, when G is a p-group of class 2 and has no Q8 section, for all non-linear irreducible characters of G say χ, we have mQ (χ) = 1. Lemma 2.3 Let G be a ﬁnite group. If the Schur index of each non-principal irreducible character is equal to m, then q(G) = mc(G). Proof : See [HB1], Corollary 3.15.

46

BEHRAVESH

Corollary 2.4 Let G be a ﬁnite p-group and G have no section isomorphic to Q8 . Then mQ (χ) = 1 for all χ ∈ Irr(G) and c(G) = q(G). Moreover when G is a ﬁnite p-group of class 2 with cyclic centre then, c(G) G has no Q8 section q(G) = 2c(G) otherwise. Proof : This follows from [I], Corollary 10.14, Lemma 2.3 and Remark (1). Theorem 2.5 Let G be a ﬁnite p-group with a unique minimal normal subgroup. Then there exists a faithful irreducible character χ. Suppose that all faithful irreducible characters of G have degree χ(1) and χ2 (1) = |G : Z(G)|. Then c(G) = χ(1) |Z(G)| = |Z(G)| |G : Z(G)|1/2 . Proof : See [HB1], Theorem 4.6. Lemma 2.6 Let G be a ﬁnite p-group of class 2 and let Z(G) be cyclic and |Z(G)| = ps for some s. Then |G| = p2r+s for some r ≥ 0, and c(G) = pr+s . Proof : See [HB1], Lemma 4.7]. Note : Let H ≤ G. Then HG =

H g is called the core of H in G.

g∈G

Lemma 2.7 Let G be a ﬁnite group with a unique minimal normal subgroup. Then p(G) is the smallest index of a subgroup with trivial core (that is, containing no non-trivial normal subgroup). Proof : See [HB1], Corollary 2.4. Lemma 2.8 Let G be a p-group and H ≤ G. Then HG = 1 if and only if Z(G) H = 1. Furthermore if G has nilpotency class 2 and HG = 1 then H is an abelian group. Proof : See [HB1], Lemma 4.2. Lemma 2.9 Let G be a ﬁnite p-group of nilpotency class 2. Let Z(G) be cyclic and let B be an abelian subgroup of G and maximal subject to B Z(G) = 1. Let A = BZ(G) and C = CG (A). Let x ∈ C and xp ∈ B. Then x ∈ A. Proof : See [HB1], Lemma 4.8. Lemma 2.10 Let G, Z(G), A, B and C be as in Lemma 2.9. Then either C/B is cyclic or C/B ∼ = Q8 . Moreover when C/B is cyclic, then C is a maximal abelian subgroup of G. Proof : By Lemma 2.9 we know that the elements of order dividing p in C/B are in A/B ∼ = Z(G) and Z(G) is cyclic, so these elements constitute a subgroup of order p. Hence by [R], 5.3.6 C/B is either cyclic or a generalized quaternion group. But when C/B is a generalized quaternion group, then by Remark (1), C/B ∼ = Q8 .

QUASI-PERMUTATION REPRESENTATIONS OF 2-GROUPS OF CLASS 2 47 Now let C/B be cyclic. As B ≤ Z(C) and C/B is cyclic, C is abelian. Let C1 be abelian and C ≤ C1 . Let y ∈ C1 \C. As [A, y] = 1 so y ∈ C. Therefore C is a maximal abelian subgroup of G. Lemma 2.11 Let G be a ﬁnite 2-group of nilpotency class 2. Let Z(G) be cyclic and let B be an abelian subgroup of G such that B Z(G) = 1. Let A = BZ(G) and C = CG (A). Let C/B ∼ = Q8 . Then |Z(G)| = 2. Proof : Since A ≤ C, then |C/B| ≥ |Z(G)|. Hence |Z(G)| = 2, 4 or 8. Also Z(G)B/B ≤ Z(C/B). So |Z(G)| = 2. Remark (2): Note that G in Lemma 2.11, is called an extraspecial 2-group. In this case G contains Q8 as a section. In Section 3 we will calculate c(G), q(G) and p(G) when G is an extraspecial 2-group. Lemma 2.12 Let G, A, B and C be as in Lemma 2.9 and let Z(G) be cyclic and |Z(G)| = ps for some s. Also let G have no Q8 section. Then |G| = p2r+s for some r ≥ 0 and |C| = pr+s . Moreover p(G) = pr+s . Proof : See [HB1], Lemma 4.11. Theorem 2.13 Let G be a ﬁnite p-group of class 2 and let Z(G) be cyclic. Then c(G) = |Z(G)| |G : Z(G)|1/2 . Moreover when G has no Q8 section, then c(G) = q(G) = p(G). Proof : This follows from Lemmas 2.6 and 2.12 and Corollary 2.4.

3

Quasi-permutation representations of an extraspecial 2-group

Let Q8 and D8 be as follows: Q8 =< a, b : a4 = 1, a2 = b2 , bab−1 = a−1 > D8 =< a, b : a4 = b2 = 1, bab−1 = a−1 > . It is easy to prove the results given in the following table. Also see [HB2] for c(G), q(G) and p(G). G Q8 D8

Z(G) < a2 > < a2 >

c(G) 4 4

q(G) = p(G) 8 4

Let G = D8 and H =< b >. Then HG = 1. Let us deﬁne Li by −1 Li =< ai , bi : a4i = b2i = 1, bi ai b−1 i = ai > .

Therefore Li ∼ = D8 .

48

BEHRAVESH

Lemma 3.1 Let G be an extraspecial 2-group. Then |G| = 22r+1 and one of the following two cases arises: (a) G is a central product of r copies of D8 or (b) G is a central product of r − 1 copies of D8 with a copy of Q8 . Proof : See [DH], Theorem 20.5. Lemma 3.2 Let G be an extraspecial 2-group. Then there exists a faithful irreducible character χ, and for all such χ χ(1) = |G : Z(G)|1/2 . Also c(G) = |Z(G)| |G : Z(G)|1/2 . Proof : See Theorem 2.5. Theorem 3.3 Let G be an extraspecial 2-group. Then c(G) if G is a central product of copies of D8 . q(G) = 2c(G) if G is a central product of copies of D8 with a copy of Q8 . So respectively q(G) = 2r+1 or 2r+2 . Proof : This follows from [HB1], Corollaries 3.14, 3.15, Theorem 2.2 and Lemma 3.2. Lemma 3.4 Let G be an extraspecial 2-group. Then (a) if G is a central product of D1 , . . . , Dr , then there exists a core-free subroup H of G, such that |H| = 2r ; (b) if G is a central product of D1 , . . . , Dr−1 and Q8 , then there exists a core-free subgroup H of G, such that |H| = 2r−1 . 2 Proof : (a) Let r G = D1 D2 . . . Dr . Since Z(Li ) =< ai > and by [DH], Proposition 19.8, Z(G) = i=1 Z(Li ) and also |Z(G)| = 2, it is easy to prove that

a21 = a22 = · · · = a2r . Now by counting the diﬀerent elements in G and also using induction on r we can deduce that b1 , . . . , br are diﬀerent elements of G. Let H = gp{b1 , . . . br }. Then it easy to prove that H is abelian and also |H| = 2r and HG = 1. So the result follows. (b) In this case let H = gp{b1 , . . . , br−1 }. The rest of the proof is the same as part (a). Theorem 3.5 Let G be an extraspecial 2-group. Then q(G) = p(G).

QUASI-PERMUTATION REPRESENTATIONS OF 2-GROUPS OF CLASS 2 49 Proof: Let r be as in Lemma 3.4. For r = 1 see the table. So let r ≥ 2. By Lemmas 2.7 and 3.4 we have 2r+1 = q(G) ≤ p(G) ≤ |G : H| = 2r+1 where G is a central product of r copies of D8 . Also 2r+2 = q(G) ≤ p(G) ≤ |G : H| = 2r+2 where G is a central product of r − 1 copies of D8 and a copy of Q8 . So the result follows. Corollary 3.6 Let G be a ﬁnite p-group of class 2 and let Z(G) be cyclic. Let B be an abelian subgroup of G which is maximal subject to B Z(G) = 1. Let A = BZ(G) and C = CG (A). Then c(G) = |Z(G)| |G : Z(G)|1/2 and also either (1) C/B is cyclic and c(G) = q(G) = p(G); or (2) C/B ∼ = Q8 , G is an extraspecial group and 2c(G) = q(G) = p(G). References [HB1] H. Behravesh, “Quasi-permutation representations of p-groups of class 2”, J. London Math. Soc. (2) 55 (1997) 251-260. [HB2] H. Behravesh, “Quasi-permutation representations of metacyclic 2-groups with cyclic center”, Bulletin of the Iranian Mathematical Society 24:1 (1998) 1-14. [BGHS] J. M. Burns, B. Goldsmith, B. Hartley, R. Sandling, “On quasi-permutation representations of ﬁnite groups”, Glasgow Math. J. 36 (1994) 301-308. [DH] K. Doerk, T. Hawkes, Finite soluble groups, de Gruyter, Berlin, 1992. [D] L. Dornhoﬀ, Group representation theory, part A, Dekker, New York, 1971. [F] Charles E. Ford, Characters of p-groups, Proceedings of the American Mathematical Society 101:4 (1987) 595-601. [G] D. Gorenstein Finite groups Harper and Row, New York, 1968. [I] I.M.Isaacs, Character theory of ﬁnite groups, Academic Press, New York, 1976. [R] D. J. S. Robinson, A course in the theory of groups Springer, New York, 1982.

GROUPS ACTING ON BORDERED KLEIN SURFACES WITH MAXIMAL SYMMETRY EMILIO BUJALANCE∗ , FRANCISCO JAVIER CIRRE∗ and PETER TURBEK†1 ∗

Departamento de Matem´ aticas Fundamentales, Facultad de Ciencias, UNED, Madrid 28040, Spain E-mail: [email protected] and [email protected] † Department of Mathematics, Computer Science and Statistics, Purdue University Calumet, Hammond, Indiana, 46323, U.S.A. Email: [email protected]

Abstract A ﬁnite group G is said to be an M∗ -group if it is the group of automorphisms of a bordered compact Klein surface with maximal symmetry. M∗ -groups play an analogous role for Klein surfaces as Hurwitz groups do for Riemann surfaces. In this survey we present a summary of results on M∗ -groups. We ﬁrst examine their properties and the known families of M∗ -groups. Then we study their structure to obtain new methods for constructing additional families. Finally we examine the relationship between Hurwitz groups, H∗ -groups and M∗ -groups.

1

Introduction

The study of Riemann and Klein surfaces with maximal automorphism groups has a long history. It is well known that a compact Riemann surface of genus g ≥ 2 admits at most 84(g − 1) automorphisms. Automorphism groups of Riemann surfaces with this maximal number of automorphisms are called Hurwitz groups. It is known that Hurwitz groups exist for inﬁnitely many values of g and also do not exist for inﬁnitely many g. The article by Conder [9] contains a nice survey of known results about Hurwitz groups. Corresponding problems concerning Klein surfaces have also received a good deal of attention and we present a summary of known results here. A Klein surface is the orbit space of a Riemann surface under the action of a symmetry, that is, an anticonformal automorphism of order two. The algebraic genus of the Klein surface is deﬁned to be the genus of its Riemann double cover. Throughout the paper, the letter p will be used exclusively to denote the algebraic genus of a Klein surface. Singerman [25] proved that the maximum number of automorphisms that a non-orientable Klein surface with empty boundary may admit is 84(p − 1). Following Singerman, we will call groups which attain this bound H∗ -groups. May [19] proved that a bordered Klein surface of algebraic genus p ≥ 2 1 The first author is partially supported by DGICYT PB98-0017 and the second author is partially supported by DGICYT PB98-0756.

GROUPS ACTING ON BORDERED KLEIN SURFACES

51

admits at most 12(p − 1) automorphisms. Groups isomorphic to the automorphism group of a bordered Klein surface with 12(p − 1) automorphisms are called M∗ -groups. Compared to Hurwitz groups, fewer things are known about M∗ -groups. In this survey we will ﬁrst examine the known families of M∗ -groups. Then we study the structure of an M∗ -group to obtain new methods for constructing additional families. Finally we examine the relationship between Hurwitz, H∗ and M∗ -groups. M∗ -groups are natural objects of study for combinatorial group theorists, however they are important from two other viewpoints. Since compact Klein surfaces correspond to real algebraic curves, M∗ -groups are the groups of birational transformations of real algebraic curves with maximal symmetry, and so they are of interest to real algebraic geometers. They also arise as automorphism groups of regular maps of surfaces of type {3, q}. Through this survey we hope to extend the interest in Klein surfaces to mathematicians working in related ﬁelds.

2

Preliminaries

A natural tool used to study automorphism groups of Klein surfaces is the theory of non-euclidean crystallographic groups as introduced in [18]. For a general background on these questions we refer the reader to [4], but we brieﬂy introduce the main ideas here. The upper half plane U endowed with the Poincar´e metric is a model for the hyperbolic plane. A non-euclidean crystallographic (NEC) group is a discrete subgroup Γ of the group PGL(2, R) of isometries of U such that the quotient space U/Γ is compact. If Γ contains only orientation preserving isometries it is called a Fuchsian group, otherwise it is a proper NEC group. A Klein surface is endowed with a dianalytic structure, see [1], and it may be non-orientable and with non empty boundary. The notion of a Klein surface is a generalization of that of a Riemann surface in the sense that Riemann surfaces may be seen as orientable Klein surfaces with empty boundary. Also, in the same way that compact Riemann surfaces of genus ≥ 2 are uniformized by Fuchsian groups, compact Klein surfaces of algebraic genus p ≥ 2 are uniformized by NEC groups. To be precise, if X is such a surface then there exists an NEC group Λ with no non-identity orientation preserving elements of ﬁnite order such that X = U/Λ. If X has non-empty boundary then Λ is called a bordered surface group. Moreover given a surface so represented, a ﬁnite group G is a group of its automorphisms if and only if there exists an NEC group Γ and an epimorphism from Γ onto G whose kernel is Λ. All groups of automorphisms of bordered Klein surfaces arise in this way. An epimorphism whose kernel is a bordered surface NEC group is called a bordered smooth epimorphism. It is well known [19] that if G = Γ∗ /Λ satisﬁes |G| = 12(p − 1), then Γ∗ has the following presentation: c0 , c1 , c2 , c3 | c2i = (c0 c1 )2 = (c1 c2 )2 = (c2 c3 )2 = (c3 c0 )3 = 1, where each ci is a reﬂection of the hyperbolic plane.

52

BUJALANCE, CIRRE, TURBEK

For each M∗ -group G there is a bordered smooth epimorphism θ : Γ∗ → G. Since Λ = ker(θ) is a bordered NEC group, at least one of {c0 , c1 , c2 , c3 } must belong to Λ. It is easy to see that neither c0 nor c3 can belong to Λ since c0 c3 has order 3. We / Λ. may assume, by a trivial change of notation if necessary, that c1 ∈ Λ; so c2 ∈ Therefore each M∗ -group G has the following (partial) presentation: α, β, γ | α2 = β 2 = γ 2 = (βγ)2 = (αγ)3 = · · · = 1,

(2.1)

where the remaining relators make the group ﬁnite. The order q of αβ is called the index of the presentation. Observe that diﬀerent epimorphisms from Γ∗ onto the same group G may yield diﬀerent Klein surfaces and diﬀerent indices. As in the case of Riemann surfaces and Hurwitz groups, it is interesting to ﬁnd the genera of the Klein surfaces with maximal symmetry. There are inﬁnitely many values of p for which there exist a genus p Klein surface with maximal symmetry, and also inﬁnitely many values for which these surfaces do not occur. However, the genus itself does not determine topologically a Klein surface, since two more data are required, namely, its orientability and the number of its boundary components. Hence the natural problem is to ﬁnd the topological types of the Klein surfaces with maximal symmetry. In this direction it is worth mentioning the computation by May [22], of the 32 diﬀerent topological types of bordered Klein surfaces with maximal symmetry of genus p ≤ 40, see also [14]. It turns out that some of these genera have more than one topological type with maximal symmetry. The same happens in higher genus, see [13], and in fact, there exists no bound (independent of p) for the number of topological types with maximal symmetry within a single genus, [23]. Other results concerning the orientability of a Klein surface with maximal symmetry have been obtained in [26]. The topological type of a Klein surface X = U/Λ with maximal symmetry can be obtained from a presentation as (2.1) of the M∗ -group Γ∗ /Λ. Indeed, the number of boundary components of X equals 6(p − 1)/q, where q is the index of the presentation, and X is orientable if and only if all the relators have an even number of letters α, β and γ, see [12]. Compact connected Klein surfaces may be viewed as real algebraic curves due to the well-known functorial equivalence between such surfaces and algebraic function ﬁelds in one variable over R, see [1]. Thus any result concerning Klein surfaces can be stated in terms of real algebraic curves and, in fact, automorphisms of Klein surfaces correspond to real birational transformations of real algebraic curves. In addition, the topological type of a Klein surface is reﬂected in the real curve in the following way. A Klein surface is orientable if and only if its corresponding real curve disconnects its complexiﬁcation. The boundary of the surface is homeomorphic to the set of real points of the curve, so the number of (empty) period cycles of the surface NEC group uniformizing the surface coincides with the number of ovals of the real curve. There is also an important correspondence between bordered Klein surfaces with maximal symmetry and regular maps, see [11], [14], [23] and [30]. A map is said to be of type {r, q} if it is composed of r-gons, with q meeting at each vertex. Suppose the M∗ -group G acts on the bordered surface X with index q. Then the surface X

GROUPS ACTING ON BORDERED KLEIN SURFACES

53

corresponds to a regular map M of type {3, q} on the surface X ∗ obtained from X by attaching a disc to each boundary component. Further G is isomorphic to the automorphism group of the map M, and the number of boundary components of X is equal to the number of vertices of M.

3

Properties of M∗ -groups

Recall that the order of an M∗ -group is of the form 12(p − 1). May proved that there are M∗ -groups for inﬁnitely many values of p [20]. In fact, given a positive integer n, there is a positive integer k such that there are at least n non-isomorphic M∗ -groups of order k, see [23]. There are restrictions which prevent a group G from being an M∗ -group and May proved [21] that there are no M∗ -groups of order 12(p − 1) for inﬁnitely many p. For example, there exists no M∗ -group of order 12q for any prime q > 5 such that q ≡ 1 mod 3. There are no solvable M∗ -groups of order 24q, 48q or 72q for any prime q > 3 and there are no solvable M∗ -groups of order 12qr for any primes q and r such that 3 ≤ q < r, see [22]. In addition, an M∗ -group is supersolvable if and only if |G| = 4 · 3r for some r, see [24]. Other properties of M∗ -groups are known. For example, if G is an M∗ -group then the index [G : G ] of its commutator is a divisor of 4, see [14]. If H is a normal subgroup of G of index greater than 6 then the quotient G/H is an M∗ -group [14]. This leads to the notion of M∗ -simple groups, which are those M∗ -groups with no normal subgroup of index greater than 6. Every M∗ -group is either M∗ -simple or an extension of a group by an M∗ -simple group. Therefore, M∗ -simple groups play an analogous role as simple groups play in the study of Hurwitz groups.

4

Families of M∗ -groups

There are not many known families of M∗ -groups and we present here the most relevant ones. The ﬁrst family was given by May in [20], where he shows that the M∗ -groups are ﬁnite quotient groups of the extended modular group PGL(2, Z). In particular, for each m ≥ 2 the principal congruence subgroup of level m is normal in PGL(2, Z) and the index is ﬁnite; so the quotient is an M∗ -group. Also in [20], using techniques analogous to those employed by Macbeath in [17] for Hurwitz groups and by Singerman in [25] for H∗ -groups, May exhibits a way to obtain inﬁnite families of M∗ -groups. The ﬁrst family of M∗ -groups explicitly described by means of generators and relations is associated to the groups Gn,q,r with generators A, B and C and deﬁning relations An = B q = C r = (AB)2 = (BC)2 = (CA)2 = (ABC)2 = 1. If we set T = BC, U = CA and V = BCA then we obtain the presentation T 2 = U 2 = V 2 = (T U )2 = (T V )n = (U V )q = (T U V )r = 1.

54

BUJALANCE, CIRRE, TURBEK

If n = 3 and the group is ﬁnite then we obtain an M∗ -group with index q. Some values of q and r which make the group ﬁnite are given in [11]. Using results of Coxeter in [10], Etayo shows in [12] that for every k the group G3,6,2k with order 12k 2 is the group of automorphisms of two Klein surfaces with maximal symmetry of algebraic genus p = 1 + k 2 , having k 2 and 3k boundary components respectively. Let q be an odd prime. The linear fractional transformations A : x → 1/(1 − x), B : x → 1/x − 1 and C : x → 1 + x, considered modulo q, generate the projective special linear group PSL(2, q) when q ≡ 1 (mod 4), and the projective general linear group PGL(2, q) when q ≡ 3 (mod 4). The orders of AB, BC, CA and ABC are all 2, while A has order 3 and C has order q. The order of B is n(q), the ordinal of the ﬁrst Fibonacci number that is divisible by q. This number divides either q + 1 or q − 1, excepting n(5) = 5, see [10]. Hence PSL(2, q) and PGL(2, q) for the above values of q satisfy the relations of G3,n(q),q and thus are M∗ -groups [14, 12]. Moreover, if G is one of these groups then the Klein surface corresponding to either group can be chosen to possess either |G|/(2n(q)) or |G|/(2q) boundary components. Also the groups PSL(2, 2m ) are quotients of G3,n,r groups [28], and therefore they are M∗ -groups. The surface associated with each of these groups can be chosen to have 2m−1 (2m + 1) boundary components, see [12]. The precise values of the prime power q for which PSL(2, q) is an M∗ -group were calculated by Singerman in [27], while those for which PGL(2, q) is an M∗ -group are given in [3]. One can also construct an M∗ -group from another M∗ -group G with special features. For example, if G has odd index q then C2 ×G is also an M∗ -group of index 2q (where we denote by Cn the cyclic group of order n); also if |G : βγ, αγ| = 2, where α, β and γ are as in (2.1) and q is not a multiple of 3 then the semidirect product C3 ϕ G is an M∗ -group, where ϕ : G → G/βγ, αγ ∼ = C2 = Aut(C3 ) is the quotient map, see [22]. The underlying idea above is to construct new families of M∗ -groups as extensions of ﬁnite groups by other known M∗ -groups. This is extensively used by Greenleaf and May in [14] where, using abelian full covers of surfaces with maximal symmetry, they construct inﬁnite families of M∗ -groups as extensions of abelian groups by a given M∗ -group. In [23] May achieved a complete presentation of two new families of M∗ -groups by using extensions which involve the simplest M∗ -group, namely, the dihedral group of order 12. For each family let n be a positive integer and let sn = n/ gcd(2, n). The ﬁrst family has the presentation u2 = x2 = z 6m = (ux)2 = (uz)2 = (xz)3 = 1, xz 6 = z 6 x, an = 1, a = xz −3 , where m divides sn . Each group has order 12n2 m and index 6m. The second family has the presentation u2 = x2 = z 6m = (ux)2 = (uz)2 = (xz)3 = 1, xz 6 = z 6 x, a3n = (zaz −1 a)n = 1, a = xz −3 , where m divides sn . Each group has order 36n2 m and index 6m.

GROUPS ACTING ON BORDERED KLEIN SURFACES

55

For the most part, the above groups were shown to be M∗ -groups by showing that they satisfy the relations (2.1). In [2], a diﬀerent point of view was adopted. One can construct M∗ -groups by beginning with groups (not necessarily M∗ -groups) which may arise as normal subgroups of index two or four in an M∗ -group, and examining which admit appropriate extensions. Concerning subgroups of index 2, it was shown that if H is a ﬁnite group generated by two elements a and b of order two and three respectively, and H admits the group automorphism γ which maps a → a and b → b−1 then the semidirect product H γ C2 is an M∗ -group. Similarly assume H is a ﬁnite group which is generated by three elements a, b and c, each of order two, such that ac and bc each has order three. If H admits the group automorphism γ which maps a → b, b → a and c → c then H γ C2 is an M∗ group. This provides a new method for constructing inﬁnite families of M∗ -groups. We illustrate it with the following examples. For any odd number q, let PSL(2, q) denote the factor group SL(2, q)/{±I}, where SL(2, q) consists of all 2 × 2 matrices of determinant 1 whose entries belong to the ring of integers mod q. It admits the following presentation [29], x, y | xq = y 2 = (xy)3 = (x(q+1)/2 yx4 y)2 = 1 . If we deﬁne a = y and b = xy, then a and b have orders two and three respectively and generate PSL(2, q). In addition, the assignment a → a and b → b−1 is an automorphism of the group. Therefore, PSL(2, q) γ C2 is an M∗ -group. If we assume now that q is an odd prime, the Klein surfaces associated to these groups can be chosen to be either orientable with (q 2 − 1)/2 boundary components, or non-orientable with |PSL(2, q)|/n boundary components where n is the order of the commutator of x and y. Another interesting example where this method is applied deals with the family of (2, 3, r; s)-groups, which has been extensively studied, see [16] and the references given there. The (2, 3, r; s) group is deﬁned by the presentation a, b | a2 = b3 = (ab)r = [a, b]s = 1. If r ≤ 6 or s ≤ 3 then this group is ﬁnite. For the other values of r and s diﬀerent to (r, s) = (13, 4), this group is ﬁnite if and only if one of the following holds: r = 7, s < 9; r = 8 or 9, s < 6; r = 10 or 11, s < 5; r ≥ 12, s < 4. Clearly it admits the automorphism γ : a → a, b → b−1 ; therefore if G is a ﬁnite (2, 3, r; s)-group, then G γ C2 is an M∗ -group. Now, let K be a ﬁnite group generated by two elements a and b each of order three. It was also shown in [2] that if K admits any two of the following automorphisms δ1 : a → a−1 , b → b−1 ,

δ2 : a → b, b → a,

δ3 : a → b−1 , b → a−1 ,

then the semidirect product K δ (C2 × C2 ) is an M∗ -group. This also provides a new method for constructing inﬁnite families of M∗ -groups. As an example, consider the group K with presentation a, b | a3 = b3 = (ab)3 = (a−1 b)n (ab−1 )m = 1 .

56

BUJALANCE, CIRRE, TURBEK

This group has order 3(n2 + mn + m2 ), see [5, pp. 413–415], and clearly admits the above automorphisms if m = n or m = 0. Therefore K δ (C2 × C2 ) is an M∗ group. The associated Klein surface can be chosen to be a torus with 3n2 boundary components if m = n, or n2 boundary components if m = 0. There is also another method which provides inﬁnite families of M∗ -groups containing M∗ -groups with certain properties as subgroups of low index. In [3] it is shown that if G is a perfect M∗ -group then G×C2 , G×C2 ×C2 , G×S3 , G×C2 ×S3 and G × S3 × S3 are also M∗ -groups. As an example, PSL(2, q) is an M∗ -group for inﬁnitely many values of q. Similarly the alternating group An is an M∗ -group for inﬁnitely many n, as we shall see in the next section. Clearly, both PSL(2, q) and An are perfect and so this gives new inﬁnite families of M∗ -groups.

5

Relations between Hurwitz, H∗ , and M∗ -groups

Although Hurwitz, H∗ and M∗ -groups can be deﬁned geometrically as the largest groups of automorphisms acting on Riemann and Klein surfaces, they can be described completely in algebraic terms. This allows one to study the relationship between them from a group theoretic point of view. It is well known that a Hurwitz group is a ﬁnite group generated by two elements A and B which satisfy the relations: A2 = B 3 = (AB)7 = 1. Singerman [25] showed that a ﬁnite group G is an H∗ -group if and only if it can be generated by three elements α, β, and γ which satisfy the relations α2 = β 2 = γ 2 = (βγ)2 = (αγ)3 = (αβ)7 = 1 and has the property that βγ and αγ generate G. It is immediate that each H∗ group is a Hurwitz group and also an M∗ -group. An M∗ -group which arises in this way, will always have index seven. The ﬁrst families of H∗ -groups were given by Hall [15] who found the conditions under which PSL(2, q) is an H∗ -group, see also [8]. For example, if q is prime then PSL(2, q 3 ) is an H∗ -group if and only if q = 2 or q ≡ 5, 9, −3, −11 mod 28. Similarly, if q ≡ 1, 13 mod 28, then PSL(2, q) is also an H∗ -group. Hence, these are both families of M∗ -groups. Conder [6] found the inﬁnitely many values of m and n for which the alternating group Am and the symmetric group Sn are generated by three elements x, y, t which satisfy x2 = y 3 = (xy)7 = t2 = (xt)2 = (yt)2 = 1. For such values, the groups Am and Sn constitute inﬁnite families of M∗ -groups [13]; see also [7], where it is shown that for each k ≥ 7, all but ﬁnitely many An and Sn are M∗ -groups of index k. In addition, Am is also an H∗ -group since it can be generated by x and y, see [8]. Fewer relationships are known between Hurwitz groups and M∗ -groups. As we saw in the previous section, if H is a ﬁnite group generated by two elements a and b of order two and three respectively, and H admits the group automorphism γ which maps a → a and b → b−1 then H γ C2 is an M∗ -group. In addition, if γ is

GROUPS ACTING ON BORDERED KLEIN SURFACES

57

an inner automorphism by an element of order two then H itself is an M∗ -group, see [3]. In particular, if H is a Hurwitz group admitting such an automorphism then H C2 and H are both M∗ -groups. References [1] N. L. Alling, N. Greenleaf, Foundations of the Theory of Klein Surfaces, Lecture Notes in Math., 219, Springer-Verlag, 1971. [2] E. Bujalance, F. J. Cirre, P. Turbek, Subgroups of M∗ -groups. Submitted. [3] E. Bujalance, F. J. Cirre, P. Turbek, An automorphism criterion for M∗ -groups. Preprint. [4] E. Bujalance, J. J. Etayo, J. M. Gamboa, G. Gromadzki, Automorphism groups of compact bordered Klein surfaces. Lecture Notes in Math. 1439, Springer-Verlag, 1990. [5] W. Burnside, Theory of groups of finite order 2nd ed., Cambridge, 1911. [6] M. D. E. Conder, Generators for alternating and symmetric groups. J. London Math. Soc., (2), 22, (1980), 75–86. [7] M. D. E. Conder, More on generators for alternating and symmetric groups. Quart. J. Math. Oxford Ser. (2) 32, (1981), no. 126, 137–163. [8] M. D. E. Conder, Groups of minimal genus including C2 extensions of PSL(2, q) for certain q. Quart. J. Math. Oxford Ser. (2), 38, (1987), no. 152, 449–460. [9] M. D. E. Conder, Hurwitz groups: a brief survey. Bull. Amer. Math. Soc., 23, (1990), (2), 359–370. [10] H. S. M. Coxeter, The abstract groups Gm,n,p , Trans. Amer. Math. Soc., 45, 73–150, (1939). [11] H. S. M. Coxeter, W. O. J. Moser, Generators and relations for discrete groups. 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 14, Springer-Verlag, 1972. [12] J. J. Etayo, Klein surfaces with maximal symmetry and their groups of automorphisms. Math. Ann., 268, (1984), 533–538. [13] J. J. Etayo, C. P´erez-Chirinos, Bordered and unbordered Klein surfaces with maximal symmetry. J. Pure Appl. Algebra, 42, (1986), (1), 29–35. [14] N. Greenleaf, C. L. May, Bordered Klein surfaces with maximal symmetry. Trans. Amer. Math. Soc., 274, (1), (1982), 265–283. [15] W. Hall, Automorphisms and coverings of Klein surfaces, Ph. D. thesis, Southampton Univ., 1977 [16] J. Howie, R. M. Thomas, Proving certain groups infinite. Geometric group theory, Vol. 1 (Sussex, 1991), 126–131, London Math. Soc. Lecture Note Ser., 181, Cambridge Univ. Press, Cambridge, 1993. [17] A. M. Macbeath, On a theorem of Hurwitz, Proc. Glasgow Math. Assoc., 5, (1961), 90–96. [18] A. M. Macbeath, The classification of non-euclidean plane crystallographic groups, Canad. J. Math., 19, (1967), 1192–1205. [19] C. L. May, Automorphisms of compact Klein surfaces with boundary. Paciﬁc J. Math., 59, (1975), 199–210. [20] C. L. May, Large automorphism groups of compact Klein surfaces with boundary I. Glasgow Math. J., 18, (1977), 1–10. [21] C. L. May, Cyclic automorphism groups of compact bordered Klein surfaces. Houston J. Math., 3, (1977), 395-405. [22] C. L. May, The species of bordered Klein surfaces with maximal symmetry of low genus. Paciﬁc J. Math., 111, (1984), 371–394.

58

BUJALANCE, CIRRE, TURBEK

[23] C. L. May, A family of M∗ -groups. Can. J. Math., 38, (1986), 1094-1109. [24] C. L. May, Supersolvable M ∗ -groups. Glasgow Math. J., 30, (1988), no. 1, 31–40. [25] D. Singerman, Automorphisms of compact non-orientable Riemann surfaces. Glasgow Math. J. 12, (1971), 50–59. [26] D. Singerman, Orientable and non-orientable Klein surfaces with maximal symmetry. Glasgow Math. J. 26, (1985), 31–34. [27] D. Singerman, PSL(2, q) as an image of the extended modular group with applications to group actions on surfaces. Proc. Edinburgh Math. Soc., 30, (1987), 143–151. [28] A. Sinkov, On generating the simple group LF(2, 2N ) by two operators of periods two and three. Bull. Amer. Math. Soc., 44, (1938), 449–455. [29] J. G. Sunday, Presentations of the groups SL(2, m) and PSL(2, m). Canad. J. Math., 24, 6, (1972), 1129–1131. [30] S. E. Wilson, Riemann surfaces over regular maps. Canad. J. Math., 30, (1978), 4, 763–782.

BREAKING POINTS IN SUBGROUP LATTICES GRIGORE CALUGAREANU AND MARIAN DEACONESCU1 Department of Mathematics and Computer Science, Kuwait University, P.O. Box 5969, Safat 13060, Kuwait

Dedicated to R.R. Khazal

Abstract The paper classifies those locally finite groups having a proper nontrivial subgroup which is comparable with any other element of the subgroup lattice.

1

Introduction

Let G be a group and let L(G) denote its subgroup lattice. The description of groups G with L(G) a chain is well-known. In a chain, every element is comparable with the others. This raises the natural question of seeing what can be said about groups G having a proper nontrivial subgroup H with the property that for every subgroup X of G one has either X H or H X. Such a subgroup H will be called a breaking point for the lattice L(G). For the sake of convenience, we shall call these groups BP-groups. Of course, BP-groups cannot be decomposed as nontrivial direct products. Moreover, if G is a BP-group with breaking point H, then every subgroup K of G strictly containing H is itself a BP-group with breaking point H. These simple considerations are valuable in what follows and we shall use them without any further reference. Standard results from abelian group theory dispose of the structure of abelian BP-groups: these are cyclic p-groups in the finite case and Pr¨ ufer p-groups Z(p∞ ) in the infinite case. This focuses the discussion on nonabelian BP-groups. As more exotic examples, the so-called extended Tarski groups, see Ol’shanskii [3], p. 344 are also BP-groups. If G is one of these groups, the largest breaking point is Z(G), which is a finite cyclic p-group (p is a rather large prime) and G/Z(G) is an infinite simple p-group of exponent p (a Tarski group). These (quasi finite) examples show that BP-groups need not be soluble, nor locally finite; the class of BP-groups is thus large enough to warrant a more serious investigation. We shall restrict ourselves here to the particular case of locally finite BP-groups; the cyclic p-groups of order at least p2 and the generalized quaternion groups are examples of finite BP-groups. As we have seen, the Pr¨ ufer p-groups exhaust the infinite abelian BP-groups - these groups are also locally finite. But there also exist infinite nonabelian locally finite BP-groups, as for example Szele’s group S discussed in [2] and [6]: S = Ax, where A is a Pr¨ ufer 2-group and x has order 1 The second named author wishes to thank Kuwait University for financial support through research contract SM09/00.

60

CALUGAREANU, DEACONESCU

four, x2 is the unique element of order two of A and x acts on the normal subgroup A by inverting all elements of A. The main result of this note shows that the examples described above exhaust all locally finite BP-groups: Theorem 1.1 A locally finite BP-group is isomorphic to one of the following groups: finite cyclic p-groups of order at least p2 , generalized quaternion groups, Pr¨ ufer groups Z(p∞ ) and Szele’s group S. The notation is standard and the proofs are elementary.

2

The proof of the Theorem

We need first some general information on BP-groups. Lemma 2.1 Let G denote a nonabelian BP-group. Then: 1) G is a p-group for some prime p and if H is a beaking point of L(G), then H is a finite cyclic group contained in Z(G). In particular, G has a unique subgroup of order p. 2) Z(G) is the largest breaking point of L(G). 3) If G is infinite, a proper infinite abelian subgroup A of G has finite index if and only if A is normal in G. 4) If G is infinite and if p is odd, then a proper infinite abelian subgroup A of G has infinite index if and only if A = NG (A). Proof 1) Note first that if H is a breaking point for L(G) and if g ∈ G \ H, then H is a proper subgroup of g, whence H is cyclic and central. We prove next that H is finite. If g ∈ G \ H, then H = h < g, so we can write h = g a for some integer a with |a| ≥ 2. Apply the same argument to the element hg = g a+1 ∈ G \ H to obtain h = g (a+1)b for some integer b with |b| ≥ 2. Suppose now that H is infinite. Then both g and hg = g (a+1) are infinite and g a = h = g (a+1)b . One must have a = (a+1)b, for otherwise g would have finite order, a contradiction. Elementary divisibility arguments force a = −2 and b = 2, thus h = g −2 . The above argument, applied to the element g −1 ∈ G \ H, gives h = g 2 , whence the equality g 2 = h = g −2 , forcing g 4 = 1, another contradiction. We are now ready to show that G is a p-group. To prove this, we show first that G is periodic. If g ∈ G \ H would be of infinite order, then g would contain a nontrivial finite subgroup, namely H = h, a contradiction. Thus G is periodic indeed. If g ∈ G \ H, then the finite cyclic group g is a BP-group with breaking point H; thus g is indecomposable, forcing g to be a p-group for some prime p. This implies at once that G is a p-group. In particular, H is a finite cyclic p-group and G has a unique subgroup of order p. This concludes the proof of 1). 2) Suppose that Z(G) is not a breaking point for L(G). Then there exists some x ∈ G \ Z(G) such that Z(G) x. The abelian group K = Z(G)x is not cyclic, nor a Pr¨ ufer group, which contradicts the fact that it must have a unique

BREAKING POINTS IN SUBGROUP LATTICES

61

subgroup of order p. Thus Z(G) is a breaking point for L(G). The maximality of Z(G) follows from 1). 3) Let A be an infinite abelian subgroup of G. Note that A is not normal in G if and only if coreG (A) is a proper subgroup of A if and only if coreG (A) is finite these follow from A being a Pr¨ ufer group. Also note that if A has finite index in G, then coreG (A) has finite index in G. Thus A being not normal in G and having finite index are contradictory. 4) Let A be an infinite proper subgroup of G, so that A is a Pr¨ ufer p-group. Then A is a maximal abelian subgroup of G (for otherwise one would find a larger Pr¨ ufer subgroup B containing A, which is impossible), so A = CG (A). Now NG (A)/A is isomorphic to a subgroup of Aut(A) and by 1) this factor group is periodic. This forces NG (A) = A since if p is odd Aut(A) has no nontrivial elements of p-power order. This completes the proof of the Lemma. ✷ We are now in a position to give a proof of the Theorem. Proof Let G be a locally finite BP-group. If G is finite, then either G is cyclic, or p = 2 and G is a generalized quaternion group. This follows from the lemma and from Satz 8.2, p. 310 of Huppert [1]. If G is infinite, then any two nontrivial elements of G generate a finite subgroup T of G which has just one minimal subgroup. For p odd, this subgroup T is cyclic. Thus G is abelian, which implies at once that G ∼ = Z(p∞ ). We reached the stage where all locally finite BP-groups were classified, except those which are infinite nonabelian 2-groups. From now on, G will denote a locally finite infinite nonabelian BP-group which is a 2-group. Since G is nonabelian, there exists a pair of non commuting elements in G which generate a nonabelian subgroup K of G, which is a generalized quaternion 2-group. Thus, if X is any finite subset of G, then X is contained in the subgroup L = X, K, which is also a generalized quaternion group. These groups are discussed in [2], p. 48, where it is proved that the unique such locally generalized quaternion group is actually Szele’s group S described in the introduction. This concludes the proof of the Theorem. ✷

3

Final remarks

1) Using our Theorem and Theorem 2.4.16 of Schmidt [4], it is easy to prove that if an infinite nonabelian BP-group G has modular lattice L(G), then G must be an extended Tarski group. 2) The Theorem implies that for an infinite BP-group G the following are equivaˇ lent: a) G is a Cernikov group; b) G is locally finite; c) G is locally nilpotent; d) G is locally soluble. 3) By parts 2) and 1) of the Lemma, it follows that if G is a nonabelian BPgroup, then there are only finitely many breaking points in L(G). Therefore, if the subgroup lattice of a group has infinitely many breaking points, then the group must be a Pr¨ ufer p-group.

62

CALUGAREANU, DEACONESCU

4) After a preliminary version of this note was written, the authors received a reprint of Prof. R. Schmidt’s recent paper [5]. In Satz 3.1 of [5], it is shown that odd order p-groups are BP-groups precisely when there exist a so-called supermodular subgroup of G (which is a breaking point of L(G)). It is also shown there (for such infinite nonabelian p-groups of odd order) that Z(G) is the largest breaking point of L(G). A number of interesting questions remain still open: a) Are there infinite BP-groups which are not locally finite and with non modular lattice L(G)? b) Are there infinite BP-groups of infinite exponent which are not locally finite? Such groups would necessarily have infinite proper abelian subgroups of infinite index. Prof A. Yu. Ol’shanskii, in a personal communication, kindly pointed out that the construction of such groups would be possible, but not very simple. This hints that a complete classification of BP-groups is far from being an easy task. References [1] B. Huppert, Endliche Gruppen I, Springer–Verlag, 1967. [2] O.H. Kegel and B.A.F. Wehrfritz, Locally finite groups, North Holland 1974. [3] A. Yu. Ol’shanskii, Geometry of defining relations in groups, Kluwer Academic Publishers, 1991. [4] R. Schmidt, Subgroup Lattices of Groups, Walter de Gruyter, 1994. [5] R. Schmidt, “Supermodulare Untergruppen von Gruppen”, Math. Kolloq. 53 (1999) 23–49. (German) [6] T. Szele, “Die unendliche Quaternionengruppe”, Bul. St. Acad. Republ. Popul. Romane vol A 1 (1949) 791-802. (German)

GROUP ACTIONS ON GRAPHS, MAPS AND SURFACES WITH MAXIMUM SYMMETRY MARSTON CONDER Department of Mathematics, University of Auckland, Private Bag 92019, Auckland, New Zealand

Abstract This is a summary of a short course of lectures given at the Groups St Andrews conference in Oxford, August 2001, on the signiﬁcant role of combinatorial group theory in the study of objects possessing a high degree of symmetry. Topics include group actions on closed surfaces, regular maps, and ﬁnite s-arc-transitive graphs for large values of s. A brief description of the use of Schreier coset graphs and computational methods for handling ﬁnitely-presented groups and their images is also given.

1

Introduction

Historically there has been a great deal of fascination with symmetry — in art, science and culture. One of the key strengths of group theory comes from the use of groups to measure and analyse the symmetries of objects, whether these be physical objects (in 2 or 3 dimensions), or more purely mathematical objects such as roots of polynomials or vectors or indeed other groups. This is now bearing unexpected fruit in areas such as structural chemistry (with the study of fullerenes for example), and interconnection networks (where Cayley graphs and other graphs constructed from groups often have ideal properties for communication systems). The aim of this paper (and the associated short course of lectures given at the Groups St Andrews 2001 conference in Oxford) is to describe a number of instances of symmetry groups of mathematical objects where the order of the group is as large as possible with respect to the genus, size or type of the object. Topics covered include Hurwitz groups (maximum order conformal automorphism groups of compact Riemann surfaces), regular maps (embeddings of graphs in surfaces having automorphism group transitive on incident vertex-edge-face triples), and ﬁnite symmetric graphs with automorphism group acting transitively on directed non-reversing walks of length s for the highest possible values of s. In each case combinatorial group theory plays a signiﬁcant role, and accordingly, a brief description of the use of some graphical and computational methods for handling ﬁnitely-presented groups and their images is also given. Please note that this paper makes no claims to be a comprehensive survey of the theme of maximum symmetry, or even of each of the topics dealt with. It is hoped, however, that the paper will provide some of the ﬂavour of research in this ﬁeld, through description of a range of recent discoveries, and a good deal of references.

64

CONDER

The paper is organised as follows. In Section 2, we describe some of the tools that have proved useful to the author and others in this area, including the low index subgroups process and Schreier coset graphs. Section 3 deals with automorphism groups of compact Riemann surfaces, with particular reference to Hurwitz groups. Section 4 concerns regular maps, on both orientable and non-orientable surfaces, and goes on to describe some recent work on the maximum number of automorphisms of a closed non-orientable surface of given genus. In Section 5 we consider ﬁnite graphs with large symmetry groups, concentrating on the special cases of 5-arc-transitive and 7-arc-transitive graphs of valency 3 and 4 respectively. Finally, Section 6 describes two instances where unexpected results have arisen, and Section 7 lists a number of open problems. A number of deﬁnitions will be useful background for the material which follows. A surface will be taken as a closed 2-manifold without boundary. The (topological) genus of an orientable surface is the number of ‘handles’ attached to a sphere to obtain it, and is related to the Euler characteristic by the formula χ = 2 − 2g (where g is the genus); for example, the sphere has genus 1, the torus has genus 2, the double torus has genus 3, etc. Analogously, the genus g of a non-orientable surface is the number of its ‘cross-caps’, and is related to the Euler characteristic by the formula χ = 2 − g ; for example, the real projective plane has genus 1, and the Klein bottle has genus 2. A graph is a combinatorial network, consisting of a pair (V, E) where V is a set (of vertices) and E is an irreﬂexive symmetric relation on V (that is, a set of unordered pairs of distinct vertices (called edges)). As such, graphs in this context are simple, with undirected edges, no loops and no multiple edges. A multigraph is a generalisation of a graph, in which multiple edges are allowed between any pair of distinct vertices. Finally, a map is a 2-cell embedding of a connected graph or multigraph into a surface (so that the connected components of the complementary space obtained by removing the graph or multigraph from the surface are all homeomorphic to open disks, called faces).

2 2.1

Methods for dealing with finitely-presented groups Computational algorithms

Several eﬃcient computational procedures have been developed over the last four decades for handling abstract groups with a small number of generators and deﬁning relations, and have been implemented in computer support packages such as Magma and GAP. Very brieﬂy, for a ﬁnitely-presented group G = X | R these include the following: (a) Coset enumeration: variants of a method due to Todd and Coxeter may be used to attempt to determine the index of a ﬁnitely-generated subgroup H in G; (b) Low index subgroups: algorithms (developed principally by Sims) enable the determination of a representative of each conjugacy class of subgroups of up to some speciﬁed index N in G;

GROUP ACTIONS WITH MAXIMUM SYMMETRY

65

(c) Reidemeister-Schreier rewriting process: this gives a deﬁning presentation for a subgroup H of ﬁnite index in G, in terms of Schreier generators; (d) Abelian quotient algorithm: this can produce the direct factors of the abelianisation H/[H, H] of a subgroup H of ﬁnite index in G; (e) p-quotient and nilpotent quotient algorithms: these produce p-quotients or nilpotent quotients (respectively) of G, of up to a given nilpotency class. Excellent descriptions of these may be found in the book by Charles Sims [51]. 2.2

Low index subgroups

The low index subgroups algorithm is especially important in the computational study of small ﬁnite images of ﬁnitely-presented groups. The basic algorithm (due to Sims) ﬁnds a representative of each conjugacy class of subgroups of index up to some speciﬁed N in a given ﬁnitely-presented group G = X | R. This involves a backtrack search through a tree, with nodes at level k in the tree corresponding to (pseudo)subgroups generated by k elements. The search begins (at level 0) with the identity subgroup, generated by the empty set, and successively adjoins and removes elements to and from the generating set for the subgroup, on a last-in ﬁrst-out basis. At each stage of the search, coset enumeration is used to deﬁne suﬃciently many right cosets of the current subgroup H, and to construct a (possibly partial) coset table for H, with rows indexed by the cosets, and columns indexed by elements of the generating set X and their inverses. This table indicates as far as possible the eﬀect of right multiplication of each generator of G on those right cosets of H which have been deﬁned. Deﬁnition of cosets is assumed to follow a systematic pattern, sometimes called normal ordering, so that to each subgroup H of ﬁnite index in G there exists exactly one coset table in normal order. In the coset enumeration procedure, deﬁnition of new cosets alternates with testing current deﬁnitions of coset numbers using the given relators for G and current generators for the (pseudo)subgroup H, and processing of any coincidences that arise. If the deﬁnitions satisfy simultaneously all tests against the relators and subgroup generators, and more than the required number of cosets have been deﬁned, then cosets may be forced to coincide. Accordingly, branches are created to new nodes at the next level of the search tree by identifying pairs of cosets: forcing Hwi = Hwj is equivalent to adjoining wi wj−1 to a set of generators for H (and therefore moving to the next level). If at any node, every entry in the coset table is ﬁlled, then the coset table is said to be closed, and a subgroup has been found. Tests are built in to avoid generating the same subgroup more than once (by rejecting sub-trees) and also to avoid conjugates of subgroups found earlier in the search tree (isomorph rejection). The algorithm stops when the whole search tree has been traversed. Example 2.1 The LowIndexSubgroups command in Magma can ﬁnd the 45991 classes of subgroups of index 120 in the Coxeter group [4, 3, 5] in less than 2 hours on a 400MHz processor.

66

CONDER

The low index subgroups algorithm may be combined with the ReidemeisterSchreier process or cohomological methods (applied to the subgroup of ﬁnite index or its core) to prove that a given ﬁnitely-presented group G is inﬁnite, or to ﬁnd lower bound on its order, or to prove a given subgroup has inﬁnite index. Numerous examples exist in the literature (for example [37]) and in documentation for Magma and GAP. Also clearly the algorithm can be used to determine all ﬁnite factor groups of G isomorphic to permutation groups of small degree (from right representations of G on cosets of subgroups H). This can be particularly helpful in a search for small concrete examples of such factor groups, which can then be used as building blocks for larger examples, as will be seen later. Another important observation to make about the low index subgroups algorithm is that distinct sub-trees can be processed independently. This provides a basis for distributed processing or parallelisation, of either the basic algorithm or special adaptations. One such adaptation involves pursuing only selected branches of the search tree: for example those which correspond to subgroups avoiding a given set of elements (and their conjugates). This has applications to searching for torsion-free subgroups of ﬁnite index, or subgroups complementary to a given ﬁnitely-generated subgroup. Example 2.2 Spherical and hyperbolic 3-manifolds tessellated by regular solids are obtainable by identifying faces of regular solids of type {p, q, r}. A complete classiﬁcation of these was obtained by Brent Everitt in his PhD thesis [35], using the observation that a typical cell ∆ is a fundamental region for a subgroup complementary to the cell-stabilizer [p, q] in the Coxeter group [p, q, r]. A similar approach was taken earlier in [18] to ﬁnd torsion-free subgroups of minimum index in particular groups which produce hyperbolic 3-manifolds and orbifolds of minimal volume, and the potential exists for further applications in geometric situations where such subgroups or complements of a given ﬁnitely-generated subgroup need to be determined. 2.3

Low index normal subgroups

Another straightforward but signiﬁcant adaptation of the low index subgroups algorithm ﬁnds all normal subgroups of up to a speciﬁed index N in a ﬁnitely presented group G = X | R, and hence can produce all ﬁnite factor groups G/K of G of order at most N . Such an adaptation of the standard low index subgroups algorithm is easy: when a coincidence between cosets Ku and Kv of the current subgroup K is forced in the branching process, all conjugates of the element uv −1 must lie in K if K is to be normal; hence in the coincidence processing and subsequent coset enumeration phases the element uv −1 should be applied to all cosets currently active (and not just the trivial coset numbered “1”). In other words, the element uv −1 is treated as an additional relator rather than an additional subgroup generator. The search still begins (at level 0) with the identity subgroup, generated by the empty set, but then successively adjoins and removes elements to and from a set of representatives

GROUP ACTIONS WITH MAXIMUM SYMMETRY

67

of conjugacy classes of G which generate the normal subgroup K, again on a last-in ﬁrst-out basis. This adaptation not only reduces the coset table more than a forced coincidence in the standard procedure at each stage, but also takes appreciably less time than ﬁnding all classes of subgroups of index up to N and eliminating those which are not normal. This in turn enables a search up to much higher index (within given computing resources). The reduction in computing time can be spectacular: Example 2.3 The modular group PSL2 (ZZ) has an abstract deﬁning presentation x, y | x2 = y 3 = 1 in terms of linear fractional transformations x : z → −1/z and y : z → (z −1)/z , and is thus isomorphic to a free product C2 ∗C3 of cyclic groups of orders 2 and 3. One way of ﬁnding all normal subgroups of index up to (say) 20 in this group is to apply the standard low-index subgroups algorithm and check each subgroup in the output for normality (using a conjugacy test), deleting all subgroups which are non-normal. On a 225Mhz processor, the standard algorithm takes about 2 minutes to ﬁnd conjugacy class representatives of all subgroups of up to index 20, while the normal subgroups adaptation described above takes only 0.05 seconds to ﬁnd all normal subgroups up to the same index. A parallel implementation of the low index subgroups algorithm (and its normal subgroups adaptation) was developed by Peter Dobcs´ anyi as part of his PhD thesis project [34]. Called Lowx, this implementation is capable of running on many par´ ka, allel hardware platforms, but its most important use to date has been on Kala a 170-node Linux cluster which he designed and built using machines in student computer laboratories during their idle time. This provided equivalent computing power to a medium-sized supercomputer, at a fraction of the cost! Applications will be described in Sections 4.5 and 5.2. 2.4

Schreier coset graphs

If G is a group with ﬁnite generating-set X = {x1 , x2 , . . . , xd }, and H is a subgroup of index n in G, then the Schreier coset graph Σ(G, X, H) is the graph with vertices labelled by the right cosets of H, and with all edges of the form Hg — Hgxi for 1 ≤ i ≤ d. This graph provides a diagrammatic representation of the action of G on cosets of H by right multiplication. Similarly, if G has a transitive permutation representation on a set Ω of size n, then we may form a graph with vertices as the points of Ω and with all edges of the form α — αxi for 1 ≤ i ≤ d; this is naturally isomorphic to the coset graph for G associated with the point-stabilizer H = Gα = {g ∈ G : αg = α}. In fact these things are essentially interchangeable: the coset table, the coset graph, and permutations induced by the group generators. See [29] for many examples. A number of observations are worth making about Schreier coset graphs. First, any path in the graph may be traced using a word w = w(X) in the generators of G, and elements of the subgroup H expressed as words in the generators of G correspond to directed circuits in the coset graph based at the vertex labelled H.

68

CONDER

Next, a Schreier transversal for H in G corresponds to a spanning tree for the coset graph: any path in a spanning tree based at the vertex H may be traced by a word w, the initial sub-words of which correspond to initial sub-paths of the given path. It follows that a Schreier generating-set for H in G corresponds to the set of edges of the coset graph not used in a spanning tree. For example, the broken edge in Figure 1 completes a circuit corresponding to the Schreier generator uxi v −1 :

Hv = Huxi H

Hu

Figure 1. Schreier generators given by edges not in the spanning tree These observations may be taken further, leading to a diagrammatic interpretation (and implementation) of the Reidemeister-Schreier process: to ﬁnd a presentation for a subgroup H of ﬁnite index in a ﬁnitely-presented group G = X | R, one can simply take a spanning tree in the coset graph Σ(G, X, H), label the unused edges with Schreier generators, and then apply the relators in R to each of the vertices in turn to obtain the relations. Schreier coset graphs have other theoretical applications, for example to the following theorem which provides a necessary condition for transitivity of a group generated by a set of permutations (due independently to Ree and Singerman): If G is the group generated by permutations x1 , x2 , . . . , xd on a set Ω of size n, such that x1 x2 . . . xd = 1, and ci is the number of orbits of xi on Ω, then G is transitive on Ω only if c1 + c2 + . . . + cd ≤ (d − 2)n + 2. This can be proved by taking a particular embedding [11] of the associated coset graph in an orientable surface of genus g ≥ 0, counting the numbers V , E and F of vertices, edges and faces (respectively), and then applying Euler’s formula 2 − 2g = χ = V − E + F . Coset graphs can also have important more practical applications. In some cases, copies of the same coset graph for a group G may be joined together to construct permutation representations of G of arbitrarily large degree, showing in particular that the group is inﬁnite. (This is related to abelianisation of the ReidemeisterSchreier process, but will not be pursued in detail here.) In other cases, two coset graphs for a given ﬁnitely-presented group G = X | R which contain a ﬁxed point of the same involutory generator (in X) can often be joined together by the insertion of an extra transposition which interchanges those two points, to produce a transitive permutation representation of G of larger degree. Necessary conditions are imposed by the relations, however the eﬀect of such composition of coset graphs on each relator (or any other word in the group gener-

GROUP ACTIONS WITH MAXIMUM SYMMETRY

69

ators) can be seen by ‘diagram chasing’, and/or using the fact that multiplication of a given permutation by a single transposition (α, β) always either splits a cycle (containing both α and β) or concatenates two diﬀerent cycles (containing α and β separately). This method of composition of coset graphs was developed by Graham Higman in proving that for all suﬃciently large n, the alternating group An is a homomorphic image of the (2, 3, 7) triangle group ∆ = x, y, z | x2 = y 3 = z 7 = xyz = 1 . Higman observed that every coset graph for this group ∆ may be drawn very simply, with small triangles and heavy dots representing 3-cycles and ﬁxed points of the permutation induced by the generator y, connected together by straight or curved lines representing 2-cycles of the permutation induced by the generator x. By convention, the vertices of each small triangle may be assumed to be permuted anti-clockwise by y, and loops representing ﬁxed points of x are omitted. Cycles of the element xy (= z −1 ) of order 7 can be traced around ‘faces’ of the drawing. If two such coset graphs both involve 7-cycles of xy which contain two ﬁxed points of x separated in the cycle by the same number of points (either 1, 2 or 3 points), then the two coset graphs may be composed together by introducing new transpositions for x, interchanging the corresponding points. In the ﬁrst case, if one has a 7-cycle (a, b, c1 , c2 , c3 , c4 , c5 ) for xy in which a and b are ﬁxed by x, and the other has a similar 7-cycle (a , b , c1 , c2 , c3 , c4 , c5 ), then introducing two new 2-cycles (a, a ) and (b, b ) to the permutation induced by x gives rise to a larger coset graph in which (a, b , c1 , c2 , c3 , c4 , c5 ) and (a , b, c1 , c2 , c3 , c4 , c5 ) are 7-cycles of xy. Other cycles of xy (and of y and x) are unaﬀected, hence the resulting graph is indeed a coset graph for the (2, 3, 7) triangle group ∆. This and the other two possibilities (which are similar) are illustrated in Figure 2.

Figure 2. Composition of coset graphs for the (2, 3, 7) triangle group Interesting things can happen when coset graphs are linked together in this way. For example, the (2, 3, 7) triangle group has permutation representations of degrees 14, 64 and 22 in which the groups generated by the permutations are isomorphic to PSL2 (13), A64 and A22 respectively. The corresponding coset graphs can be

70

CONDER

composed to create a transitive permutation representation of degree 100, in which the permutations generate the Hall-Janko simple group J2 , of order 604800. Another representation as PSL2 (13) on 42 points has three cycles of xy available for Higman’s composition technique, and copies of the coset graph can be linked together in a circuit to produce permutations which generate extensions by PSL2 (13) of an abelian group of any given exponent. The cycle structure of the commutator [x, y] in the latter representation is 13 133 . This coset graph can be composed with another coset graph for ∆ on 36 points, in which the commutator [x, y] has cycle structure 11 42 51 112 , so that in the resulting transitive permutation representation of ∆ on 78 points, the commutator [x, y] has cycle structure 14 42 51 111 122 132 . In particular, by choice of representations, each of the unique 5- and 11-cycles of [x, y] contains points α and β such that αx = α and αy = β. Now if B were a block of imprimitivity containing α, then B would be preserved by the 11-cycle [x, y]780 , but then B would be preserved by x (as αx = α ∈ B) and by y (as αy = β ∈ B), forcing |B| = 78, and hence the action must be primitive. By Jordan’s theorem (on primitive groups containing prime-length cycles) [58], it follows that the permutations generate A78 . Chains of additional copies of the coset graph on 42 points and/or one or two of the graph on 14 points mentioned earlier can also be joined to the ﬁrst (on 42 points), to prove that the alternating group An is a homomorphic image of ∆ for all n of the form 14k + 78 with k ≥ 0. Similarly other coset graphs of various shapes and sizes can be tacked on, to prove the following reﬁnement of Higman’s theorem [9], published by the author in 1980: Theorem 2.1 The alternating group An is a homomorphic image of the (2, 3, 7) triangle group, for all n ≥ 168. Incidentally, this has been taken much further recently by Higman’s academic grandson, Brent Everitt, who has proved in [36] that a similar result holds not only for every hyperbolic (p, q, r) triangle group x, y, z | xp = y q = z r = xyz = 1 with 1/p + 1/q + 1/r < 1, but also for every Fuchsian group (see Section 3.1): Theorem 2.2 Every Fuchsian group has all but ﬁnitely many alternating groups An among its homomorphic images.

3 3.1

Automorphism groups of compact Riemann surfaces Hurwitz’s theorem

The theory of Riemann surfaces is very well-developed, and described nicely in a book by Jones and Singerman on complex functions [39]. A Riemann surface is a connected 2-manifold endowed with a complex analytic structure (called an atlas) that allows local coordinatisation — somewhat analogous to a book of maps of the planet Earth. An automorphism of a Riemann surface X is a homeomorphism f : X → X which preserves the local analytic structure. As usual, automorphisms form a group under composition, known as the automorphism group of X and denoted by Aut X.

GROUP ACTIONS WITH MAXIMUM SYMMETRY

71

A Riemann surface X may also be identiﬁed with the orbit space U/Λ of the action of a normal subgroup Λ of ﬁnite index in some discrete subgroup Γ of the group PSL2 (IR), acting on the upper-half complex plane U. The quotient group Γ/Λ is then isomorphic to the automorphism group Aut X. Associated with the action of the discrete group Γ on U is a fundamental region D = D(Γ): this is a closed set whose images under Γ have disjoint interiors and cover the whole of U. If the Riemann surface X = U/Λ is compact, then a fundamental region for Γ has ﬁnitely many sides. In this case the group Γ has a ﬁnite presentation in terms of elliptic generators X1 , X2 , . . . , Xr and hyperbolic generators A1 , B1 , . . . , Aγ , Bγ (where γ is called the underlying genus, determined by Λ), and subject to deﬁning relations X1m1 = X2m2 = . . . = Xrmr = 1 and X1 X2 . . . Xr [A1 , B1 ] . . . [Aγ , Bγ ] = 1. Such a discrete group Γ is called a Fuchsian group, and is said to have signature (γ; m1 , m2 , . . . , mr ). The parameters mi are the orders of branch points. The area µ(D) of the fundamental region D = D(Γ) is given by the formula µ(D) = 2π (2γ − 2 + ri=1 (1 − 1/mi ) ). The celebrated Riemann-Hurwitz formula states that |Γ/Λ| = 2π(2g−2) µ(D) , where g is the topological genus of the surface X, and this easily converts to the more customary form of 2g − 2 = |Aut X| (2γ − 2 +

r

(1 − 1/mi ) ).

i=1 1 The bracketed expression has minimum positive value of 42 , which is attained precisely when γ = 0, r = 3 and {m1 , m2 , m3 } = {2, 3, 7}. This leads to:

Theorem 3.1 (Hurwitz, 1893) If X is a compact Riemann surface of genus g > 1, then |Aut X| ≤ 84(g − 1), and moreover, the upper bound on this order is attained if and only if Aut X is a homomorphic image of the (2, 3, 7) triangle group ∆ = X, Y, Z | X 2 = Y 3 = Z 7 = XY Z = 1 . Because of this theorem, non-trivial ﬁnite quotients of the (2, 3, 7) triangle ∆ are known as Hurwitz groups. 3.2

Hurwitz groups

Every Hurwitz group G is perfect (that is, G coincides with its commutator subgroup G ), since abelianisation of the (2, 3, 7) relations gives 1 = z −7 = (xy)7 = x7 y 7 = xy and so x = y −1 , which implies x and y are both trivial. It follows that every Hurwitz group has a nonabelian simple quotient, and hence it is natural to look among the nonabelian simple groups for examples of Hurwitz groups. In the 1960s Murray Macbeath [45] used matrix and number-theoretic arguments to prove that the projective special linear group PSL2 (q) is Hurwitz if and only if • q = 7, or • q = p for some prime p ≡ ±1 mod 7, or • q = p3 for some prime p ≡ ±2 or ±3 mod 7.

72

CONDER

As noted earlier (in Section 2.4), the author of this paper used Graham Higman’s method of composition of coset graphs to prove in [9] that the alternating group An is Hurwitz for all n ≥ 168 (and for all but 64 smaller values of n as well). Several other simple groups (and families of simple groups) have been shown to be Hurwitz using character-theoretic techniques. In any ﬁnite group G with known character table, the number of pairs (x, y) of elements such that x has order 2, y has order 3, and xy has order 7 can be calculated using the structure constants cijk =

|Ki ||Kj ||Kk | |G|

χ∈Irr(G)

χ(gi )χ(gj )χ(gk ) χ(1)

where Ki , Kj and Kk denote conjugacy classes of elements of orders 2, 3 and 7, and χ(gr ) is the value of the irreducible character χ on a representative gr of the conjugacy class Kr . Good knowledge of the maximal subgroup structure and local analysis can often be used to account for subgroups generated by these pairs, and hence to determine whether or not G itself is so generated. (In fact by Philip Hall’s theory of M¨ obius inversion on lattices, all one needs to know are the numbers of pairs that lie in intersections of maximal subgroups; see [41]). Chih-han Sah proved in [50] that certain Ree groups 2 G2 (3e ) are Hurwitz, and Gunter Malle later showed that 2 G2 (3e ) is Hurwitz for all odd e > 1 (see [47], and [40]). Also Malle proved that the Chevalley groups G2 (q) are Hurwitz for all q > 4, as well as the twisted simple groups 3D4 (q) for all q = 4 or 3s (for any s) and 2F4 (22n+1 ) for all n ≡ 1 mod 3 (see [47, 48]). Particular attention has been paid to the sporadic simple groups in this context. In a series of papers in the 1990s by Rob Wilson and Andy Woldar and the author (whose involvement was relatively minor), it has been established that exactly 12 of the 26 sporadic ﬁnite simple groups are Hurwitz. The ﬁnal (and most spectacular) step in this process was the very recent proof by Rob Wilson [59] that the Monster can be generated by two elements of orders 2 and 3 whose product has order 7, as a result of some highly innovative computational approaches to investigating subgroups of the Monster using its 196882-dimensional representation over GF(2). In particular, the Monster is a group of automorphisms of some compact Riemann surface X for which equality is attained in the upper bound on |Aut X| given by Hurwitz’s theorem. (It is also the orientation-preserving subgroup of the group of automorphisms of a regular map on a surface of the same genus; see Section 4.) The sporadic simple groups which are Hurwitz are now known to be J1 , J2 , He, Ru, Co3 , F i22 , HN , Ly, T h, J4 , F i24 and the Monster M . The other 14 sporadic ﬁnite simple groups are not Hurwitz (although many of them can still be generated by two elements of orders 2 and 3). Also recently, Andrea Lucchini, Chiara Tamburini and John Wilson have taken a diﬀerent approach to show that ‘most’ ﬁnite simple classical groups of suﬃciently large dimension are Hurwitz groups [43, 44]. In fact what they prove in [44] is that if R is any ﬁnitely-generated ring with at least one generator t having the property that 2t − t2 is a unit of ﬁnite multiplicative order, and En (R) is the group of invertible n × n matrices generated by the set { In + reij | r ∈ R, 1 ≤ i = j ≤ n } of elementary transvections, then En (R) can be generated by two matrices X and

GROUP ACTIONS WITH MAXIMUM SYMMETRY

73

Y such that X 2 = Y 3 = (XY )7 = 1, for all but ﬁnitely many n. The proof uses the permutation matrices corresponding to Hurwitz generators for the alternating group An (as provided in [9]), with modiﬁcation of the generator of order 2 in order to obtain En (R). Similar methods are applied in [43]. As a consequence of this work by Lucchini, Tamburini and Wilson, the following are Hurwitz groups, in addition to many others: • the special linear group SLn (q) for all n ≥ 287 and every prime-power q; • the symplectic group Sp2n (q) for all n ≥ 371 and all q; • the orthogonal groups Ω+ 2n (q) for all q and Ω2n+7 (q) for all odd q, for n ≥ 371; • the unitary groups SU2n (q) for all q and SU2n+7 (q) for all odd q, for n ≥ 371. In particular, the simple projective quotients of these groups are all Hurwitz also; hence for example, PSLn (q) is a Hurwitz group for all n ≥ 287 and every prime-power q. In addition, it follows from the main theorem of [44] and previous work by John Wilson that there are 2ℵ0 inﬁnite simple groups which are factor groups of the (2, 3, 7) triangle group. Next, we note that there are several ways of constructing larger Hurwitz groups from given examples. Such constructions include direct products of Hurwitz groups (with diﬀerent presentations), semi-direct products (of abelian groups by simple Hurwitz groups for example), and central products (of special linear groups for example). Some of these were described in the author’s determination of all Hurwitz groups of order up to 1 million [10] and in his survey article [15]. The central product construction can also be applied to the (2, 3, 7)-generation of SLn (q) to show that the centre of a Hurwitz group can be any ﬁnite abelian group, fully answering a question posed by John Leech in the 1960s (see [27]). Finally, we note some non-existence results. Jeﬀrey Cohen [8] showed in 1981 that PSL3 (q) is Hurwitz only when q = 2, and hence the same holds for SL3 (q). Very recently, Di Martino, Tamburini and Zalesskii proved that many other linear groups of small degree are not Hurwitz, including SLn (q) and SUn (q 2 ) for several n ≤ 19 and various q (see [31]), using Leonard Scott’s matrix group analogue of the Ree-Singerman theorem on a necessary condition for generation by a given subset.

4 4.1

Regular maps Definitions and background

Regular maps may be viewed as generalisations of the Platonic solids. As deﬁned earlier, a map is a 2-cell embedding of a connected graph (or multigraph) into a closed surface without boundary. Such a map M is composed of a vertex-set V = V (M ), an edge-set E = E(M ), and a set of faces which we will denote by F = F (M ). The map is orientable or non-orientable according to whether the underlying surface (on which the graph is embedded) is orientable or non-orientable. The faces of M are the simply-connected components of the complementary space obtained by removing the embedded graph from the surface; alternatively, in the orientable case, they can be deﬁned more directly by considering just the underlying graph together with a ‘rotation’ at each vertex (see [38] or [60]).

74

CONDER

Associated also with any map is a set of darts (or arcs), which are the incident vertex-edge pairs (v, e) ∈ V × E. Each dart is made up of two blades, one corresponding to each face containing the edge e (except in degenerate situations where an edge lies in just one face, but these will not concern us much here.) An automorphism of a map M is a permutation of its blades, preserving the properties of incidence, and as usual these form a group under composition, called the automorphism group of the map, and denoted by Aut M . From connectedness of the underlying graph, it follows that every automorphism is uniquely determined by its eﬀect on any blade, and hence the number of automorphisms of M is bounded above by the number of blades, or equivalently, |Aut M | ≤ 4|E|. Now if there exist automorphisms R and S with the property that R cyclically permutes the consecutive edges of some face f (in single steps around f ), and S cyclically permutes the consecutive edges incident to some vertex v of f (in single steps around v), then following Steve Wilson [60] we may call M a rotary map. Under more currently accepted terminology, M is also called a regular map (in the sense of Brahana, who generated early interest [2] in such objects in the 1920s). In this case, again by connectedness, Aut M acts transitively on vertices, on edges, and on faces of the map M , and it follows that M is combinatorially regular, with all its faces bordered by the same number of edges, say p, and all its vertices having the same degree, say q. The pair {p, q} is known as the type of the map M . (Note that the converse does not hold: a map can be combinatorially regular without being regular; indeed coset graphs for the (2, 3, 7) triangle group can be used to prove [20] that for every g > 1 there exists a combinatorially regular map of type {3, 7} on an orientable surface of genus g, with trivial automorphism group.) When M is rotary, R and S may be chosen (by replacing one of them by its inverse if necessary) so that the automorphism RS interchanges the vertex v with one of its neighbours along an edge e (on the border of f ), interchanging f with the other face containing e in the process. The three automorphisms R, S and RS may thus be viewed as rotations which satisfy the relations Rp = S q = (RS)2 = 1. a S e v

R f Figure 3. Local action associated with a blade (v, e, f ) in a regular map If a rotary map M admits also an automorphism a which (like RS) ‘ﬂips’ the edge e but (unlike RS) preserves the face f , then we say the regular map M is reﬂexible. This automorphism a is may be thought of geometrically as a reﬂection,

GROUP ACTIONS WITH MAXIMUM SYMMETRY

75

about an axis passing through the midpoints of the edge e and the face f . Similarly, the automorphisms b = aR and c = bS may also be thought of as reﬂections, and the following relations are satisﬁed: a2 = b2 = c2 = (ab)p = (bc)q = (ca)2 = 1. In this case, Aut M is transitive (indeed regular) on blades, and can be generated by the three reﬂections a, b and c. If the map M is orientable, then the elements R = ab and S = bc generate a normal subgroup of index 2 in Aut M , consisting of all elements expressible as a word of even length in {a, b, c}, called the rotation subgroup Aut+ M . In this case the elements of Aut+ M are precisely those automorphisms which preserve the orientation of the underlying surface, while all those in Aut M \ Aut+ M are orientation-reversing. In the non-orientable case, however, there are no true reﬂections: every ‘reﬂection’ is a product of rotations. In particular, each of a, b and c is expressible as a word in the rotations R and S, and hence R, S = ab, bc has index 1 in Aut M . On the other hand, if no such automorphism a exists, then the rotary map M is called chiral , and its automorphism group is generated by the rotations R and S. Chiral maps are necessarily orientable. Also chiral maps occur in opposite pairs, with one member of each pair obtainable from the other by reﬂection. Further details and some historical background may be found in [29, 38, 60]. 4.2

Genus calculation

The genus of a map M is deﬁned as the genus of the surface on which M is embedded, and is given by the usual formula in terms of the Euler characteristic: 2 − 2g if M is orientable χ(M ) = |V | − |E| + |F | = 2−g if M is non-orientable. For regular maps of type {p, q}, counting the number of blades containing a given edge e yields |Aut M | = 2|E| if the rotary map M is chiral, or |Aut M | = 4|E| when M is reﬂexible. Also in both cases, counting the number of darts incident with a given vertex, edge or face gives q|V | = 2|E| = p|F |. These together with the formula above make the calculation of the genus straightforward:   |Aut M |(1/8 − 1/4p − 1/4q) + 1 if M is orientable and reﬂexible g = g(M ) = |Aut M |(1/4 − 1/2p − 1/2q) + 1 if M is orientable but chiral  |Aut M |(1/4 − 1/2p − 1/2q) + 2 if M is non-orientable. As similarly observed for Hurwitz’s theorem, in all cases the bracketed expression attains its smallest positive value when {p, q} = {3, 7}, and thus regular maps of types {3, 7} and {7, 3} have the largest possible symmetry groups. Indeed: Theorem 4.1 If X is a reﬂexible regular map of genus g > 1, then 168(g − 1) if M is orientable |Aut X| ≤ 84(g − 2) if M is non-orientable, and moreover, the upper bound on this order is attained if and only if Aut X is a homomorphic image of the extended (2, 3, 7) triangle group A, B, C | A2 = B 2 = C 2 = (AB)2 = (BC)3 = (CA)7 = 1 .

76

CONDER

Note that the topological dual of a regular map M , denoted by M ∗ and obtainable by taking V (M ∗ ) = F (M ), E(M ∗ ) = E(M ) and F (M ∗ ) = V (M ) and the same relations of incidence, will also be regular, with the same automorphism group as M , and of type {q, p}. Hence types {3, 7} and {7, 3} are equivalent. Similarly the orders 2, 3 and 7 of the pairwise products AB, BC and CA in the above presentation for the extended (2, 3, 7) triangle group can be permuted among themselves while still deﬁning the same group. 4.3

Group theoretic construction of regular maps

In the background analysis described in Section 4.1, the three reﬂections a, b and c generating the automorphism group of a regular map M were chosen so that with respect to the given blade (v, e, f ), the automorphism a stabilises the edge e and the face f but moves the vertex v, while b = aR ﬁxes v and f but moves e, and c = aRS ﬁxes v and e but moves f . Accordingly, V = b, c is the stabilizer in G = Aut M of the vertex v, while E = a, c is the stabilizer in G of the edge e, and F = a, b is the stabilizer in G of the face f. Also vertices, edges and faces of M can be identiﬁed with (right) cosets of these subgroups V , E and F , with incidence corresponding to non-trivial intersection. This background theory can be exploited to produce a purely group-theoretic method of construction of examples of reﬂexible regular maps. Suppose G is any group generated by three involutions a, b and c such that ac has order 2, and ab and bc have orders greater than 2, say p and q respectively. Then the vertices, edges and faces of a map M = M (a, b, c) may be taken as the right cosets in G of the subgroups V = b, c, E = a, c and F = a, b respectively, and incidence deﬁned by non-empty intersection of these cosets. Then the group G = a, b, c acts as a group of automorphisms of M , and transitively (and hence regularly) on its blades. Unless degenerate, this map M is regular of type {p, q}. Also M is orientable if the subgroup ab, bc of G has index 2 in G, and nonorientable if this index is 1. Now using this correspondence between regular maps and generators for their automorphism groups, one can set about ﬁnding regular maps on surfaces of up to given genus by determining groups with the appropriate properties — or more speciﬁcally, non-degenerate ﬁnite homomorphic images of the extended (2, p, q) triangle groups A, B, C | A2 = B 2 = C 2 = (AB)2 = (BC)p = (CA)q = 1 . As an illustration, we have the following well-known construction for regular maps on orientable surfaces of every possible genus: Example 4.1 For any positive integer m, let G be the dihedral group of order 8m, generated by elements u and v such that u2 = v 4m = (uv)2 = 1. Taking involutions a = u, b = uv, and c = uv 2m , we have pairwise products ab = v and bc = v 2m−1 of order 4m, and ca = v 2m of order 2. The corresponding regular map M = M (a, b, c) is orientable since ab, bc = v has index 2 in G. Its Euler characteristic is χ = |D4m |(1/8m − 1/4 + 1/8m) = 2 − 2m. Thus for every m ≥ 1 there exists a regular map of type {4m, 4m} on an orientable surface of genus m.

GROUP ACTIONS WITH MAXIMUM SYMMETRY

77

Also regular maps of type {3, 7} (with the maximum possible number of symmetries for given genus) can be constructed from non-degenerate quotients of the extended (2, 3, 7) triangle group. In fact there are inﬁnitely many orientable regular maps of type {3, 7}, and also inﬁnitely many non-orientable regular maps of type {3, 7}. Macbeath’s 1969 theorem [45] provides inﬁnitely many orientable examples with Aut M ∼ = PGL2 (q) or PSL2 (q) × C2 for various q, and inﬁnitely many non-orientable examples with Aut M ∼ = PSL2 (q) for some q, depending on the choice of an involution which inverts the Hurwitz generators of PSL2 (q) in each case. Similarly the author’s 1980 theorem [9] provides orientable examples with Aut M ∼ = Sn or An × C2 for all n ≥ 168, and also non-orientable examples with Aut M ∼ = An for all n ≥ 168, as both An and Sn are obtainable as homomorphic images of the extended (2, 3, 7) triangle group in such a way that the ordinary (2, 3, 7) triangle group maps onto An , for all n ≥ 168, and many smaller n besides. In addition, we note here that Macbeath and Singerman developed ways of constructing inﬁnite families of examples of covering maps of a given example of a regular map of type {3, 7}; see [52] for details. A related method works as follows, to produce an inﬁnite family of semi-direct products of cyclic groups by a given rotation group under certain circumstances: Construction 4.1 Suppose H is any ﬁnite group which can be generated by two elements x and y of orders 2 and p respectively, where p is even, such that y ∈ xy, y 2 . Also let K be a cyclic group of arbitrary order n, generated say by z. Now form the semi-direct product KH of K by H, with H acting on K so that x and y both invert z, and let G be the subgroup generated by X = zx and Y = y. Because xzx = z −1 , we see that X has order 2, and clearly Y has order p. Further, if xy has order q then (XY )q = (zxy)q = z q (xy)q = z q , so XY has order qm, where m = n/ gcd(n, q), the order of z q . Thus G is a (2, p, qm)-generated group, having the original (2, p, q)-generated group H as a quotient. For increasing n, we obtain a family of such groups, with orders in arithmetic progression. This construction (given in [16]) has numerous applications. For example, taking H as the dihedral group D4 = x, y | x2 = y 4 = (xy)2 = 1 gives a family of rotation groups of order 8n acting on orientable rotary maps of type {4, 2n} and genus n − 1, for every n > 1; these are known as Accola-Maclachlan groups, and will be referred to again in Section 4.6. 4.4

Non-orientable regular maps

Under similar conditions, the construction described above can be taken further, to provide a means of taking a regular map M whose automorphism group is a quotient H of the extended (2, p, q) triangle group, for even p, and producing from this an inﬁnite family of regular maps of type {p, qm} for increasing m, each being an m-fold cover of the base map M of type {p, q} associated with H. For example, if H is the octahedral group S4 , which is a quotient of the extended (2, 3, 4) triangle group A, B, C | A2 = B 2 = C 2 = (AB)2 = (BC)3 = (CA)4 = 1 via a homomorphism which takes A to (1,2)(3,4), B to (3,4) and C to (2,3), this

78

CONDER

construction gives a family of groups of order 24m which are the automorphism groups of non-orientable regular maps of type {4, 3m} and genus 3m − 2, for every m > 1. In particular, this shows that regular maps exist on non-orientable surfaces of every genus g ≡ 1 mod 3. Further details are given in [19], where it is shown how the same method of construction (with variable choice of H) can be used to prove that there exist ﬁnite regular maps on non-orientable surfaces of over 77.5% of all possible genera. The complete genus spectrum of non-orientable regular maps is not known. Apart from genus 2 and 3 (which are somewhat trivial exceptions), it is known that no such maps exist on non-orientable surfaces of genus 18, 24, 27, 39 or 48 (by unpublished work of Antonio Breda and Steve Wilson). Also recently Wilson and the author have shown there is no such map of genus 87. This may be contrasted with orientable regular maps, which are known to exist for all possible genera (see Example 4.1, although it should be noted here that the underlying graphs of maps in this family have multiple edges). We will return to this matter later. 4.5

Regular maps of small genus

A slightly diﬀerent way of looking at the automorphism groups of reﬂexible regular maps is to consider them as non-degenerate ﬁnite images of the extended (2, ∞, ∞) triangle group Φ = A, B, C | A2 = B 2 = C 2 = (AB)2 = 1 , under a homomorphism θ taking A to c, B to a, and C to b. By Theorem 4.1, the automorphism group of any such map M is a homomorphic image of Φ of order at most 168(g − 1) if M is orientable of genus g > 1, or 84(g − 2) if M is non-orientable of genus g > 2. If G = a, b, c is any such image, then of course we take p and q to be the orders of the images ab and bc (of BC and CA) in order to obtain the type (and hence also the genus) of the map, and we still consider the index of the image ab, bc of the subgroup BC, CA in order to determine orientability. Note that if the latter index is 1, then each of a, b and c is expressible as a word in ab and bc, or equivalently, these three involutory generators of G satisfy some relation in which the total number of occurrences of a, b and c is odd, and hence the kernel of the homomorphism θ : Φ → G contains some word in the generators A, B, C of Φ of odd length. Conversely, if ker θ contains such an odd-length word, then the corresponding word in the generators of G will be trivial and so one (and hence all) of a, b, c will lie in the subgroup ab, bc. These observations have been used together with a search for normal subgroups of low index in the group Φ (and related groups) in the determination of all orientable regular maps of genus 2 to 15 inclusive, and all non-orientable regular maps ´ ka system (as of genus 3 to 30 inclusive, with the help of Peter Dobcsanyi’s Kala described in Section 2.3). In fact rather than search for all normal subgroups of index up to 168×14 = 2352 in the extended (2, ∞, ∞) triangle group Φ, the search was broken down into four sub-searches, for normal subgroups of index up to 2352 in the extended (2, 3, 7) triangle group, index up to 1344 in the extended (2, 3, ∞) triangle group, index up to 1120 in the extended (2, 4, ∞) triangle group, and index up to 560 in Φ itself.

GROUP ACTIONS WITH MAXIMUM SYMMETRY

79

The same approach works for chiral maps — which are orientable and have the maximum possible number of rotational symmetries but no reﬂective symmetries — by considering homomorphic images of the ordinary (2, ∞, ∞) triangle group x, y, z | x2 = xyz = 1 which take y and z to the rotational symmetries R and S described in Section 4.1. Here one needs a way of determining chirality (or irreﬂexibility), which is equivalent to the non-existence of an involutory automorphism of the rotation group G+ = R, S inverting each of R and S (as in mirror reﬂection). This however is quite strightforward to check, by replacing all occurrences of R and S in the deﬁning relations for G+ by their inverses, and checking whether or not the resulting words remain as relations. If all the new words are relations, then the map is reﬂexible and so can be eliminated, while on the other hand if some relation becomes a non-relation under this substitution, then no reﬂection exists and so the map is chiral. Such a test can easily be built into a post-processing phase of the normal subgroups adaptation of the low index subgroups process, if desired. The details and results for all three types of regular map of small genus may be found in [25].

4.6

Group actions on non-orientable surfaces

Hurwitz’s theorem gives an upper bound of 84(g − 1) on the number of conformal automorphisms of a compact orientable surface X of given genus g > 1. This maximum is achieved for inﬁnitely many but relatively few genera. It is also interesting to ask for a lower bound on the maximum number of conformal automorphisms of a compact orientable surface of given genus. The answer to this question was obtained independently by Accola [1] and Maclachlan [46], who proved that if µ(g) denotes the largest number of conformal automorphisms of a compact Riemann surface of genus g, then µ(g) ≥ 8g + 8 for all g > 1, and this lower bound on µ(g) is sharp for inﬁnitely many g. For non-orientable surfaces, David Singerman [52] proved the following analogue of Hurwitz’s theorem in 1969: Theorem 4.2 If X is a compact non-orientable surface of genus p > 2, then |Aut X| ≤ 84(p − 2), and moreover, the upper bound on this order is attained if and only if there exists a homomorphism from the extended (2, 3, 7) triangle group onto Aut X which maps the ordinary (2, 3, 7) triangle group also onto Aut X. It is also natural to ask for an analogue of the Accola-Maclachlan theorem: what is a lower bound on the maximum number of automorphisms of a compact nonorientable surface X of given genus p > 2? A partial answer has been provided by the author in joint work with Colin Maclachlan and Steve Wilson: if the maximum number is ν(p), then ν(p) ≥ 4p if p is odd, while ν(p) ≥ 8(p − 2) if p is even. Further reﬁnements are also possible

80

CONDER

for genus p in speciﬁc residue classes  8(p + 2) if    8(p − 2) if ν(p) ≥ 6(p + 1) if    4p if

mod 12. Indeed: p≡1 p≡2 p≡9 p≡3

mod mod mod mod

3 3 or p ≡ 0 mod 6 12 12.

This work has not yet been published, but has been the subject of a recent PhD thesis [53] by Sanja Todorovic-Vasiljevic, who has proved that each of these bounds is sharp for inﬁnitely p in the corresponding residue class mod 12, with the possible (but unlikely) exception of the last case, of genus p ≡ 3 mod 12. To explain this further, some more background material is needed. If X is a compact non-orientable surface of genus p > 2, then X ∼ = U/Λ, where U is the upper-half complex plane, and Λ is a subgroup of PGL2 (IR) containing both conformal and anti-conformal homeomorphisms of U, and acting on U without ﬁxed points. Also Aut X ∼ = Γ/Λ where Γ is a discrete subgroup of PGL2 (IR), equal to the normaliser of Λ in PGL2 (IR), and known as a non-Euclidean crystallographic group (or NEC group). Every NEC group is either Fuchsian (contained in PSL2 (IR)) or proper (not contained in PSL2 (IR)). As Λ contains anticonformal homeomorphisms of U, both Λ and Γ are proper NEC groups, and the natural homomorphism from Γ to Γ/Λ ∼ = Aut X maps the index 2 subgroup Γ+ = Γ ∩ PSL2 (IR) onto G = Aut X. Conversely, if Γ is any proper NEC group, and θ is a homomorphism from Γ to any ﬁnite group G such that the kernel of θ is a non-orientable surface group and θ maps Γ+ = Γ ∩ PSL2 (IR) onto G, then the orbit space X = U/ ker θ is a nonorientable surface on which the group G acts faithfully as a group of automorphisms. The genus of the surface X depends on the signature of Γ, corresponding to the analytic structure which is determined largely by ﬁxed circles of reﬂections in Γ and branch points of Γ+, of the form (γ; ±; [m1 , . . . , mτ ]; {(ni1 , . . . , nisi ) : 1 ≤ i ≤ k}). Each signature determines a deﬁning presentation for the NEC group Γ in terms of certain elements, the orders of some of which must be preserved by the homomorphism θ : Γ → G. That being the case, the genus p and the Euler characteristic χ of the associated surface X are given by the Riemann-Hurwitz equation 2 − p = χ = |G|ξ, where in the + case ξ = 2 − 2γ − k −

τ i=1

(1 − 1/mi ) −

si k

(1 − 1/nij )/2 ,

i=1 j=1

while in the − case, 2 − 2γ − k is replaced by 2 − γ − k. For example, if p is any odd integer > 2 then there is a homomorphism from the NEC group with signature (0; +; [−]; {(2, 2, 2, p)}) to the dihedral group D2p of order 4p, preserving the orders of appropriate elements, and mapping the conformal subgroup onto D2p . Thus D2p acts faithfully on a non-orientable surface of characteristic χ = 4p(−1/4 + 1/2p) = 2 − p and genus p. This gives the lower bound ν(p) ≥ 4p for odd p, and the other bounds listed above can be obtained in a similar fashion.

GROUP ACTIONS WITH MAXIMUM SYMMETRY

81

Also if p ≡ 3 mod 6 and p − 2 is prime, then there exists a homomorphism from the NEC group with signature (0; +; [2, 3]; {(1)}) onto the semi-direct product Cp−2 · C6 having the required properties, and hence there exits a compact nonorientable surface of genus p with an automorphism group of order 6(p − 2). When p ≡ 3 mod 6 and p − 2 = m2 , a perfect square, a similar homomorphism onto (Cm ×Cm )·C6 provides another such group of order 6(p−2), and so ν(p) ≥ 6(p−2) whenever p − 2 is a product of integer squares and primes congruent to 1 mod 6. This, however, still leaves inﬁnitely many p congruent to 3 modulo 12 for which the bound ν(p) ≥ 4p appears to be the best possible. To prove sharpness of this (or other bounds listed above), we may assume the bound on ν(p) is exceeded, which in turn gives a lower bound on the Euler characteristic χ in terms of the order of the group G = Aut X. This severely restricts the possibilities for the signature of the NEC group Γ associated with X. For example assuming |G| > 4p gives χ > −|G|/4, which restricts the signature to one of the following: (a)

(1; −; [2, 3]; {})

(b)

(0; +; [2, 3]; {(1)})

(c)

(0; +; [2]; {(n1 , n2 )}) where 1/2 + 1/p < 1/n1 + 1/n2 < 1

(d)

(0; +; [3]; {(2, 2)})

(e)

(0; +; [m]; {(n)}) where 1/4 + 1/p < 1/m + 1/2n < 1/2

(f)

(0; +; [−]; {(2, n1 , n2 )}) where 1/p < 1/n1 + 1/n2 < 1/2

(g)

(0; +; [−]; {(3, n1 , n2 )}) where 1/6 + 1/p < 1/n1 + 1/n2 < 2/3

(h)

(0; +; [−]; {(4, n1 , n2 )}) where 1/4 + 1/p < 1/n1 + 1/n2 < 3/4

(i)

(0; +; [−]; {(5, n1 , n2 )}) where 3/10 + 1/p < 1/n1 + 1/n2 < 4/5

(j)

(0; +; [−]; {(2, 2, 2, n)}) where 3 ≤ n < p

(k)

(0; +; [−]; {(2, 2, 3, n)}) where 3 ≤ n ≤ 5.

Many of these cases can be eliminated in a number of diﬀerent ways for inﬁnitely many p of the form M q + 2 where M and q are primes, each congruent to 11 mod 12, with M ﬁxed (and small) and q large and variable, and satisfying certain other conditions such as q ≡ 1 mod M . Frequently the Riemann-Hurwitz equation gives |G| = µ(p − 2) = µM q where µ is a rational number whose denominator fails to divide M q, or gives |G| as an integer which fails to be divisible by the orders (periods) of the generators prescribed by the corresponding signature. Also if |G| = 6(p − 2) = 6M q then by choice of M and q and Sylow theory the group G has a cyclic normal subgroup of order 3M q = 3(p − 2), which is contrary to Bujalance’s 1983 theorem on maximal cyclic group actions [7]; thus |G| = 6(p − 2), eliminating signature types (a), (b) and (d). In some cases more advanced methods are required. For example, often Sylow theory shows G has a cyclic normal subgroup K of order q, with CG (K) of low index, and then by the Schur-Zassenhaus theorem CG (K) has a quotient of order q, while the Reidemeister-Schreier process shows that all subgroups of low index are generated by elements of order coprime to q, making this impossible. Similarly a

82

CONDER

theorem of Schur on the transfer (to the eﬀect that the exponent of the commutator subgroup G divides the index |G : Z(G)| of the centre of a group G) may be used to limit q to ﬁnitely many possibilities, for certain signatures. The main diﬃculty lies with case (f), involving signature (0; +; [−]; {(2, m, n)}) for large m and n. Here the corresponding NEC group presentation is nothing other than A, B, C | A2 = B 2 = C 2 = (AB)2 = (BC)m = (CA)n = 1 , which is the extended (2, m, n) triangle group, associated with regular maps of type {m, n}. Hence proving sharpness of the bound ν(p) ≥ 4p for p ≡ 3 mod 12 is intimately associated with the question of non-existence of regular maps on non-orientable surfaces of inﬁnitely many genera congruent to 3 mod 12. See [28, 53] for further details.

5 5.1

Symmetric graphs Definitions and background

Let X = (V, E) be an undirected simple graph. A symmetry (or automorphism) of X is a permutation of its vertices preserving adjacency, that is, a bijection π : V → V with the property that {π(x), π(y)} ∈ E whenever {x, y} ∈ E. Under composition, the symmetries of a graph X form a group called the automorphism group of X, and denoted by Aut X. Finite graphs with maximum symmetry are very easy to classify: the largest possible number of automorphisms of a graph on n vertices is n!, and this is achieved only by the null graph Nn (which has no edges) and its complement the complete graph Kn (in which every two vertices are joined by an edge). These examples, however, are rather uninteresting, and graphs of more frequent attention are those which lie in between these two extremes but have an automorphism group which acts transitively on vertices, edges, arcs, or directed walks of a given length. If Aut X has a single orbit on vertices, then the graph X is said to be vertextransitive. Similarly if Aut X is transitive on the edges of X, or on arcs (directed edges) of X, then X is edge-transitive or arc-transitive respectively. Taking this further, an s-arc in a graph X is deﬁned as an ordered (s + 1)tuple (v0 , v1 , . . . , vs ) of vertices of X such that any two consecutive vi are adjacent in X and any three consecutive vi are distinct, that is, {vi−1 , vi } ∈ E(X) for 1 ≤ i ≤ s and vi−1 = vi+1 for 1 ≤ i < s. If Aut X has just a single orbit on s-arcs then X is said to be s-arc-transitive. Thus 0-arc-transitivity is the same as vertex-transitivity, and 1-arc-transitivity the same as arc-transitivity. Connected 1-arc-transitive graphs are also called symmetric. Note that symmetric graphs are necessarily vertex-transitive, and therefore regular (in the sense that every vertex has the same degree, or valency). Also note that under the assumption of connectedness, s-arc-transitivity implies (s − 1)-arc-transitivity, for all s ≥ 1. Examples 5.1 • The complete graph Kn is vertex-, edge- and arc-transitive, for all n ≥ 3, but is 2-arc-transitive only when n = 3 (as there are two types of 2-arc when n ≥ 4); • the simple cycle Cn is s-arc-transitive for all s ≥ 0;

GROUP ACTIONS WITH MAXIMUM SYMMETRY

83

• the 1-skeleton of a cube is 2-arc- but not 3-arc-transitive; • the complete bipartite graph Kn,n is 3-arc- but not 4-arc-transitive; • the Petersen graph is 3-arc- but not 4-arc-transitive; • the Heawood graph (the incidence graph of the Fano plane) is 4-arc- but not 5-arc-transitive; • Tutte’s 8-cage (pictured in Figure 4) is 5-arc- but not 6-arc-transitive.

Figure 4. Tutte’s 8-cage The degree of symmetry of a non-null arc-transitive graph X can be measured by the stabilizer H = Gv = {g ∈ G : v g = v} of a vertex v in its automorphism group G = Aut X. Vertices of X can be labelled with cosets of H, and if w is any neighbour of v then there exists an automorphism a ∈ G reversing the edge {v, w}, from which it follows that the vertex v (labelled H) is adjacent to all vertices of the form wh = v ah (labelled Hah) for h ∈ H, as illustrated in Figure 5. Ha H

Hah

Hah Figure 5. Local action (on neighbourhood of H) The vertex labelled Hx is adjacent to the vertex labelled Hy if and only if xy −1 lies in the double coset HaH, and from this it follows that X is connected if and only if HaH generates G. Note also that a2 ∈ Gv = H.

84

CONDER

As with regular maps, this background theory can be exploited to produce a group-theoretic method of construction of examples of symmetric graphs. Given any group G containing a subgroup H and an element a such that a2 ∈ H, we may construct a graph Γ = Γ(G, H, a) on which G acts as an arc-transitive group of automorphisms, as follows: take as vertices of Γ the right cosets of H in G, and join two cosets Hx and Hy by an edge in Γ whenever xy −1 ∈ HaH. Deﬁned in this way, Γ is an undirected graph on which the group G acts as a group of automorphisms under the action g : Hx → Hxg for each g ∈ G and each coset Hx in G. The stabilizer in G of the vertex H is the subgroup H itself, and as this acts transitively on the set of neighbours of H (which are all of the form Hah for h ∈ H), it follows that Γ is symmetric. Furthermore, the degree of any vertex of Γ is equal to |H : H ∩ a−1 Ha|, the number of right cosets of H in HaH, and the graph Γ is connected if and only if the elements of HaH generate G. Also the background theory can be taken much further, to produce some very strong conditions on maximum symmetry. For symmetric graphs of valency 3 (often called trivalent, or cubic), if the automorphism group acts transitively on s-arcs then the order of the vertex-stabiliser must be divisible by 3 × 2s−1 . In 1947 Tutte proved the following remarkable theorem by local analysis: Theorem 5.1 (Tutte [54, 55]) If X is a ﬁnite trivalent graph with arc-transitive automorphism group G, then G acts regularly on the s-arcs of X for some s ≤ 5, and in particular, |Gv | ≤ 48 for all v ∈ V (X). This may be contrasted starkly with the 4-valent case, where generalisations of the graph in Figure 6 show that the stabiliser of a vertex can be arbitrarily large.

Figure 6. An arc-transitive 4-valent graph with large vertex-stabiliser Nevertheless there is still an upper limit on s-arc-transitivity for ﬁnite 4-valent graphs, and indeed for ﬁnite symmetric graphs of valency greater than 2 in general. If X is 2-arc-transitive, then the stabiliser of a vertex v is doubly-transitive on the neighbourhood X(v) of v. Using this observation and the classiﬁcation of doublytransitive ﬁnite permutation groups (based in turn on the classiﬁcation of ﬁnite simple groups), Richard Weiss proved the following spectacular generalisation of Tutte’s theorem in 1981:

GROUP ACTIONS WITH MAXIMUM SYMMETRY

85

Theorem 5.2 (Weiss [56]) If X is a ﬁnite s-arc-transitive graph of degree d > 2, then s ≤ 7, and moreover, if s = 7 then d = 3m + 1 for some positive integer m. In particular, there are no ﬁnite 8-arc-transitive graphs of valency greater than 2. 5.2

The trivalent case

By the work of Tutte, Goldschmidt, Sims, Djokovi´c and others, the automorphism group of every arc-transitive ﬁnite trivalent graph is a factor group of one of seven ﬁnitely-presented groups which can be listed as G1 , G12 , G22 , G3 , G14 , G24 and G5 . Here the group Gs or Gis corresponds to a regular action of the automorphism group on s-arcs, with i = 1 or 2 depending on whether or not the arc-reversing automorphism a described in Section 5.1 can be taken as an involution (see [33]). Presentations for these seven groups may be taken as follows (see [13]): G1 = h, a | h3 = a2 = 1

(the modular group)

G12

= 1, php = h−1 , a−1 pa = p

= h, p, a |

h3

=

p2

=

a2

G22 = h, p, a | h3 = p2 = 1, a2 = p, php = h−1 , a−1 pa = p G3 = h, p, q, a | h3 = p2 = q 2 = a2 = 1, pq = qp, php = h, qhq = h−1 , a−1 pa = q G14 = h, p, q, r, a | h3 = p2 = q 2 = r2 = a2 = 1, pq = qp, pr = rp, (qr)2 = p, h−1 ph = q, h−1 qh = pq, rhr = h−1 , a−1 pa = p, a−1 qa = r G24 = h, p, q, r, a | h3 = p2 = q 2 = r2 = 1, a2 = p, pq = qp, pr = rp, (qr)2 = p, h−1 ph = q, h−1 qh = pq, rhr = h−1 , a−1 pa = p, a−1 qa = r G5 = h, p, q, r, s, a | h3 = p2 = q 2 = r2 = s2 = a2 = 1, pq = qp, pr = rp, ps = sp, qr = rq, qs = sq, (rs)2 = pq, h−1 ph = p, h−1 qh = r, h−1 rh = pqr, shs = h−1 , a−1 pa = q, a−1 ra = s . Conversely, every non-degenerate homomorphic image of one of these seven groups acts arc-transitively on a connected trivalent graph whose vertices may be identiﬁed with cosets of a certain subgroup, namely h in the case of G1 , or h, p in the case of G12 and G22 , or h, p, q in the case of G3 , or h, p, q, r in the case of G14 and G24 , or h, p, q, r, s in the case of G5 . These observations were exploited in [13] to produce the ﬁrst known examples of ﬁnite trivalent symmetric graphs of the types corresponding to G22 and G24 (having no involutory automorphism ﬂipping an edge). More recently, using the normal subgroups adaptation of the low index subgroups algorithm described in Section 2.3, Peter Dobcs´ anyi and the author have determined all ﬁnite trivalent symmetric graphs on up to 768 vertices (see [26]). In particular, this extends the Foster census [6] of such graphs of order up to 512 (compiled largely by hand by by R.M. Foster), conﬁrming the census for graphs of order up to 402 (with just one omission corrected in its publication), and ﬁlling some of the gaps from 408 to 512. Also as a bonus discovery, one of the graphs found in [26] but not in [6], on 448 vertices and of type G22 , is the smallest ﬁnite arc-transitive trivalent graph having no arc-reversing automorphism of order 2.

86

CONDER

Examples of ﬁnite 5-arc-transitive trivalent graphs include Tutte’s 8-cage (on 30 vertices) and Wong’s graph (on 234 vertices), which are described in [3] along with a method (due to John Conway) for constructing an inﬁnite number of covers of any given example. Further examples include many of the sextet graphs constructed by Biggs and Hoare in [4], using projective linear groups PGL2 (p) and PΓL2 (p2 ) for certain primes p. For some time these examples (and their covers) and other bipartite examples were the only ﬁnite 5-arc-transitive trivalent graphs known, until the author of this paper adapted the construction of Tutte’s 8-cage to produce an example on 75600 vertices with S10 as its automorphism group, and then used techniques of coset graph composition to prove that for all but ﬁnitely many positive integers n, examples may be constructed with the alternating group An and the symmetric group Sn as automorphism groups (see [12]). The key to the latter construction comes from the fact that the automorphism group of any ﬁnite 5-arc-transitive trivalent graph is a homomorphic image of the group G5 given above, with the stabiliser of a vertex being the image of the subgroup H = h, p, q, r, s , of order 3 × 24 = 48, and with the image of the element a reversing an edge. The subgroup A = p, q, r, s is normalised by the involution a, has index 3 in H, and its image is the stabiliser of an arc. It can now be observed that in any transitive permutation representation of G5 , all orbits of H have lengths dividing 48, and decompose into orbits of A which are linked together by cycles of the permutations induced by h and a. This observation makes it easy to construct multitudes of transitive permutation representations of G5 of arbitrarily large degree, in a similar way to representations of the (2, 3, 7) triangle group, and hence multitudes of 5-arc-transitive cubic graphs. 5.3

Finite 7-arc-transitive graphs

Weiss’s theorem [56] shows that ﬁnite symmetric graphs of valency greater than 2 are at most 7-arc-transitive, and that 7-arc-transitivity can occur only in cases where the valency is of the form 3m + 1 for integer m. Examples exist with the maximum possible symmetry — indeed the incidence graph of the generalised hexagon associated with the simple group G2 (3m ) is a 7-arc-transitive graph of valency 3m + 1, for all m ≥ 1. The smallest such example is the one associated with G2 (3), on 728 vertices, and larger examples can be constructed as covers of given examples under certain conditions (see [30, 57]). Further, as in the case of ﬁnite 5-arc-transitive cubic graphs, for each k of the form 3m + 1 there exists a generic inﬁnite but ﬁnitely-presented group Rk,7 , with generators prescribed in terms of speciﬁc types of symmetries, such that if X is any ﬁnite 7-arc-transitive graph of valency k, then its automorphism group Aut X must be a homomorphic image of Rk,7 (see [57]). Also conversely, every non-degenerate homomorphic image of Rk,7 acts 7-arc-transitively on a connected k-valent graph whose vertices may be identiﬁed with cosets of a certain subgroup. In response to a challenge by Norman Biggs, the author used this information to prove the following, in joint work with Cameron Walker [21]:

GROUP ACTIONS WITH MAXIMUM SYMMETRY

87

Theorem 5.3 For all but ﬁnitely many positive integers n, both the alternating group An and the symmetric group Sn may be represented as 7-arc-transitive groups of automorphisms of ﬁnite connected 4-valent graphs. The proof is based on a careful selection of permutation representations of the generic group R4,7 as building blocks for constructing transitive permutation representations of arbitrarily large degree, as in [12]. The group R4,7 itself may be taken to have generators p, q, r, s, t, u, v, h and b, subject to the following deﬁning relations: h4 = p3 = q 3 = r3 = s3 = t3 = u3 = v 2 = b2 = 1, (hu)3 = (uv)2 = (huv)2 = [h2 , u] = [h2 , v] = 1, [p, q] = [p, r] = [p, s] = [p, t] = [q, r] = [q, s] = [q, t] = [r, s] = [r, t] = 1, [s, t] = p, h−1 ph

= p, h−1 qh = q −1 r, h−1 rh = qr, h−1 sh = pq −1 r−1 s−1 t−1 ,

h−1 th

= p−1 qr−1 s−1 t,

u−1 pu = p, u−1 qu = q, u−1 ru = q −1 r, u−1 su = s, u−1 tu = pqrst, vpv = p−1 , vqv = q −1 , vrv = r, vsv = s, vtv = t−1 , bpb = q −1 , bqb = p−1 , brb = s−1 , bsb = r−1 , btb = u−1 , bub = t−1 , bvb = v, and bh2 b = h2 v. The role of vertex-stabiliser is played by the subgroup H = h, p, q, r, s, t, u, v , which is a semi-direct product of the normal subgroup M = p, q, r, s, t of order 35 by the complementary subgroup L = h, u, v ∼ = GL2 (3) of order 48. In particular, H has order 11664. The generator b takes the role of the arc-reversing involution a in the construction described in Section 5.1, and this normalises the index 4 subgroup K = h2 , p, q, r, s, t, u, v , indeed K = H ∩ b−1 Hb. In any permutation representation of R4,7 , orbits of the subgroup H can be decomposed into orbits of the subgroup K = H ∩ b−1 Hb, which are linked together by 2-cycles of the permutation induced by b, and 2- and 4-cycles of the permutation induced by h. For our construction in [21], we chose as building blocks two transitive permutation representations A and B of R4,7 on 2912 and 8825 points respectively, each having two points ﬁxed by both K and b. Each block in turn was made up of representations of smaller degree, linked together by multiple transpositions of b. Now any suﬃciently large n can be written in the form 2912k + 8825l where k and l are positive integers, since 2912 and 8825 are relatively prime. Taking k copies of the block A and l copies of B, we may link these together into a chain to produce a transitive permutation representation of R4,7 on n points. If the order in which the blocks are linked is chosen carefully, then the permutation induced by bh will have a single cycle of length 107, and lengths of all other cycles will be relatively prime to 107. With the help of Jordan’s theorem, this is enough to show the permutations generate Sn . Similarly by linking a copy of the trivial permutation representation of the subgroup H to one end of the chain, we may obtain An+1 as a non-degenerate factor group of R4,7 as well, proving the theorem.

88

6

CONDER

Some unexpected results/surprises

In this Section we brieﬂy describe two instances of unexpected results of research on graphs with large symmetry groups. The ﬁrst arose in answer to a question posed by Norman Biggs in his continuation of work begun by John Conway, on the result of inserting an extra relator into a generic partial presentation for a group of automorphisms of a 4- or 5-arc-transitive 3-valent graph (corresponding to the presence of a circuit of speciﬁc type). Somewhat surprisingly, it turns out that adjoining the extra relation (ha)12 = 1 to the presentation h, p, q, r, a | h3 = p2 = q 2 = r2 = a2 = 1, pq = qp, pr = rp, (qr)2 = p, h−1 ph = q, h−1 qh = pq, rhr = h−1 , a−1 pa = p, a−1 qa = r given for the group G14 in Section 5 produces a group which is isomorphic to the semi-direct product SL3 (ZZ) : θ, where θ is the inverse-transpose automorphism; see [14] and [30]. In particular, this produces an unexpectedly succinct presentation for the group SL3 (ZZ). In a personal communication, Peter Neumann has given a partial explanation for the isomorphism, associated with a 3-valent graph of incidence between quadrangles and quadrilaterals in a ﬁnite projective plane. The second arose in the construction of a family {Xn } of arc-transitive 4-valent graphs in which the orders of vertex-stabilisers in vertex-transitive subgroups of Aut Xn of smallest possible order form a strictly increasing sequence [22]. Here Sierpinski’s gasket (Pascal’s triangle modulo 2) plays an important role, and recognition of the underlying pattern has led to a direct (closed form) deﬁnition [23] of the binary reﬂected Gray codes, simply in terms of binomial coeﬃcients modulo 2.

7

Some open problems

We conclude with a number of open problems in this area of research: Problem 1: Complete the determination of those ﬁnite simple groups which are Hurwitz. (Note: only groups of Lie type remain to be considered.) Problem 2: What is the complete genus spectrum of non-orientable regular maps? Problem 3: Is it true that for every positive integer g there exists a regular map on an orientable surface of genus g such that the underlying graph is simple? (Note: the underlying graphs of the families of examples customarily used to show orientable maps exist for all possible genera have multiple edges.) Problem 4: Prove sharpness of the lower bound of 4p on the maximum number of automorphisms of a non-orientable surface of given genus p, for inﬁnitely many p ≡ 3 modulo 12. (Note: this will involve proving there are inﬁnitely many such p for which there is no non-orientable map of genus p.) Problem 5: Obtain a classiﬁcation of all ﬁnite 2-arc-transitive graphs. (Note: considerable progress has been made on this by Cheryl Praeger.) Problem 6 (proof of the Weiss conjecture): Prove that the order of the vertexstabiliser in any vertex-transitive and locally primitive group of automorphisms of

GROUP ACTIONS WITH MAXIMUM SYMMETRY

89

a ﬁnite, connected, non-bipartite graph is bounded by a function of the valency. More speciﬁcally, prove there exists an integer-valued function f such that if G is any group of automorphisms of a ﬁnite, connected, non-bipartite graph X such that G is transitive on the vertices of X and the stabilizer in G of a vertex v is primitive on the neighbourhood X(v) of v, then |Gv | < f (d) where d = |X(v)|. (Note: Cheryl Praeger and Cai Heng Li and have reduced this problem to the case where the socle soc(G) is simple and vertex-transitive on X; see [24].) Acknowledgement The author is grateful to the organisers of the Groups St Andrews conference for the invitation to speak at the Oxford conference, and to the N.Z. Marsden Fund for its ﬁnancial support. Electronic Availability An implementation of the normal subgroups adaptation of the low index subgroups process is available (together with tables of results obtained from it) at http://www.scitec.auckland.ac.nz/∼peter. References [1] R.D.M. Accola, On the number of automorphisms of a closed Riemann surface, Trans. Amer. Math. Soc. 131 (1968), 398–408. [2] H.R. Brahana, Regular maps and their groups, Amer. J. Math. 49 (1927), 268–284. [3] N.L. Biggs, Algebraic Graph Theory, Cambridge Univ. Press (London, 1974). [4] N.L. Biggs and M.J. Hoare, The sextet construction for cubic graphs, Combinatorica 3 (1983), 153–165. [5] W. Bosma, J. Cannon & C. Playoust, The Magma Algebra System I: the user language, J. Symbolic Comput. (24) (1997), 235–265. [6] I.Z. Bouwer (ed.), The Foster Census, Charles Babbage Research Centre (Winnipeg, 1988). [7] Emilio Bujalance, Cyclic groups of automorphisms of compact non-orientable Klein surfaces without boundary, Paciﬁc J.Math. 109 (1983), 279–289. [8] J. Cohen, On non-Hurwitz groups and noncongruence subgroups of the modular group, Glasgow Math. J. 22 (1981), 1–7. [9] M.D.E. Conder, Generators for alternating & symmetric groups, J. London Math. Soc. (2) 22 (1980), 75–86. [10] M.D.E. Conder, The genus of compact Riemann surfaces with maximal automorphism group, J. Algebra 108 (1987), 204–247. [11] M.D.E. Conder & J. McKay, A necessary condition for transitivity of a ﬁnite permutation group, Bull. London Math. Soc. 20 (1988), 235–238. [12] M.D.E. Conder, An inﬁnite family of 5-transitive cubic graphs, Ars Combinatoria 25A (1988), 95–108. [13] M.D.E. Conder & P.J. Lorimer, Automorphism groups of symmetric graphs of valency 3, J. Combinatorial Theory, Series B 47 (1989), 60–72. [14] M.D.E. Conder, A surprising isomorphism, J.Algebra 129 (1990) 494–501. [15] M.D.E. Conder, Hurwitz groups: a brief survey, Bull. American Math. Society 23 (1990), 359–370. [16] M.D.E. Conder & R.S. Kulkarni, Inﬁnite families of automorphism groups of Riemann surfaces, in: Discrete Groups and Geometry (ed. W.J. Harvey & C. Maclachlan),

90

CONDER

L.M.S. Lecture Note Series 173 (Cambridge University Press, 1992), pp. 47–56. [17] M.D.E. Conder, R.A. Wilson and A.J. Woldar, The symmetric genus of sporadic groups, Proc. Amer. Math. Soc. 116 (1992), 653–663. [18] M.D.E. Conder & G.J. Martin, Cusps, triangle groups and hyperbolic 3-folds, J. Austral. Math. Soc. (Series A) 55 (1993), 149–182. [19] M.D.E. Conder & B.J. Everitt, Regular maps on non-orientable surfaces, Geometriae Dedicata 56 (1995), 209–219. [20] M.D.E. Conder, Asymmetric combinatorially-regular maps, J. Algebraic Combinatorics 5 (1996), 323–328. [21] M.D.E. Conder & C.G. Walker, The inﬁnitude of 7-arc-transitive graphs, Journal of Algebra 208 (1998), 619–629. [22] M.D.E. Conder & C.G. Walker, Vertex-transitive graphs with arbitrarily large vertexstabilizers, J. Algebraic Combinatorics 8 (1998), 29–38. [23] M.D.E. Conder, Explicit deﬁnition of the binary reﬂected Gray codes, Discrete Math. 195 (1999), 245–249. [24] M.D.E. Conder, Cai Heng Li & C.E. Praeger, On the Weiss conjecture for ﬁnite locally primitive graphs, Proc. Edinburgh Math. Soc. (2) 43 (2000), 129–138. [25] M.D.E. Conder & P. Dobcs´ anyi, Determination of all regular maps of small genus, J. Combin. Theory Ser. B 81 (2001), 224–242. [26] M.D.E. Conder & P. Dobcs´ anyi, Trivalent symmetric graphs on up to 768 vertices, J. Combin. Math. Combin. Comput., to appear. [27] M.D.E. Conder, Hurwitz groups with given centre, preprint. [28] M.D.E. Conder, C. Maclachlan, S. Todorovic Vasiljevic & S.E. Wilson, Bounds for the number of automorphisms of a compact non-orientable surface, preprint. [29] H.S.M. Coxeter & W.O.J. Moser, Generators and Relations for Discrete Groups, 4th ed. Springer-Verlag (Berlin), 1980. [30] A. Delgado & R. Weiss, On certain coverings of generalized polygons, Bull. London Math. Soc. 21 (1989), 235–242. [31] L. Di Martino, M.C. Tamburini & A. Zalesskii, On Hurwitz groups of low rank. Comm. Algebra 28 (2000), 5383–5404. [32] M.J. Dinneen & P.R. Hafner, New results for the degree/diameter problem, Networks 24 (1994), 359–367. [33] D.Z. Djokovi´c & G.L. Miller, Regular groups of automorphisms of cubic graphs, J. Combinatorial Theory Ser. B 29 (1980), 195–230. [34] P. Dobcs´anyi, Adaptations, Parallelisation and Applications of the Low-index Subgroups Algorithm, PhD thesis, University of Auckland, 1999. [35] B.J. Everitt, Images of Hyperbolic Reﬂection Groups, PhD thesis, University of Auckland, 1994. [36] B.J. Everitt, Alternating quotients of Fuchsian groups, J. Algebra 223 (2000), 457– 476. [37] D.F. Holt & W. Plesken, A cohomological criterion for a ﬁnitely presented group to be inﬁnite, J. London Math. Soc. (2) 45 (1992), 469–480. [38] G.A. Jones & D. Singerman, Theory of maps on orientable surfaces, Proc. London Math. Soc. (3) 37 (1978), 273–307. [39] G.A. Jones & D. Singerman, Complex functions: An algebraic and geometric viewpoint. Cambridge University Press (Cambridge, 1987). [40] G.A. Jones, Ree groups and Riemann surfaces, J. Algebra 165 (1994), 41–62. [41] G.A. Jones, Characters and surfaces: a survey, The atlas of ﬁnite groups: ten years on (Birmingham, 1995), London Math. Soc. Lecture Note Ser., vol. 249, Cambridge Univ. Press, 1998, pp. 90–118. [42] J. Leech, Generators for certain normal subgroups of (2, 3, 7), Proc. Cambridge Philos.

GROUP ACTIONS WITH MAXIMUM SYMMETRY

91

Soc. 61 (1965), 321–332. [43] A. Lucchini and M.C. Tamburini, Classical groups of large rank as Hurwitz groups, J. Algebra 219 (1999), 531–546. [44] A. Lucchini, M.C. Tamburini and J.S. Wilson, Hurwitz groups of large rank, J. London Math. Soc. (2) 61 (2000), 81–92. [45] A.M. Macbeath, Generators of the linear fractional groups, Number theory (Proc. Sympos. Pure Math., Vol XII, 1967), Amer. Math. Soc., 1969, pp. 14–32. [46] C. Maclachlan, A bound for the number of automorphisms of compact Riemann surface, J. London Math. Soc. 44 (1969), 265–272. [47] Gunter Malle, Hurwitz groups and G2 (q), Canad. Math. Bull. 33 (1990), 349–357. [48] Gunter Malle, Small rank exceptional Hurwitz groups, Groups of Lie type and their geometries (Como, 1993). London Math. Soc. Lecture Note Ser., vol.207, Cambridge Univ. Press (Cambridge, 1995), pp. 173–183. [49] C.E. Praeger, Highly arc transitive digraphs, European J. Combinatorics 10 (1989), 281–292. [50] Chih-han Sah, Groups related to compact Riemann surfaces, Acta Math. 123 (1969), 13–42. [51] C.C. Sims, Computation with ﬁnitely presented groups. Encyclopedia of Mathematics and its Applications, vol. 48. Cambridge University Press (Cambridge, 1994). [52] David Singerman, Automorphisms of compact nonorientable Riemann surfaces, Glasgow Math. J. 12 (1971), 50–59. [53] Sanja Todorovic Vasiljevic, Bounds on the Number of Automorphisms of a Compact Non-orientable Surface of Given Genus, PhD thesis, University of Auckland, 2001. [54] W.T. Tutte, A family of cubical graphs, Proc. Camb. Phil. Soc. 43 (1947), 459–474. [55] W.T. Tutte, On the symmetry of cubic graphs, Canad. J. Math. 11 (1959), 621–624. [56] Richard Weiss, The non-existence of 8-transitive graphs, Combinatorica 1 (1981), 309–311. [57] Richard Weiss, Presentations for (G, s)-transitive graphs of small valency, Math. Proc. Cambridge Philos. Soc. 101 (1987), 7–20. [58] H. Wielandt, Finite Permutation Groups (Academic Press, 1964). [59] R.A. Wilson, The Monster is a Hurwitz group, preprint. [60] S.E. Wilson, Operators over regular maps, Paciﬁc J. Math. 81 (1979), 559–568.

ON DUAL PRONORMAL SUBGROUPS AND FITTING CLASSES A. D’ANIELLO

∗

† ´ and M.D. PEREZ-RAMOS

1

∗

Dipartimento di Matematica e Applicazioni. Universit` a di Napoli, Via Claudio 21, 80125 Napoli (Italia) † ` Departament d’Algebra, Universitat de Val`encia, C/ Doctor Moliner 50, 46100 Burjassot (Val`encia), Spain

1

Introduction

All groups considered in this note are ﬁnite. The analysis of the possible embedding properties of the subgroups in a group is a ﬁrst way of entering into its structure. Normality and subnormality are the most elementary ones. From the study of conjugacy classes of subgroups the property of pronormality arises. A subgroup H of a group G is said to be pronormal in G if, for every element g of G, H and H g are conjugated in their join H, H g . In [17] it was proved that in fact this condition is equivalent to their conjugacy in H, H g N , the smallest normal subgroup of H, H g with nilpotent factor group. This fact motivated the concept of dual pronormality. A subgroup H of a group G is said to be dual pronormal in G if, for every element g of G, F (H, H g ), the Fitting subgroup of H, H g , is contained in H. This property emerges both as a weaker condition than normality and as a dual concept to pronormality. Its inﬂuence on the structure of the groups was initially studied in a series of papers ([4], [5], [6]). Dual pronormal subgroups are close to N -injectors, for the class N of nilpotent groups, and, in this study, groups containing several relevant classes of dual pronormal subgroups were also taken into consideration. This development shows how far dual pronormality is from normality and subnormality and how dual pronormality can provide additional information. In particular, groups in which Carter subgroups are dual pronormal have been studied. This leads to groups in which Carter subgroups and N -injectors coincide. In various previous papers Huppert [12], Mann [16] and Schmidt [19] had investigated groups whose n-maximal subgroups, n > 1, are respectively normal, subnormal and modular. In our context the case when these subgroups are dual pronormal has also been studied. This investigation is taken further in the context of Fitting classes by means of the extension to F-dual pronormality, for a Fitting class F, in [7], [8] and [18]. This property appears in a natural way from the observation that the Fitting subgroup of a group is the radical for the class of nilpotent groups. Fdual pronormal subgroups ﬁnd a fruitfuil setting here. Fischer F-subgroups, and in particular F-injectors, associated to a Fitting class F, in the soluble universe, are F-dual pronormal subgroups. In fact the following property holds: The Fradical of a group G normalizes every F-dual pronormal subgroup of G, whenever 1 Supported by Proyecto PB 97-0674-C02-02 of DGICYT, Ministerio de Educaci´ on y Ciencia, Spain.

DUAL PRONORMAL SUBGROUPS AND FITTING CLASSES

93

either G is soluble or F is subgroup-closed. This allows Fischer F-subgroups of soluble groups, for a Fitting class F, to be characterized as F-dual pronormal F-maximal subgroups. On the other hand, we recall that normal Fitting classes were originally deﬁned in [2] as Fitting classes whose injectors are normal in every soluble group. Later on the study of Fitting classes whose injectors enjoy other embedding properties, such as normal embedding or permutability, has been of interest (see [9]). We also ﬁnd certain diﬀerent local normality concepts between Fitting classes motivated by diﬀerent characterizations of normal Fitting classes in the soluble universe. More precisely we refer to the concepts of normality, strong normality, A-normality and quasinormality (See Deﬁnitions 3.1, 3.3, 3.4, 3.5, and Theorem 3.2). I turns out now that F-dual pronormal subgroups can play the role of injectors in such a way that A-normality appears as a natural generalization of normality corresponding to its original deﬁnition via injectors. To be more exact, initially a Fitting class F of soluble groups was proved to be normal if, and only if, every F-dual pronormal subgroup has respectively one of the following embedding properties: normality, subnormality, pronormality, normal embedding. Notice that normality and A-normality coincide in this case. Subsequently a localization of these results for arbitrary Fitting classes in the ﬁnite universe was carried out (see Theorem 3.6). Normality fails in this development, even in the soluble case. As mentioned, the extension of normality that works in this case is A-normality. These results are involved in a detailed study on the relation among the diﬀerent local normality concepts, in answer to a question proposed by K. Doerk and T.O. Hawkes in ([9], p. 718). Our aim here is to provide an up-to-date account of the achievements regarding dual pronormal and F-dual pronormal subgroups, for a Fitting class F. We refer to[9] for unexplained notation and concepts.

2

Relevant families of subgroups being F-dual pronormal

We refer to [4], [7] and [18] for the next concepts and results. Definition 2.1 Let F be a Fitting class. A subgroup H of a group G is said to be F-dual pronormal in G, and we write H F-dpn G for short, if H, H g F is contained in H, for each g ∈ G. When F = N is the class of nilpotent groups, the N -dual pronormal subgroups are the dual pronormal subgroups. The most signiﬁcant property of F-dual pronormal subgroups is the following one: Theorem 2.2 Let F be a Fitting class and let H be an F-dual pronormal subgroup of a group G. Assume that either G is soluble or F is subgroup-closed. Then GF normalizes H.

´ D’ANIELLO, PEREZ-RAMOS

94

In particular, for any group G, F (G) normalizes every dual pronormal subgroup of G. This property does not characterize F-dual pronormal subgroups. A cyclic subgroup of order 4 of Sym(4), the symmetric group of degree 4, is an example for this. In soluble groups, Fischer F-subgroups and, in particular, F-injectors, associated to a Fitting class F, are F-dual pronormal subgroups. In fact, from Theorem 2.2 the following characterization for Fischer F-subgroups is derived. Theorem 2.3 Let F be a Fitting class and let G be a soluble group. For a subgroup H of G the following statements are equivalent: (i) H is a Fischer F-subgroup of G; (ii) H is an F-dual pronormal F-maximal subgroup of G. For dominant Fitting classes in S, the class of soluble groups, something more can be said: Proposition 2.4 Assume that F is a dominant Fitting class in S. Let G be a soluble group and H an F-dual pronormal subgroup of G such that H ∈ F. Then H I for some F-injector I of G. Assume now that F is a subgroup-closed Fitting class of soluble groups. Then F is a saturated formation (see [9], XI.(1.1)). In this case Fischer F-subgroups and F-injectors coincide and the above characterization holds also for the last ones. Under this hypothesis, the situation is completely diﬀerent if we require every Fprojector to be F-dual pronormal. In fact, if F-projectors are F-dual pronormal subgroups in a soluble group, then F-projectors and F-injectors coincide. It is known that this happens in all soluble groups precisely when F = Sπ for a suitable set of primes π (see [11], p.34). More information can be obtained on a particular group if the members of some relevant family of its subgroups are F-dual pronormal. Theorem 2.5 Let F be a Fitting class containing N and let G be a soluble group. The following statements are equivalent: (i) G is nilpotent; (ii) every Sylow subgroup of G is F-dual pronormal in G; (iii) every maximal subgroup of G is F-dual pronormal in G. Carter subgroups This is taken from [4]. It is known that the Carter subgroups of a soluble group are exactly the N projectors. In particular the following result holds: “A Carter subgroup C of a soluble group G is dual pronormal in G if, and only if, C is an N -injector of G”.

DUAL PRONORMAL SUBGROUPS AND FITTING CLASSES

95

This fact takes us to groups in which N -injectors and N -projectors coincide. It is not possible to give a complete description of such groups because, as P. Hauck pointed out, every group G can be embedded in a group K satisfying this property. Such a group K can be constructed from a Theorem of Alperin and Thompson (see [13], VI.13.7) which states the following: “Given a group G, there exists a group H such that: (1) G can be embedded in H; (2) the Carter subgroups of H coincide with the Sylow p-subgroups of H, for a prime p.” Now if K = Zp H is the regular wreath product of Zp with H, then G is embedded in K and the N -projectors and the N -injectors of K coincide. The following result however provides a characterization of those soluble groups in which the Carter subgroups are dual pronormal and maximal. Theorem 2.6 A Carter subgroup of a soluble group G is maximal and dual pronormal in G if, and only if, there exists a normal subgroup R of G such that R > Z∞ (G), |R/Z∞ (G)| = q γ , G/R is a primitive group of order pα q β and degree pα (p,q are diﬀerent primes, β > 0). Now the following consequences are derived. Corollary 2.7 Let G be a soluble group, G ≤ G such that G = Z(G) × G (in particular if Φ(G) = 1). G possesses a maximal and dual pronormal Carter subgroup if, and only if, there exists R G, with |R| = q γ , G/R is a primitive group of order pα q β and degree pα (p,q are diﬀerent primes, γ, β > 0). Corollary 2.8 Let G be a group with a Sylow tower. If Carter subgroups of G are dual pronormal, then G is a nilpotent group. Corollary 2.9 Let G be a group such that the derived subgroup is nilpotent. If Carter subgroups of G are dual pronormal, then G is a nilpotent group. n-maximal subgroups As mentioned in the introduction, groups in which all n-maximal subgroups, n > 1, enjoy some particular embedding property have been investigated by several authors. Next we collect the results involving groups whose n-maximal subgroups, n > 1, are dual pronormal. This is taken from [5]. The groups considered in this part are soluble. We recall that for a group G, n-maximal subgroups are deﬁned recursively: if U is a maximal subgroup of G, U is said to be 1-maximal in G; for n > 1, a subgroup U is said to be n-maximal in G, if U is (n − 1)-maximal in a maximal subgroup M of G. Theorem 2.10 If every 2-maximal subgroup of a group G is dual pronormal, then either:

´ D’ANIELLO, PEREZ-RAMOS

96 (i) G is nilpotent, or

(ii) G is minimal non nilpotent, |G| = pq β , Gp G, q|p − 1. In particular G is supersoluble and every 2-maximal subgroup is normal in G. With respect to 3-maximal subgroups the following information is obtained. Theorem 2.11 The following statements are equivalent: (a) Every 3-maximal subgroup of a group G is dual pronormal. (b) Every 3-maximal subgroup of a group G is normal. In [19], Satz 3, R. Schmidt proved that the same is true for modularity: more precisely he describes non supersoluble groups in which every 3-maximal subgroup is modular discovering that such groups are exactly groups in which every 3-maximal subgroup is normal, while this equivalence is not true for the case of maximal or 2-maximal subgroups. From the previous theorem and Schmidt’s result the groups whose 3-maximal subgroups are dual pronormal can be described. Corollary 2.12 Let G be a group in which every 3-maximal subgroup is dual pronormal, then one of the following holds: (a) G is supersoluble; (b) |G| = p2 q, p and q are primes or (c) G is the semidirect product of the quaternion group of order 8 and the cyclic group of order 3. Vice versa the groups in b) and c) have normal 3-maximal subgroups. The explicit description of such supersoluble groups is also known. Theorem 2.13 Let G be a supersoluble group. If every 3-maximal subgroup of G is dual pronormal then one of the following holds: (a) G nilpotent with normal 3-maximal subgroups. (b) G minimal non nilpotent, |G| = pq β , Gp G. (c) |G| = rpq β , r = p = q primes, Gr Gq and Gp Gq minimal non nilpotent, Gi Gi Gq , for i = r, p, and if β > 1 Gp , Gr are normal in G. (d) |G| = p2 q β , Gq cyclic, Φ(Gq ) ≤ Z(G), and if β > 1 the subgroups of Gp are normal in G. (e) |G| = pq β+1 , Gp G and one of the following holds: (i) Gq cyclic, [Gq : CGq (Gp )] = q 2 ; (ii) Gq abelian of type (q, q β ), CGq (Gp ) abelian of type (q, q β−1 ); β β−1 (iii) Gq ∼ = a, b|aq = bq = 1, ab = a1+q , β ≥ 4, minimal non q abelian, CGq (Gp ) = a , b; (iv) Gq ∼ = Q8 , |CGq (Gp )| = 4. Conversely the groups in (a), (b), (c), (d) and (e) have normal 3-maximal subgroups.

DUAL PRONORMAL SUBGROUPS AND FITTING CLASSES

97

In particular, for the groups under consideration the number of prime divisors of their order is limited unless G is nilpotent. This fact holds also for n-maximal subgroups, for an arbitrary n. Corollary 2.14 If every n-maximal subgroup of a group G is dual pronormal, then: (a) |π(G)| ≥ n ⇒ G is supersoluble. (b) |π(G)| ≥ n + 1 ⇒ G is nilpotent. C. Casolo and, independently, K. Doerk have proved that in fact theorem 2.11 holds if 3-maximal subgroup is replaced by n-maximal subgroup, for an arbitrary n. Normality and subnormality We refer to [6], [7] and [18] for this part. It is known that a subgroup H of a group G is normal in G if, and only if, H is subnormal and pronormal in G. This is not true if pronormality is replaced by dual pronormality. It is enough to consider the semidirect product [Z3 × Z3 ]SL(2, 3), with respect to the natural action, and the subgroup [Z3 × Z3 ]H, where H is a cyclic subgroup of SL(2, 3) of order 4. Subnormality and F-dual pronormality cannot be compared: an F-dual pronormal subgroup can be non-subnormal, indeed every maximal subgroup containing the F-radical is F-dual pronormal, for any Fitting class F. Notice also that a subnormal subgroup can be non-F-dual pronormal; this happens for instance for a subgroup of order two in Alt(4), and F = N . It is however true that subnormal subgroups containing the F-radical are F-dual pronormal, for any Fitting class F. The following results provide information about groups in which F-dual pronormal subgroups are either normal or subnormal. Proposition 2.15 Let F be a Fitting class and G a group. Suppose that either G is soluble or F is subgroup-closed. Then the following statements are equivalent: (i) Whenever GF ≤ H F-dpn G, then H G. (ii) Whenever H F-dpn G, then H G. (iii) G/GF is a Dedekind group. Proposition 2.16 Let F be a Fitting class and G a group. Suppose that either G is soluble or F is subgroup-closed. Then the following statements are equivalent: (i) Whenever GF ≤ H F-dpn G, then H sn G. (ii) Whenever H F-dpn G, then H sn G. (iii) G/GF is a nilpotent group. Proposition 2.17 Let F be a Fitting class containing N and let G be a soluble group. The following statements are equivalent: (i) G is a t-group (i.e., every subnormal subgroup of G is normal in G); (ii) every subnormal subgroup of G is F-dual pronormal in G.

´ D’ANIELLO, PEREZ-RAMOS

98

As a consequence, soluble t-groups are charaterized as groups in which F-dual pronormality is a transitive relation, for any Fitting class F containing N . We remark that the conclusion of the above proposition is not true in general in the ﬁnite universe. Take for instance G = Alt(5) Z2 , the regular wreath product of Alt(5) with Z2 , and F = N . Also in this result the hypothesis N ⊆ F is necessary. We can consider F = Sp , the class of all p-groups for a prime p, and G a soluble non-t-group with Op (G) = 1.

3

Local normality concepts between Fitting classes

This part is taken from [8] and [18]. We recall ﬁrst some concepts and previous results. Definition 3.1 ([9], IX.(2.13)(b)) A Fitting class F is said to be normal in a class X of ﬁnite groups (or X -normal) if (1) = F ⊆ X and the F-radical GF is F-maximal in G for all G in X . The next result collects the characterizations of the normal Fitting classes in the soluble universe, which motivated the local normality concepts mentioned in the introduction and recalled afterwards. Theorem 3.2 ([9], X.(3.7)) Let π be a non-empty set of primes. The following statements about a Fitting class F of ﬁnite soluble π-groups are equivalent: (a) F is normal in Sπ ; (b) ([15]) For each p ∈ π and G ∈ F, there exists a natural number n ≥ 1 such that Gn Zp ∈ F; (c) ([14]) F ∗ = Sπ ; (d) ([2]) G/GF is abelian for all G ∈ Sπ . Theorem 3.3 Let F and G be Fitting classes with F ⊆ G. The following statements are equivalent: (a) F ∗ = G ∗ , that is, F and G belong to the same Lockett section; (b) [G, Aut(G)] ≤ GF for all G ∈ G (i.e. F is strongly normal in G; see [3]). In particular, F ∗ ⊆ FA. Definition 3.4 ([10], (5.16)) Let F ⊆ G be Fitting classes. F is said to be Anormal in G if G/GF is abelian for all G ∈ G (i.e. G ⊆ FA). It is known that A-normality does not imply strong normality even in the soluble universe. See for instance [1]. On the other hand, consider the class F of all groups G such that the non-abelian component of the socle of G is a direct factor of G, (see [9], p. 714). Then F is normal but not A-normal in E, the class of all ﬁnite groups. Even in the soluble case, notice that N is normal but not A-normal in N 2 .

DUAL PRONORMAL SUBGROUPS AND FITTING CLASSES

99

Definition 3.5 ([10], (5.1)) Let F ⊆ G be Fitting classes. F is said to be quasinormal in G if the following condition holds: If G ∈ F, p is a prime number and G Zp ∈ G, then there exists a natural number m ≥ 1 such that Gm Zp ∈ F. The main results regarding F-dual pronormal subgroups can be summarized as follows: Theorem 3.6 Let F ⊆ G be Fitting classes. The following statements are pairwise equivalent: (i) F is A-normal in G; (ii) whenever GF ≤ H F-dpn G ∈ G, then H G; (iii) whenever GF ≤ H F-dpn G ∈ G, then H is normally embedded in G. If F ⊆ G ⊆ S, then (i),(ii), (iii) are also equivalent to (iv) whenever GF ≤ H F-dpn G ∈ G, then H is pronormal in G. If G ∗ Sp = G ∗ for some prime p, then (i),(ii), (iii) are also equivalent to (iv) whenever GF ≤ H F-dpn G ∈ G, then H is pronormal in G; (v) F is strongly normal in G; i.e., F ∗ = G ∗ . If G = Eπ for some set of primes π, then the above hypothesis holds, and so (i), (ii), (iii) are equivalent to (iv), (v) and also to (vi) whenever GF ≤ H F-dpn G ∈ G, then HsnG. Moreover if either (1) F ⊆ G ⊆ S or (2) at least one of F, G is subgroup closed, then (i),(ii),(iii) are also equivalent to (ii’) whenever H is F-dual pronormal in G ∈ G, then H G. Quasinormality turns out to be a weaker hypothesis. In fact A-normality implies quasinormality but neither normality implies quasinormality nor the converse holds. (See [10], (5.20), (5.3)(b), and [18], (3.20)). In the soluble universe the following result is recovered. Theorem 3.7 For a Fitting class F of soluble groups, the following statements are equivalent: (a) F is normal in S; (b) every F-dual pronormal subgroup of a soluble group is normal; (c) every F-dual pronormal subgroup of a soluble group is subnormal; (d) every F-dual pronormal subgroup of a soluble group is normally embedded; (e) every F-dual pronormal subgroup of a soluble group is pronormal.

´ D’ANIELLO, PEREZ-RAMOS

100 References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]

J. C. Beidleman and B. Brewster, Strict normality in Fitting classes II. J. Algebra 51 (1978), 218-227. ¨ D. Blessenohl and W. Gasch¨ utz, Uber normale Schunck- und Fittingklassen, Math. Z. 118 (1970), 1-8. R.A. Bryce and J. Cossey, A problem in the theory of normal Fitting classes, Math. Z. 141 (1975), 99-110. A. D’Aniello and A. Leone, Su alcune classi di sottogruppi pronormali duali, Boll. Un. Mat. Ital. D (6) 5, no. 1 (1986), 135-144. A. D’Aniello, Groups in which n-maximal subgroups are dual pronormal, Rend. Sem. Mat. Univ. Padova 84 (1990), 83-90. A. D’Aniello, Pronormalit` a duale e normalit`a nella classe dei gruppi ﬁniti risolubili, Rendiconti Istituto Lombardo sez. A (Sc. Mat. Appl.) 131, fasc. 1 (1997). A. D’Aniello, Dual pronormality and Fitting classes, Comm. Algebra 26 (2) (1998), 425-433. A. D’Aniello, On normal Fitting classes, Arch. Math. 71 (1998), 1-4. K. Doerk and T. O. Hawkes, Finite soluble groups (de Gruyter, 1992). P. Hauck, Zur Theorie der Fittingklassen endlicher auﬂ¨ osbarer Gruppen. Ph. D. thesis. Universit¨ at Mainz (1977). T. Hawkes, Finite soluble groups, Group Theory, Essay for P. Hall, edited by K.W. Gruenberg and J.E. Roseblade, Academic Press (1984), 13-60. B. Huppert, Normalteiler und maximale Untergruppen endlicher Gruppen, Math. Z. 60 (1954), 409-434. B. Huppert, Endliche Gruppen I, Springer-Verlag, Berlin 1967. P. Lockett, The Fitting class F ∗ , Math. Z. 137 (1974), 131-136. A.R. Makan, Fitting classes with the wreath product property are normal, J. London Math. Soc. 8 (2) (1974), 245-246. A. Mann, Finite groups whose n-maximal subgroups are subnormal, Amer. Math. Soc. 132 (1968), 395-405. N. M¨ uller, F-Pronormale Untergruppen Endlich Auﬂ¨ osbarer Gruppen, Diplomarbeit, Universit¨ at Mainz (1985). M. D. P´erez-Ramos, On A-normality, strong normality and F-dual pronormal subgroups in Fitting classes, J. Group Theory 3 (2000), 127-145. R. Schmidt, Endliche Gruppen mit vielen modularen Untergruppen, Abh. Math. Sem. Hamburg 34 (1969), 115-125.

(p, q, r)-GENERATIONS OF THE SPORADIC GROUP O’N M. R. DARAFSHEH ∗ , A. R. ASHRAFI

∗∗

and G. A. MOGHANI

†

∗ Department of Mathematics and Computer Science, University of Tehran, Tehran, Iran E-mail: [email protected] ∗∗ Department of Mathematics, Faculty of Science, University of Kashan, Kashan, Iran E-mail: [email protected] † Department of Mathematics, Iran University of Science and Technology, Tehran, Iran E-mail: [email protected]

Abstract A group G is (l, m, n)-generated if it is a quotient group of the triangle group T (l, m, n) = x, y, z|xl = y m = z n = xyz = 1. In this paper we ﬁnd all (p, q, r)generations, p, q and r are distinct primes with p < q < r for the sporadic group O’N. 2000 AMS Subject Classification: Primary 20D08, 20F05. Keywords: O’Nan group, generator, triangle group, sporadic simple group.

1 Introduction A group G is said to be (l, m, n)-generated if it can be generated by two elements x and y such that o(x) = l, o(y) = m and o(xy) = n. In this case G is the quotient of the triangle group T (l, m, n) and for any permutation π of S3 , the group G is also ((l)π, (m)π, (n)π)-generated. Therefore we may assume that l ≤ m ≤ n. By [1], if the non-abelian simple group G is (l, m, n)-generated, then either G ∼ = A5 or 1 1 1 l + m + n < 1. Hence for a non-abelian ﬁnite simple group G and divisors l, m, n 1 + n1 < 1, it is natural to ask if G is a (l, m, n)of the order of G such that 1l + m generated group. The motivation for this question came from the calculation of the genus of ﬁnite simple groups [16]. It can be shown that the problem of ﬁnding the genus of a ﬁnite simple group can be reduced to one of generations(for details see [13]). In a series of papers, [7], [8], [9], [10] and [11] Moori and Ganief established all possible (p, q, r)-generations, where p, q, r are distinct primes, of the sporadic groups J1 , J2 , J3 , HS, M cL, Co3 , Co2 , and F22 . Authors in [4] and [5] solved the similar problem for the sporadic group Co1 . The motivation for this study is outlined in these papers and the reader is encouraged to consult these papers for background material as well as basic computational techniques. Throughout this paper we use the same notation as in [4] and [5]. In particular, ∆(G) = ∆(lX, mY, nZ) denotes the structure constant of G for the conjugacy

102

DARAFSHEH, ASHRAFI, MOGHANI

classes lX, mY, nZ, whose value is the cardinality of the set Λ = {(x, y)|xy = z}, where x ∈ lX, y ∈ mY and z is a ﬁxed element of the conjugacy class nZ. In Table II, we list the values ∆(pX, qY, rZ), p, q and r distinct prime divisors of |ON |. These values are calculated by using the character table of O’Nan. Also, ∆ (G) = ∆G (lX, mY, nZ) and Σ(H1 ∪ H2 ∪ · · · ∪ Hr ) denote the number of pairs (x, y) ∈ Λ such that G = x, y and x, y ⊆ Hi (for some 1 ≤ i ≤ r), respectively. The number of pairs (x, y) ∈ Λ generating a subgroup H of G will be given by Σ (H) and the centralizer of a representative of lX will be denoted by CG (lX). A general Conjugacy class of a subgroup H of G with elements of order n will be denoted by nx. Clearly, if ∆ (G) > 0, then G is (lX, mY, nZ)-generated and (lX, mY, nZ) is called a generating triple for G. The number of conjugates of a given subgroup H of G containing a ﬁx element z is given by χNG (H) (z), where χNG (H) is the permutation character of G with action on the conjugates of H(cf. [14]). In most cases we will calculate this value from the fusion map from NG (H) into G stored in GAP, [12]. Now we discuss techniques that are useful in resolving generation type questions for ﬁnite groups. We begin with a result of [2] that, in certain situations, is very eﬀective at establishing non-generations. Lemma 1.1 Let G be a finite centerless group and suppose lX, mY and nZ are G-conjugacy classes for which ∆ (G) = ∆G (lX, mY, nZ) < |CG (z)|, z ∈ nZ. Then ∆ (G) = 0 and therefore G is not (lX, mY, nZ)-generated. Lemma 1.2 [15] Let G be a finite simple group and H a maximal subgroup of G containing a fixed element x. Then the number h of conjugates of H containing x is χH (x), where χH is the permutation character of G with action on the conjugates of H. In particular, h=

m |CG (x)| |CH (xi )| i=1

where x1 , x2 , · · · , xm are representatives of the H-conjugacy classes that fuse to the G-conjugacy class of x. In the present paper we investigate the (p, q, r)-generations, p, q and r are distinct primes with p < q < r, for the O’Nan group O’N. We prove the following result: Theorem The O’Nan group O’N is (pX, qY, rZ)-generated for all p, q, r ∈ {2, 3, 5, 7, 11, 19, 31} with p < q < r, except when (p, q, r) = (2, 3, 5) and (2, 3, 7).

2 (p, q, r)-Generations for O’N In this section we obtain all of the (pX, qY, rZ)-generation of the O’Nan group O’N. Since 31A−1 = 31B, hence, the group O’N is (pX, qY, 31A)-generated if and only if it is (pX, qY, 31B)-generated. Therefore, it is enough to investigate the (pX, qY, 31A)-generation of O’N. We will use the maximal subgroups of O’N listed in the ATLAS extensively, especially those with order divisible by 19. We listed in Table II, all the maximal

(p, q, r)-GENERATIONS OF THE SPORADIC GROUP O’N

103

subgroups of O’N and in Table III, fusion maps of these maximal subgroups into O’N (obtained from GAP) that will enable us to evaluate ∆ON (pX, qY, rZ), for prime classes pX, qY and rZ. In this table h denotes the number of conjugates of the maximal subgroup H containing a ﬁxed element z (see Lemma 1.2). For basic properties of the group O’N and information on its maximal subgroups the reader is referred to [3]. It is a well known fact that O’N has exactly 13 conjugacy classes of maximal subgroups, as listed in Table II. , T h, Woldar, in [16] determined which sporadic groups other than F i22 , F23 , F24 J4 , B and M are Hurwitz groups, i.e. generated by elements x and y with order o(x) = 2, o(y) = 3 and o(xy) = 7. In fact, G is a Hurwitz group if and only if G is (2, 3, 7)-generated. By his result, O’N is not a Hurwitz group and so O’N is not (2, 3, 7)-generated. Theorem 1 The group O’N is (2X, pY, qZ)-generated if and only if (p, q) = (3, 5) and (3, 7). ∼ A5 . Also as we Proof Obviously O’N is not (2,3,5)-generated since T (2, 3, 5) = mentioned above O’Nan is not (2,3,7)-generated. Set A = {(11, 31), (19, 31), (7, 11), (7, 31)}. By Table II, if (p, q) ∈ A then there is no maximal subgroup of O’N which intersect the triple (2X, pY, qZ), hence the group O’N is (2X, pY, qZ)-generated. Now our main proof will consider a number of cases. In what follows, we mention only one case in detail and the rest will follow similarly. Case (2A, 5A, 7B). From the list of maximal subgroups of O’N, we observe that, up to isomorphisms, J1 and two non-conjugate subgroups isomorphic to A7 are the only maximal subgroups of O’N that admit (2A, 5A, 7A)-generated subgroups. From the structure constants, Table I, we calculate ∆(ON ) = 10682, Σ(J1 ) = 98 and Σ(A7 ) = 28. Thus, ∆ (ON ) ≥ 10682 − 98 · 7 − 28 · 14 − 28 · 14 > 0. This shows that the O’Nan group O’N is (2A, 5A, 7B)-generated. Theorem 2 The group O’N is (3A, pY, qZ)-generated, for all prime classes pY and qZ. Proof Set A = {(7, 31), (11, 31), (19, 31)}. First of all, if (p, q) ∈ A then there is no maximal subgroup of O’N that admit (3A, pY, qZ)-generated subgroups. Now for such a triple (3A, pY, qZ), ∆(ON ) = 0 and so O’N is (3A, pY, qZ)-generated. We can investigate other triples case by case. Here, we mention only one case in detail and the rest will follow similarly. Case (3A, 5A, 7B). From the list of maximal subgroups of O’N we observe that, up to isomorphisms, J1 and two non-conjugate subgroups isomorphic to A7 , are the only maximal subgroup of O’N that admit (3A, 5A, 7B)-generated subgroups. From the structure constants we calculate ∆(ON ) = 554974, Σ(J1 ) = 378 and Σ(A7 ) = 112. Thus, ∆ (ON ) ≥ 554974 − 378 · 7 − 112 · 14 − 112 · 14 > 0. This shows that the O’Nan group O’N is (3A, 5A, 7B)-generated. Theorem 3 The group O’N is (pX, qY, rZ)-generated, p ≥ 5, for all prime classes pX, qY and rZ.

104

DARAFSHEH, ASHRAFI, MOGHANI

Proof Set A = {(7, 31), (11, 31), (19, 31)} and B = {(11, 31), (19, 31)}. First of all, if (p, q) ∈ A then there is no maximal subgroup of O’N that admit (5A, pY, qZ)generated subgroups. Now for such a triple, ∆(ON ) = 0 and so O’N is (5A, pY, qZ)generated. Similarly, if (p, q) ∈ B then there is no maximal subgroup of O’N that admit (7X, pY, qZ)-generated subgroups, and since ∆(ON ) = 0, O’N is (7X, pY, qZ)generated. We can investigate other triples case by case. Here, we mention only one case in detail and the rest will follow similarly. Case (5A, 7B, 19Z), Z ∈ {A, B, C}. Amongst the maximal subgroups of O’N with order divisible by 5 × 7 × 19, the only maximal subgroups of O’N that may contain (5A, 7B, 19Z)-generated proper subgroups are isomorphic to J1 , Z ∈ {A, B, C}. Moreover, ∆(ON ) = 52246656 and Σ(J1 ) = 1672. On the other hand, ∆ (ON ) ≥ 52246656 − 1672 > 0 and so O’N is (5A, 7B, 19A)-, (5A, 7B, 19B)- and (5A, 7B, 19C)-generated.

Table I The Structure Constants of the Group O’N pX 7A 7B 11A 19A 19B 19C 31A pX 19A 19B 19C 31A pX 7A 7B 11A 19A 19B 19C 31A pX 31A

∆(2A, 3A, pX) 869 874 874 874 837 ∆(2A, 11A, pX) 259711 259711 259711 258075 ∆(3A, 5A, pX) 546056 554974 556050 556320 556320 556320 554590 ∆(3A, 19A, pX) 7484950

∆(2A, 5A, pX) 12691 10682 11088 11058 11058 11058 11129 ∆(2A, 19A, pX) 150381 ∆(3A, 7A, pX) 100936 103664 103664 103664 105679 ∆(3A, 19B, pX) 7484950

∆(2A, 7A, pX) 1936 2090 2090 2090 2356 ∆(2A, 19B, pX) 150381 ∆(3A, 7B, pX) 2893044 2902592 2902592 2902592 2916914 ∆(3A, 19C, pX) 7484950

∆(2A, 7B, pX) 57387 58311 58311 58311 59489 ∆(2A, 19C, pX) 150381 ∆(3A, 11A, pX) 12930678 12930678 12930678 12922257 -

(p, q, r)-GENERATIONS OF THE SPORADIC GROUP O’N

105

Table I (Continued) pX 11A 19A 19B 19C 31A pX 31A pX 19A 19B 19C 31A pX 31A

∆(5A, 7A, pX) 1868108 1865952 1865952 1865952 1862976 ∆(5A, 19B, pX) 134747514 ∆(7A, 11A, pX) 30536154 30536154 30536154 30489864 ∆(7A, 19B, pX) 17676758

∆(5A, 7B, pX) 52259592 52246656 52246656 52246656 52244052 ∆(5A, 19C, pX) 134747514 ∆(7B, 11A, pX) 854949403 854949403 854949403 854723289 ∆(7B, 19B, pX) 494965933

∆(5A, 11A, pX) 232742970 232742970 232742970 232777512 ∆(7A, 19A, pX) 17676758 ∆(7A, 19C, pX) 17676758

∆(5A, 19A, pX) 134747514 ∆(7B, 19A, pX) 494965933 ∆(7B, 19C, pX) 494965933

Table II The Maximal Subgroups of O’N Group L3 (7).2 J1 ON M 5 L2 (37) 43 .L3 (2) ON M 11 A7

Order 26 .32 .73 .19 23 .3.5.7.11.19 26 .34 .5 25 .3.5.31 29 .3.7 24 .32 .5.11 23 .32 .5.7

Group ON M 2 4 − 2.L3 (4).2 − 1 34 .21+4 .D10 ON M 8 M11 A7

Order 26 .32 .73 .19 29 .32 .5.7 26 .34 .5 25 .3.5.31 24 .32 .5.11 23 .32 .5.7

106

DARAFSHEH, ASHRAFI, MOGHANI

Table III Partial Fusion Maps into O’N L3 (7).2-class → O’N h L3 (7).2-class → O’N L3 (7).2-class → O’N h ON M 2-class → O’N h ON M 2-class → O’N ON M 2-class → O’N h J1 -class → O’N h J1 -class → O’N h A7 -class → O’N h

2a 2A

2b 2A

3a 3A

4a 4B

4b 4A

6a 6A

6b 6A

7a 7A

8a 8A 19a 19A 1 2a 2A

8b 8A 19b 19B 1 2b 2A

8c 8A 19c 19C 1 3a 3A

12a 12A 28a 28A

14a 14A 28b 28B

14b 14A

16a 16A

16b 16B

4a 4B

4b 4A

6a 6A

6b 6A

7a 7A

8a 8B 19a 19A 1 2a 2A

8b 8B 19b 19B 1 3a 3A

8c 8B 19c 19C 1 5a 5A

12a 12A 28a 28A

14a 14A 28b 28B

14b 14A

16a 16C

5b 5A

6a 6A

7a 7B 7

15b 15B

19a 19A 1 3a 3A

19b 19B 1 3b 3A

19c 19C 1 4a 4B

5a 5A

6a 6A

2a 2A

7b 7A 15 16c 16B

7c 7B 1 16d 16A

16b 16D

7b 7A 15 16c 16D

7c 7B 1 16d 16C

10a 10A

10b 10A

11a 11A

15a 15A

7a 7B

7b 7B 14

(p, q, r)-GENERATIONS OF THE SPORADIC GROUP O’N

107

Table III (Continued) 42 .L3 (4) : 21 -class → O’N 42 .L3 (4) : 21 -class → O’N h 42 .L3 (4) : 21 -class → O’N ON M 5-class → O’N ON M 5-class → O’N ON M 5-class → O’N 34 .21+4 .D10 -class → O’N 34 .21+4 .D10 -class → O’N L2 (37)-class → O’N L2 (37)-class → O’N h A7 -class → O’N h

2a 2A 5a 5A

2b 2A 6a 6A

2c 2A 6b 6A

3a 3A 6c 6A

12a 12A 2a 2A 4d 4B 8d 8B 2a 2A 6a 6A 2a 2A 15d 15B

14a 14A 2b 2A 4e 4B 10a 10A 2b 2A 8a 8A 3a 3A 16a 16A

16a 16A 2c 2A 4f 4B 12a 12A 3a 3A 8b 8B 4a 4B 16b 16B

16b 16B 3a 3A 4g 4B 12b 12A 4a 4A 10a 10A 5a 5A 16c 16A

4a 4A 7a 7A 49 16c 16C 3b 3A 5a 5A 12c 12A 4b 4B 10b 10A 5b 5A 16d 16B

2a 2A

3a 3A

3b 3A

4a 4B

5a 5A

4b 4B 8a 8A

4c 4B 8b 8B

4d 4B 8c 8A

4e 4A 8d 8B

4f 4B 10a 10A

16d 16D 3c 3A 6a 6A 15a 15A 4c 4B 12a 12A 8a 8A 31a 31A 1 6a 6A

20a 20A 3d 3A 6b 6A 15b 15B 4d 4B 12b 12A 8b 8A 31b 31B 1 7a 7B

20b 20B 4a 4A 8a 8A 20a 20A 4e 4B

28a 28A 4b 4A 8b 8A 20b 20B 5a 5A

28b 28B 4c 4B 8c 8B

15a 15A

15b 15B

15c 15A

7b 7B 14

5b 5A

108

DARAFSHEH, ASHRAFI, MOGHANI

Table III (Continued) ON M 8-class → O’N ON M 8-class → O’N h 43 .L3 (2)-class → O’N h 43 .L3 (2)-class → O’N M11 -class → O’N h ON M 11-class → O’N h

2a 2A 15d 15B

3a 3A 16a 16C

4a 4B 16b 16D

5a 5A 16c 16C

5b 5A 16d 16D

8b 8B 31b 31A 1 6a 6A

15a 15A

15b 15B

15c 15A

4b 4B

8a 8B 31a 31B 1 4c 4B

2a 2A

2b 2A

3a 3A

4a 4A

7a 7B

7b 7B 14

8a 8A

8b 8B 2a 2A

12a 12A 3a 3A

12b 12A 4a 4B

16a 16A 5a 5A

16b 16B 6a 6A

16c 16D 8a 8A

16d 16C 8b 8A

11a 11A

5a 5A

6a 6A

8a 8B

8b 8B

11a 11A

11b 11A 2 11b 11A 2

2a 2A

3a 3A

4a 4B

References [1] M. D. E. Conder, Some results on quotients of triangle groups, Bull. Austral. Math. Soc. 30 (1984), 73-90. [2] M. D. E. Conder, R. A. Wilson and A. J. Wolder, The symmetric genus of sporadic groups, Proc. Amer. Math. Soc. 116 (1992), 653-663. [3] J. H. Conway, R. T. Curtis, S. P. Norton and R. A. Wilson, Atlas of Finite Groups, Oxford Univ. Press (Clarendon), Oxford, 1985. [4] M. R. Darafsheh and A. R. Ashraﬁ, (2, p, q)-Generation of the Conway group Co1 , Kumomoto J. Math. 13 (2000), 1-20. [5] M. R. Darafsheh ; A. R. Ashraﬁ and G. A. Moghani, (p, q, r)-Generations of the Conway group Co1 , for odd p, Kumomoto J. Math. 14 (2001), 1-20. [6] M. R. Darafsheh ; A. R. Ashraﬁ and G. A. Moghani, (p, q, r)- and nX-Complementary Generations of the Sporadic Group O’N, Submitted. [7] S. Ganief and J. Moori, (p, q, r)-Generations of the smallest Conway group Co3 , J. Algebra 188 (1997), 516-530. [8] S. Ganief and J. Moori, Generating pairs for the Conway groups Co2 and Co3 , J. Group Theory 1 (1998), 237-256. [9] S. Ganief and J. Moori, 2-Generations of the Fourth Janko Group J4 , J. Algebra 212 (1999), 305-322. [10] J. Moori, (p, q, r)-Generations for the Janko groups J1 and J2 , Nova J. Algebra and Geometry, Vol. 2, No. 3 (1993), 277-285. [11] J. Moori, (2, 3, p)-Generations for the Fischer group F22 , Comm. in Algebra 2(11) (1994), 4597-4610. [12] M. Schonert et al., GAP, Groups, Algorithms and Programming, Lehrstuhl De fur Mathematik, RWTH, Aachen, 1992. [13] A. J. Woldar, Representing M11 , M12 , M22 and M23 on surfaces of least genus, Comm. in Algebra 18 (1990), 15-86.

(p, q, r)-GENERATIONS OF THE SPORADIC GROUP O’N

109

[14] A. J. Woldar, Sporadic simple groups which are Hurwitz, J. Algebra 144 (1991), 443-450. [15] A. J. Woldar, 3/2-Generation of the Sporadic simple groups, Comm. in Algebra 22(2) (1994), 675-685. [16] A. J. Woldar, On Hurwitz generation and genus actions of sporadic groups, Illinois Math. J. (3) 33 (1989), 416-437.

COMPUTATIONS WITH ALMOST-CRYSTALLOGRAPHIC GROUPS KAREL DEKIMPE∗ and BETTINA EICK† ∗ †

Katholieke Universiteit Leuven, Campus Kortrijk, B–8500 Kortrijk, Belgium Fachbereich f¨ ur Mathematik, Universit¨ at Kassel, D–34109 Kassel, Germany

Abstract Recently, algorithmic approaches to construct and investigate almost crystallographic groups and a library of almost crystallographic groups of small Hirsch length have been made available in the Aclib package of Gap. Here we present a survey of these methods and we illustrate a variety of their applications.

1

Introduction

Almost crystallographic groups have ﬁrst been discussed in the theory of actions on connected and simply connected nilpotent Lie groups L. In this setting L Aut(L) acts aﬃnely on L via l(m,α) = lα · m for l, m ∈ L and α ∈ Aut(L). If C is a maximal compact subgroup of Aut(L), then a subgroup G of LC is almost crystallographic if the action of G on L is properly discontinuous and the quotient space L/G is compact. Almost crystallographic groups can also be characterized as those ﬁnitely generated nilpotent-by-ﬁnite groups whose normal torsion subgroup is trivial. One of the most fundamental observations on almost crystallographic groups is that for a given ﬁnitely generated torsion-free nilpotent group N there exist only ﬁnitely many almost crystallographic groups having N as Fitting subgroup. This property can be used as a basis for a classiﬁcation of almost crystallographic groups. In fact, in [2] this approach has been exploited to determine a library of almost crystallographic groups of Hirsch length at most 4. Recently, this library of almost crystallographic groups has been made available in electronic form in the package Aclib [3] of the computer algebra system Gap [20]. In addition, a variety of algorithms to compute with almost crystallographic groups are implemented in the Aclib package. This setup now forms a powerful tool to exploit and construct almost crystallographic groups. Our aim in this paper is to give a survey on the algorithms of the Aclib package and to illustrate a variety of their applications in computations with almost crystallographic groups. In particular, we consider the construction of almost crystallographic groups with certain properties and we perform structural investigations of such groups.

2

Investigating almost crystallographic groups

Recall that a group G is polycyclic if it has a polycyclic series; that is, a subnormal series G = G1 . . . Gn Gn+1 = 1 with cyclic factors Gi /Gi+1 . The number of inﬁnite cyclic factors in such a series is an invariant of G which is called its Hirsch

COMPUTING WITH ALMOST-CRYSTALLOGRAPHIC GROUPS

111

length denoted by hl(G). A variety of algorithms to compute with polycyclic groups has been introduced recently in [6] and implementations of many of these methods are available as part of the Polycyclic package [10] of Gap. Let G be an almost crystallographic group with Fitting subgroup N . By construction, the factor G/N is a ﬁnite group which is called the holonomy group of G. The subgroup N is ﬁnitely generated nilpotent and torsion-free. Thus, in particular, N is polycyclic. Further, G is polycyclic if and only if G/N is polycyclic. This is an often arising case in which we obtain that hl(G) = hl(N ). More generally, we deﬁne the Hirsch length of G as the Hirsch length of N . In our applications we want to compute with polycyclic almost crystallographic groups using the methods of the Polycyclic and the Aclib packages. For this purpose we ﬁrst consider in Section 2.1 representations for such groups which are suitable for this purpose. Then, in Section 2.2, we discuss applications of our methods. 2.1

Representations of polycyclic almost-crystallographic groups

The computationally most useful representation for a polycyclic group is by a (consistent) polycyclic presentation; that is, a presentation of the form G = g1 , . . . , gn

|

g

gi j = wi,j (gj+1 , . . . , gn ) for 1 ≤ j < i ≤ n, g −1

gi j = wj,i (gj+1 , . . . , gn ) for 1 ≤ j < i ≤ n, j ∈ I giri = wi,i (gi+1 , · · · , gn )

for i ∈ I

where I ⊆ {1, . . . , n}, the exponents ri ∈ N for i ∈ I and the right hand sides wi,j of the relations are words in the considered generators and their inverses. In particular, the methods for polycyclic groups implemented in the Polycyclic package of Gap require such a presentation for the considered groups. Thus, if we want to use these methods for our purposes, then we need to determine such a presentation for a given almost crystallographic group as a ﬁrst step. Almost crystallographic groups are often described as rational matrix groups. In this case we can use the methods of [18] to check whether a considered group is polycyclic and, if so, then we can determine a polycyclic presentation for it. For example, the almost crystallographic groups in [2] are described as rational matrix groups. In fact, these groups are always unipotent-by-ﬁnite which simpliﬁes the determination of a polycyclic presentation signiﬁcantly. Another important representation for almost crystallographic groups is by ﬁnite presentations. In general, it is an algorithmically undecidable problem to check whether a given ﬁnitely presented group is polycyclic. If a considered group is known to be polycyclic, then the algorithm of [16] can be used to determine polycyclic presentations for it. 2.2

Polycyclically presented almost crystallographic groups

Suppose that a polycyclically presented group G is given. Then the methods of the Polycyclic package apply to G. As a ﬁrst application, we can check whether G is almost crystallographic. For this purpose we • determine the normal torsion subgroup T N (G) and check that T N (G) = 1,

112

DEKIMPE, EICK

• determine the Fitting subgroup F it(G) and check that [G : F it(G)] < ∞. Methods to determine of the order, the Hirsch length or the index of a subgroup of a polycyclically presented group are described in [19] and algorithms to compute T N (G) and F it(G) are introduced in [7] and [8]. A summary of these methods is provided in [6]. Further, the following theorem can be used to replace the determination of F it(G) by a more eﬀective approach. Recall that each free abelian subfactor A of a group G is a ZG-module. We call such a module rationally semisimple if Q ⊗ A is semisimple as a QG-module. Theorem 2.1 Let G be a polycyclic group and let G = G1 . . . Gl Gl+1 = 1 be a normal series of G whos e factors Fi = Gi /Gi+1 are either ﬁnite or free abelian and rationally semisimple. Then [G : F it(G)] is ﬁnite if and only if Ci = CG (Fi /Fip ) centralizes Fi for each free abelian factor Fi and for an arbitrary odd prime p. Proof As shown in Theorem 9.11 of [6], the Fitting subgroup F it(G) is the centralizer of a series of G whose factors are either semisimple elementary abelian or rationally semisimple free abelian. The series used here can be reﬁned to such a series by reﬁning its ﬁnite factors. The centralizers of ﬁnite factors of G have ﬁnite index in G. Thus F it(G) has inﬁnite index in G if and only if there exists a rationally semisimple free abelian factor Fi such that CG (Fi ) has inﬁnite index in G. This is the case if and only if Ci does not centralize Fi , since the index of the ✷ centralizer of Fi in Ci is either trivial or inﬁnite by Remark 7.8 of [6]. Now we suppose that G is almost crystallographic. First, as described in [19], we can determine polycyclic presentations for factor groups of G. In particular, we can easily construct such a presentation for the ﬁnite holonomy group G/F it(G) and thus we can investigate this ﬁnite factor. Further, a group is called crystallographic if and only if it is ﬁnitely generated abelian-by-ﬁnite and has no non-trivial normal torsion subgroup. Hence crystallographic groups are a special case of almost crystallographic groups. Another important class of almost crystallographic groups are the almost Bieberbach groups: these are almost crystallographic and torsion-free. For both special cases we can check eﬀectively whether the given group G is of this type: • G is crystallographic if and only if F it(G) is abelian, and • G is almost Bieberbach if and only if T (G) = {g ∈ G | |g| < ∞} = 1. Using the method of [7] (also described in [6]), we can check whether T (G) forms a subgroup of G and, if so, then we can compute it. Further, Polycyclic provides methods to determine all conjugacy classes of ﬁnite subgroups of a polycyclic group. As ﬁnal part of this section we outline an explicit example computation illustrating the methods described above. For this purpose we consider a group from the classiﬁcation in [2] and examine it. We note that polycyclically presented groups are printed by Gap via their “orders” only; these “orders” are the orders of the factors in a polycyclic series, where a 0 indicates an inﬁnite cyclic factor. If desired, then more detailed information such as generators of the determined subgroups can also be computed in Gap.

COMPUTING WITH ALMOST-CRYSTALLOGRAPHIC GROUPS

113

# define a group of type 175 ([1], page 213) as a rational matrix group gap> G := AlmostCrystallographicGroup( 4, "175", [ 1, -4, 1, 2, -3, 2 ] ); <matrix group of size infinity with 6 generators> # determine a polycyclic presentation for G gap> iso := IsomorphismPcpGroup( G );; gap> P := Image( iso ); Pcp-group with orders [ 2, 6, 0, 0, 0, 0 ] # check that it really is almost crystallographic gap> IsAlmostCrystallographic(P); true # determine its Fitting subgroups and check that $G$ is not crystallographic gap> F := FittingSubgroup(P);; gap> IsAbelian(F); false # show that F is of class 2 and observe that F/Z(F) = Z^2 = Z(F) gap> Length( UpperCentralSeries(F) ) - 1; 2 gap> Centre(F); Pcp-group with orders [ 0, 0 ] gap> F / Centre(F); Pcp-group with orders [ 0, 0 ] # show that G is not almost Bieberbach gap> IsTorsionFree(P); false # show that the finite subgroups of G have orders 4, 2 and 1 gap> FiniteSubgroupClasses(P); [ Pcp-group with orders [ 2, 2 ]^G, Pcp-group with orders [ 2 ]^G, Pcp-group with orders [ 2 ]^G, Pcp-group with orders [ 2 ]^G, Pcp-group with orders [ 2 ]^G, Pcp-group with orders [ ]^G ] # finally, observe that G/Fit(G) is abelian and determine its isomorphism type gap> H := P/FittingSubgroup(P);; gap> IsAbelian(H); true gap> AbelianInvariants(H); [ 2, 2, 3 ]

3

Constructing almost crystallographic groups

One of the aims of Dekimpe’s book [2] is the classiﬁcation of many almost crystallographic groups of Hirsch length 4; In particular, all almost Bieberbach groups are determined in [2]. The groups are constructed as extensions of (almost) crystallographic groups of smaller Hirsch length as outlined in the following lemma. This lemma has been introduced by Lee [13] and is also proved in [2], Lemma 2.4.2. Lemma 3.1 Let G be an almost crystallographic group whose Fitting subgroup N has class c. Then γc (N ) is the last non-trivial subgroup in the lower central series of N and we deﬁne I(N ) by I(N )/γc (N ) = T (N/γc (N )) the torsion subgroup of N/γc (N ). (The subgroup I(N ) is also called the isolator of γc (N ) in N .) Then we obtain that • I(N ) is a normal, free abelian subgroup of G and I(N ) ≤ Z(N ). • G/I(N ) is almost crystallographic with the Fitting subgroup N/I(N ) and N/I(N ) is of class c − 1.

114

DEKIMPE, EICK

As indicated by the above lemma, we use the class and the Hirsch length of the Fitting subgroup as the primary invariants to construct and classify almost crystallographic groups. Clearly, the almost crystallographic groups with Fitting subgroup of class 1 are exactly the crystallographic groups. Their Hirsch length is equivalent to their dimension. Classiﬁcations of the crystallographic groups of small dimensions are known. In particular, explicit lists of the isomorphism type representatives of the at most 4-dimensional crystallographic groups have been determined in [1] and they are available in the Crystcat package [11] of Gap. Moreover, various data libraries and algorithms to compute crystallographic groups in higher dimensions have been determined. We mention the methods available in Gap, its Crystgap package [9] and the Carat system [12]. Let G be an almost crystallographic group whose Fitting subgroup N has class c > 1 and Hirsch length l. Then G is an extension of I ∼ = Zd by an almost crystallographic group G whose Fitting subgroup N has class c − 1 and Hirsch length l − d by the above Lemma. In that setup, N centralizes I. We investigate extensions with these properties in more detail in the next lemma. Lemma 3.2 Let G be an almost crystallographic group with Fitting subgroup N of class c − 1. Let I ∼ = Zd be a G-module which is centralized by N and let G be an arbitrary extension of I by G. a) G is almost crystallographic. b) N = F it(G) fulﬁlls N/I = N and hl(N ) = hl(N ) + d. c) The class of N is c − 1 or c. If the class is c, then I(N ) ≤ I. Proof Since I is centralized by N , we obtain that the subgroup N of G with N/I = N is nilpotent. Since N is maximal nilpotent in G, this yields that N = F it(G). Hence G is ﬁnitely generated nilpotent-by-ﬁnite. Suppose that U is a normal torsion subgroup of G. Then U I/I is a normal torsion subgroup of G and hence U I/I = 1. Thus U ≤ I. Since I ∼ = Zd , we obtain that U = 1 and G is almost crystallographic proving a). Part b) follows directly. For part c) we observe that N is an extension of a nilpotent group N of class c − 1 by a central module I. Thus N has class c or c − 1. Also, γc (N ) ≤ I, since N/I has class c − 1. Since N/I is torsion-free, we readily obtain I(N ) ≤ I. ✷ This lemma yields the following approach towards determining almost crystallographic groups G whose Fitting subgroup N has class c and Hirsch length l. We consider an almost crystallographic group G whose Fitting subgr oup N has class c − 1 and Hirsch length n < l and determine extensions of I ∼ = Zd for d = l − n by G such that I(N ) = I in the following steps. 1) Compute all actions of G on I such that N centralizes I. 2) For each obtained action determine the corresponding second cohomology group H 2 (G, I). Recall that the elements of this group correspond to the equivalence classes of extensions of I by G. 3) Discard those cocycles in H 2 (G, I) which yield extensions G whose Fitting subgroup N has class c − 1 or yields I(N ) < I at class c.

COMPUTING WITH ALMOST-CRYSTALLOGRAPHIC GROUPS

115

For the ﬁrst step, we need to determine all homomorphisms G/N → GL(d, Z). Here we can use the classiﬁcations of the ﬁnite subgroups of GL(d, Z) for d ≤ 31. This classiﬁcation is published in various places and we refer to [17] for further details. Usually, we want to apply this method for very small dimension d. In particular, in the often arising case d = 1 we have GL(1, Z) ∼ = C2 and thus we only need to determine the subgroups of index 2 in the factor G/N in this case. In the second step we assume that G is polycyclic and then we apply the methods of Polycyclic to determine the ﬁnitely generated abelian group H 2 (G, I) explicitly. This cohomology group method uses the correspondence between the second cohomology group and the equivalence classes of extensions for its purposes. Each extension G of I by G can be described by a polycyclic presentation which is obtained by extending a polycyclic presentation of G by the natural presentation of the free abelian group I. The obtained relations of G are of three types: the relations of I, the relations expressing the G-module structure of I and, ﬁnally, the relations of G extended by a word (“tail”) in the generators of I. It is obvious that it is exactly the set of tails which determines the speciﬁc extension G. If the presentation of G has m relations, then the presentation of G incorporates m tails and these tails can be viewed as an element of I m . The Polycyclic package computes the second cohomology group H 2 (G, I) via such tails as a subfactor of I m . Thus each element of H 2 (G, I) is of the form tB 2 (G, I) for some t ∈ I m and t is the sequence of tails which translates the relations of G to relations of the extension G. In particular, a polycyclic presentation of G can be read oﬀ from this description of the considered cocycle. The third step is the most diﬃcult part of the construction. If an extension G with Fitting subgroup N is given, then we can determine the isolator I(N ) and thus we can check whether I(N ) = I. However, it seems to be less straightforward to determine explicitly the subset of H 2 (G, I) which yields the desired extensions. In the following lemma we outline descriptions for this subset in some interesting special cases. These cases arise in the study of almost Bieberbach groups of small Hirsch length. In particular, the classiﬁcation of almost crystallographic groups as obtained in [2] can be rebuilt and extended using this lemma. We refer to [4] for more details. ∼ Lemma 3.3 Let G be a polycyclic almost crystallographic group acting on I = Zd in such a way that N = F it(G) acts trivially on I. Let G be an extension corresponding to an element ρ ∈ H 2 (G, I) and let N = F it(G). If N has class c − 1, then a) cl(N ) = 1 if and only if cl(N ) = 1 and ρ has ﬁnite order. b) Suppose that d = 1. Then I(N ) = I if and only if γc (N ) = 1. The latter condition can be read oﬀ from the description of H 2 (G, I) via tails if the given presentation of G exhibits γc−1 (N ). Proof Part a) is essentially proved in [4], Lemma 5.5. For part b) we note that I/γc (N ) is cyclic. Thus it is ﬁnite if and only if γc (N ) = 1 and in this case I(N ) = I. Further, γc (N ) is generated by commutators [a, b] where a is a generator of N and b is a generator of γc−1 (N ). If γc−1 (N ) is exhibited by the presentation of G, then this presentation contains relations ba = b. This translates to a relation ba = bti in

116

DEKIMPE, EICK

G and the element ti ∈ I is an entry of the tail vector t for G. Thus it remains to ﬁnd those tail vectors which have a non-trivial entry at the position corresponding ✷ to a relation ba = b. Example applications of the ideas developed here are outlined in Section 3.2. 3.1

Constructing almost Bieberbach groups

As a special case of the above method, we consider its restriction to compute almost Bieberbach extensions only. As above, we assume that we have a polycyclic almost crystallographic group G acting on I ∼ = Zd and we want to determine extensions of I by G which are almost Bieberbach. Basically, we can apply the same approach as above, with an additional step in which we calculate those cocycles in H 2 (G, I) corresponding to a torsion-free extension. If I ∼ = Z, then we can use the eﬀective method of [4] for this purpose. This method can be generalized to I ∼ = Zd for arbitrary d ∈ N using the following lemma. Lemma 3.4 Let G be an extension of I ∼ = Zd by a polycyclic group G. Let H 1 , . . . , H s be a set of conjugacy class representatives for the subgroups of prime order in G and denote their preimages in G by Hi . a) G contains torsion if and only if there exists an i ∈ {1, . . . , s} such that Hi contains torsion. b) Let Hi = h, I ≤ G and denote p = [Hi : I]. Then Hi contains torsion if and only if hp ∈ I a where −a = hp−1 + . . . + h + 1 ∈ ZHi . Note that I is a right G-module via the conjugation action of G on I. Thus I is also a right ZG-module and hence I a is well-deﬁnied for a ∈ ZHi . Proof a) Suppose that G contains torsion. Then there exists an element g ∈ G of prime order p. Since I = Zd , we obtain that g ∈ I and gI is an element of order p in G. Thus gI is conjugate to some subgroup H i and Hi contains torsion. The converse is obvious. b) First, Hi contains torsion if and only if Hi splits over I. If this is the case, then there exists an element of the form hm in Hi with m ∈ I such that (hm)p = 1. ✷ Rewriting this yields hp ma = 1 and thus hp ∈ I a . First, to use the approach indicated by the above lemma, we need to compute representatives for the conjugacy classes of subgroups of prime order in G. Since G is polycyclic, we can use a restricted version of the algorithm to compute the conjugacy classes of ﬁnite subgroups for this purpose. Then, we consider each computed representative H in turn and we want to determine those elements of H 2 (G, I) in whose corresponding extension the preimage of H does not contain torsion. Let H = h and denote by Mh the matrix action of h on I ∼ = Zd ; that is, p−1 Mh ∈ GL(d, Z). Denoting |H| = p, we can compute M = Mh + . . . + Mh + 1 ∈ Md×d (Z). Since M acts on Zd as the element a of Lemma 3.4 acts on I, we obtain that I a can be computed easily as the image of the matrix M . It remains to check if hp ∈ I a where H is a preimage of H in an extension G. Since we

COMPUTING WITH ALMOST-CRYSTALLOGRAPHIC GROUPS

117

cannot list all extensions G explicitly, but only have implic it descriptions of these extensions via the elements of the group H 2 (G, I), this is the more diﬃcult part of the computation. We solve this problem by describing the extensions of I by G via parameterized polycyclic presentations as outlined in [4]. In summary, these presentations incorporate certain indeterminates T1 , . . . , Tm which can take values in I. They correspond to the tails used to describe the elements of H 2 (G, I) and they can be applied to compute with the extensions of I by G in a symbolic form. In particular, we can compute a word representing hp in a parameterized am with a ∈ polycyclic presentation and we obtain a word of the form T1a1 · · · Tm i ZG. Once this word is given, it remains to determine those values t1 , . . . , tm ∈ Z 2 (G, I) with ta11 · · · tamm ≡ 1 mod I a as observed in Lemma 3.4. If we switch to additive notation, then this equation translates into a linear map. Thus its kernel forms a subspace S ≤ Z 2 (G, I) which can be computed as nullspace of a system of linear equations. Its setwise complement is the set of those cocycles whose corresponding extensions have a torsion-free preimage of H. If we perform this computation for each representative H1 , . . . , H s of the conjugacy classes of subgroups of prime order in G, then we obtain a set of subgroups S1 , . . . , Ss of Z 2 (G, I). The extensions corresponding to cocycles not contained in any Si are the desired torsion-free extensions. These cocycles can be explicitely computed, since each Si has ﬁnite index in Z 2 (G, I), and thus also S = si=1 Si has ﬁnite index in Z 2 (G, I). Therefore, the computation of the diﬀerence Z 2 (G, I) \ si=1 Si of inﬁnite sets, can be reduced to a ﬁnite problem: the determination of the diﬀerence Z 2 (G, I)/S \ si=1 Si /S. 3.2

An example application

We consider the determination of almost crystallographic and almost Bieberbach extensions of a group from the Aclib catalogue. # choose a group from the library, take a module and compute its cohomology gap> Gbar := AlmostCrystallographicPcpGroup( 4, 30, [ 3, -1, 1, 1 ] ); Pcp-group with orders [ 2, 2, 0, 0, 0, 0 ] gap> I := AllActionsHolonomy( Gbar );; gap> CR := CRRecordByMats( Gbar, I[1] );; gap> two := TwoCohomologyCR( CR );; gap> two.factor.rels; [ 2, 4, 0 ] # take a cocycle of infinite order and determine its extension gap> cc := two.factor.prei[3]; [ 0, 0, 0, 0, 1, -1, 0, -6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -8, -4, 0, 0, 0, 0, 0, 0, -2, 2, 0, 0, 0, 0 ] gap> G := ExtensionCR( CR, cc ); Pcp-group with orders [ 2, 2, 0, 0, 0, 0, 0 ] # check that G is almost crystallographic gap> IsAlmostCrystallographic( G ); true # more detailed: get I(Fit(G)) and check that I(Fit(G)) = I gap> N := FittingSubgroup( G ); Pcp-group with orders [ 0, 0, 0, 0, 0 ]

118

DEKIMPE, EICK

gap> ser := LowerCentralSeries( N ); [ Pcp-group with orders [ 0, 0, 0, 0, 0 ], Pcp-group with orders [ 0, 0 ], Pcp-group with orders [ 0 ], Pcp-group with orders [ ] ] gap> Iso := IsolatorSubgroup( ser[3] ); Pcp-group with orders [ 0 ] gap> Iso = G.module; true gap> Index( Iso, ser[3] ); 12 # in fact, G is almost Bieberbach gap> TorsionSubgroup( G ); Pcp-group with orders [ ] # we can also check directly that G has almost Bieberbach extensions gap> HasTorsionFreeExtension( G, CR ); true

4

Further applications

There exists a variety of further applications of our methods. In this section we want to include references for some of these applications. First, the Aclib package can be used to compute Betti numbers of closed manifolds K(G, 1) for polycyclic groups G. More precisely, to determine the Betti numbers of K(G, 1), we construct a polycyclic presentation of the underlying group G and then we apply the methods outlined in [4]. These can be used to determine the Betti numbers βi (G) for i ∈ {0, 1, 2, n − 2, n − 1, n} where n is the dimension of K(G, 1) (or the Hirsch length of G). Further, if n = 6, then we can compute all Betti numbers of K(G, 1). Secondly, the Aclib package can be used to investigate infra-nilmanifolds. The fundamental groups of these manifolds are exactly the almost Bieberbach groups and many questions concerning infra-nilmanifolds can be reduced to purely algebraic questions on their fundamental groups. As an example, in [4] we have considered infra-nilmanifolds modeled on the 5-dimensional Heisenberg Lie group. The holonomy groups of their fundamental groups have been of interest in [14] and [15] where it has been proved that they have order at most 24. Using our algorithmic methods, we can now easily determine all possible holonomy groups; Their orders are 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 16, 18, 24 and an explicit list of their isomorphism types can be found in [4]. In particular, we observed that there exist two isomorphism types of holonomy groups of order 24: the groups C2 × C12 and C3 Q8 . Finally, we brieﬂy mention that our algorithmic tools can also be used to compute P -localizations of almost crystallographic groups. These have been considered recently in [5] where, among others, examples for P -localizations of almost crystallographic groups have been computed. These examples can also be obtained using our methods as shown in the Aclib manual.

COMPUTING WITH ALMOST-CRYSTALLOGRAPHIC GROUPS

119

References [1] H. Brown, R. B¨ ulow, J. Neub¨ user, H. Wondratschek, and H. Zassenhaus. Crystallographic Groups of Four-Dimensional Space. Wiley, 1978. [2] K. Dekimpe. Almost-Bieberbach Groups: Aﬃne and Polynomial Structures, volume 1639 of Lecture Notes in Mathematics. Springer, 1996. [3] K. Dekimpe and B. Eick. Aclib, 2000. A GAP share package, see [20]. [4] K. Dekimpe and B. Eick. Computational aspects of group extensions and their applications in topology. To appear in Exp. Math., 2001. [5] A. Descheemaeker and W. Malfait. P -localization of relative groups. Journal of Pure and Applied Algebra, 159(1):43–56, 2001. [6] B. Eick. Algorithms for polycyclic groups. Habilitationsschrift, Universit¨ at Kassel, 2001. [7] B. Eick. Computing with inﬁnite polycyclic groups. In A. Seress and W. M. Kantor, editors, Groups and Computation III (1999), pages 139 – 153, 2001. [8] B. Eick. On the Fitting subgroup of a polycyclic-by-ﬁnite group and its applications. J. Alg., 242:176 – 187, 2001. [9] B. Eick, F. G¨ ahler, and W. Nickel. CrystGap, 1998. A GAP share package, see [20]. [10] B. Eick and W. Nickel. Polycyclic, 2000. A GAP share package, see [20]. [11] V. Felsch and F. G¨ ahler. CrystCat, 1998. A GAP share package, see [20]. [12] W. P. J. Opgenorth and T. Schulz. Crystallographic algorithms and tables. Acta Cryst A, 54:517–531, 1998. [13] K. B. Lee. Aspherical manifolds with virtually 3–step nilpotent fundamental group. Amer. J. Math., 105:1435–1453, 1983. [14] K. B. Lee. Infranil-manifolds modeled on heis5 . In preparation, 2000. [15] K. B. Lee and A. Szczepa´ nski. Maximal holonomy of almost bieberbach groups for heis5 . To appear in Geom. Dedicata., 2000. [16] E. H. Lo. A polycyclic quotient algorithm. J. Symb. Comput., 25:61 – 97, 1998. [17] G. Nebe. Finite subgroups of GLn (Q) for 25 ≤ n ≤ 31. Comm. Alg., 24 (7):2341– 2397, 1996. [18] G. Ostheimer. Practical algorithms for polycyclic matrix groups. J. Symb. Comput., 28:361 – 379, 1999. [19] C. C. Sims. Computation with ﬁnitely presented groups. Cambridge University Press, Cambridge, 1994. [20] The GAP Group. GAP – Groups, Algorithms and Programming. http://www.gapsystem.org, 2000.

RANDOM WALKS ON GROUPS: CHARACTERS AND GEOMETRY PERSI DIACONIS Departments of Mathematics and Statistics, Stanford University, Stanford, CA 94305 USA

Introduction These notes tell two stories. The ﬁrst is an overview of a general approach to studying random walk on ﬁnite groups. This involves the character theory of the group and the geometry of the group in various generating sets. The second is the life and times of a single example: random transpositions on the symmetric group. This was the ﬁrst example where sharp estimates were obtained. We will prove it takes 12 n log n transpositions to mix up n cards. This example gives rise to a rich comparison theory allowing general walks to be studied. Its history goes back to Hurwitz’s 1891 work on counting branched covers. There are generalizations to ﬁnite groups of Lie type, various deformations (Jack symmetric functions and Hecke algebras) and applications to diﬀusion, phylogeny and coagulation processes in chemistry. Let us begin with a deﬁnition. Let G be a ﬁnitegroup. Let {Q(g)}g∈G , be a Q(g) = 1. This is the basic probability distribution on G. Thus Q(g) ≥ 0 and g

data. Deﬁne convolution by Q Q(g) = Q gh−1 Q(h),

Qk (g) = Q(k−1) Q(g).

h

Thus QQ(g) is the chance that a random walk on G generated by picking elements repeatedly with weight Q(g) and multiplying is at g after two steps; some element h must have been chosen ﬁrst, followed by gh−1 . Similarly, Qk (g) is the chance that the walk is at g after k steps. All walks start at the identity. Under a mild restriction (the support of Q is not contained in a coset of a subgroup) the random walk converges to the uniform distribution u(g) = 1/|G|. Qk (g) →

1 |G|

as k → ∞.

The question is how fast? We will measure convergence by the total variation distance k 1 k 1 Q (A) − |A| . = max (0.1) Q (g) − ||Qk − u|| = A⊆G 2 g |G| |G| We thus arrive at a well-posed math problem: Given a ﬁnite group G, a probability measure Q(g) and > 0, how large should k be so that ||Qk − u|| ≤ ?

RANDOM WALKS ON GROUPS: CHARACTERS AND GEOMETRY

121

For group with an element theorists, this may be rephrased as follows. Identify Q g Q(g) of the group algebra. The coeﬃcient of g in q k is Qk (g). High q = g 1 powers of q converge to |G| g. We are asking for the speed of convergence in the g

1 norm. Here is our leading example, said more carefully. Example (Random transpositions). Imagine a deck of n playing cards face down in an ordered row with card 1 at the left, card n at the right. The cards are repeatedly mixed by the following operation. The left hand touches a random card. The right hand touches a random card (so left = right with probability 1/n). These cards are transposed (nothing is done if left = right). It is intuitively clear that after many switches the row of cards is all mixed up. More mathematically, on the symmetric group Sn , let   1/n if π = id 2/n2 if π is a transposition Q(π) =  0 otherwise

(0.2)

Repeated switches are modeled by Qk (π). The following theorem was proved in joint work with Shahshahani [26]. Theorem A For the random walk generated by (0.2) on the symmetric group Sn , if k = 12 n(log n + c) for c > 0, then ||Qk − u|| ≤ 6e−c

(0.3)

conversely, if k = 12 n(log n − c) there is > 0 such that ||Qk − u|| > for all n. Remarks ˙ to make the distance in (0.3) smaller than 1/100, 1. When n = 52, 12 n log n=103; requires k =270 ˙ switches. 2. The theorem shows that convergence to stationarity has a cutoﬀ or threshold at 12 n log n. The transition from order to chaos happens as c varies. This often happens for random walk on non-commutative groups. See [9], [65] for further discussion; it is one of the important open problems of this subject to understand the cutoﬀ phenomenon. 3. The original motivation for studying random transpositions is worth mentioning. In the course of a large scale study of optimal strategies in experiments with feedback ([13]), a Monte Carlo study involving hundreds of millions of random permutations in S52 was carried out. The ﬁnal results “looked funny.” In the course of checking the programming, the method of generating the permutations came under scrutiny. The usual way of generating a random permutation on a computer is to put numbers 1, 2, . . . , n into nregisters. Then choose a random number I1 uniformly from one to n and transpose registers 1 and I1 . Next, choose I2 randomly from two to n and

122

DIACONIS

transpose registers 2 and I2 . Continuing n − 1 steps gives a perfectly random permutation. This is the subgroup algorithm [27]. The programmer had thought this “too fussy” and instead carried out 60 random transpositions (as in (0.2)). The mathematicians involved complained and asked that the simulation be redone (three hours of computer time on a powerful processor). The programmer (and her boss) complained, but in the end, the simulation was redone using the proper algorithm and satisfactory results were obtained. All of this left me wanting to know how many random transpositions are required to mix up 52 cards. Section 1 introduces a basic upper bound on the distance between Q∗k and the uniform distribution. The bound is most useful for class functions (Q(s−1 ts) = Q(t)) and involves a detailed knowledge of characters. This is illustrated for random walks on the symmetric group using random transpositions and for the Drunkard’s walk on the circle. Section 2 introduces various norms and quadratic form used to bound eigenvalues via the minimax characterization. The basic bounds on distance to uniform using eigenvalues are set out. Then, comparison theorems are developed which allow analysis of a random walk based on one generating set (usually a small or messy set) in terms of a walk based on a nice generating set. The analysis involves relating the geometry of the two Cayley graphs. This is illustrated by giving a sharp analysis of the walk on the symmetric group using a transposition and an n-cycle. In Section 3, developments of the random transpositions result are outlined. These go back to Hurwitz work on coverings of the sphere and deform to interesting walks on Hecke algebras with application to the Metropolis algorithm of statistical physics. The ﬁnal section sets out some open problems. Who cares about this stuﬀ? Of course, one can take the high road and answer that mathematics should be judged by its own internal naturalness and beauty. It is resoundingly true that mathematics developed ‘just because it was there’ has a remarkable record of turning out useful. For example, Frobenius’s development of character theory was based on the strange question of understanding why the determinants of circulant matrices are a product of linear forms in the entries. The random transposition results that are the basis of the present paper are completely derivative of Frobenius’s development. Of course, certain random walks arise in daily life when people shuﬄe cards. I have written a survey of this subject in [10]. Ordinary random walk on Rd is a mainstay of parts of biology, chemistry, physics, and ﬁnance. It is natural to seek appropriate generalization to more general groups. Hughes [41, 42] and Saloﬀ-Coste [66] show how natural these questions can be, even for mathematics. A very satisfactory answer comes from problems generated from within group theory. Modern algorithms to manipulate and study large ﬁnite groups need a source of random elements. These are generated by a variety of random walk algorithms. One of the most popular is the product replacement algorithm. The mathematics in the present paper can be used to study these algorithms. Igor

RANDOM WALKS ON GROUPS: CHARACTERS AND GEOMETRY

123

Pak has done wonderful work along these lines. His survey Pak [59] contains an extensive review. On diﬀerent lines, the mathematical questions that arise in studying random walk on groups are often quite diﬀerent than classical questions. They have led to some interesting new group theory. Perhaps the most compelling current motivation is the ‘Markov chain Monte Carlo revolution’. Scientists in every walk of life are computing quantities of interest by running generalized random walks called Markov Chains. Liu [52] gives a nice introduction to this subject. Random walks on groups are special cases of Markov chains and the extra group structure along with years of hard work by group theorists can allow sharp results. The comparison theory explained below allows transfer to more general chains. More basically, techniques developed for groups can sometimes be extended to general Markov chains; again, the comparison theory is a good example. (See Diaconis and Saloﬀ-Coste [18].)

Acknowledgement It is the greatest pleasure to acknowledge my co-authors Mehadad Shahshahani, David Aldous, Dave Bayer, Ken Brown, Lou Billera, Fan Chung, Ron Graham, Susan Holmes, Jim Fill, Arun Ram and Laurent Saloﬀ-Coste for their co-development of the subject. I have had the good fortune to work on random walks with wonderful graduate students, Farid Bassiri, Eric Belsley, Carl Dou, Martin Hildebrand, Andy Greenhaugh, Jason Fulman, Nathan Lulov, Igor Pak, Peter Mathews, Robin Pemantle, Francis Su, Jeﬀrey Rosenthal, Elizabeth Wilmer and Thomas Yan. Finally, I’m truly thankful to the organizers and participants of Group St. Andrews, Oxford Branch, for making an outsider feel welcome.

1

Random walk and representation theory

Let G be a ﬁnite group. A representation ρ : G → GLdρ (V ) assigns matrices to group elements in such a way that ρ(st) = ρ(s)ρ(t). Here the dimension of V is denoted dρ . Background in representation theory and the probabilistic developments discussed here may be found in my book [8]. If Q(s) is a probability distribution on G, the Fourier transform of Q at ρ is deﬁned as Q(s)ρ(s). Q(ρ) = s

As usual, Fourier transforms turn convolution into products 2

2 (ρ) = Q(ρ) . Q

124

DIACONIS

Further, for irreducible representations, the uniform distribution u(s) = 1/|G| is characterized by its Fourier transform

1 if ρ is the trivial representation 0 if ρ is non-trivial irreducible.

u (ρ) =

Repeated convolutions may be shown to converge to the uniform distribution by k → 0 for ρ non-trivial irreducible. A quantitative version of showing that Q(ρ) this follows from the Fourier inversion and Planchenel theorems. Theorem 1.1 Let f be a complex function defined on G. Then a) f (s) = b) ||f ||2 =

1 dρ tr f (ρ) ρ s−1 |G| ρ 1 dρ ||f (ρ)||2 (trace norm). |G| ρ

Proof For (a), both sides are linear in f . Take f (s) = δt (s). Then f (ρ) = ρ(t) and (a) asserts 1 dρ χρ (st−1 ). δt (s) = |G| ρ This assertion holds as the right hand sum is the character of the regular representation which is |G| or zero as st−1 is the identity or not. For (b), we show f1 , f2 =

1 f1 (s)f¯2 (s) = dρ tr f 1 (ρ)f 2 (ρ) . |G| ρ

Again, both sides are linear in f2 . Taking f2 = δt , we must show f1 (τ ) =

1 dρ tr f 1 (ρ)ρ (τ ) . |G| ρ

This follows from (a) since without loss of generality, ρ is unitary so ρ (τ ) = ρ(τ −1 ) The basic upper bound lemma was developed to study random transpositions in joint work with Shahshahani. Lemma 1.2 (Upper bound lemma) Let Q be a probability on the finite group G. Then, for the distance defined in (0.1), 4||Qk − u||2 ≤

ρ=1

2k dρ ||Q(ρ)|| .

RANDOM WALKS ON GROUPS: CHARACTERS AND GEOMETRY

125

Proof From the deﬁnitions 2 2 k 2 k 4||Q − u|| = ≤ |G| Q (s) − u(s) Qk (s) − u(s) s

=

s

∗ k (ρ)k ≤ Q dρ tr (Q(ρ) dρ ||Q(ρ)|| . k

ρ=1

ρ=1

There, the ﬁrst inequality is from Cauchy-Schwarz, the second equality is from Plancherel, the last inequality is ||AB|| ≤ ||A|| ||B||. ✷ Example 1.3 (Drunkard’s walk) Let Cn be the integers modulo n, and let Q(±1) = 12 with Q(j) = 0 otherwise. Thus Qk (j) is the chance that a simple random walk is at j after k steps. To avoid periodicity problems, suppose n is odd. The irreducible representations are one-dimensional and given by ρh (j) = e2πijh/n . Here Q(h) = 12 e2πijh/n + 12 e−2πih/n = cos(2πh/n). The upper bound lemma gives 4||Q − u||2 ≤

n−1

cos

h=1

2πh n

2

(n−1)/2

=2

cos

h=1

2πh n

2 .

This last sum must be bounded by calculus arguments. One way to proceed is to use cos(x) ≤ e

−x2 2

for 0 ≤ x ≤ π/2. Thus ||Q − u||2 ≤

(n−1)/2 1 −π2 j 2 /n2 e 2 j=1

∞

≤

1 −π2 /n2 −π2 (j 2 −1)/n2 e e 2

≤

1 −π2 /n2 −3π2 j/n2 e e 2

=

1 e−π /n . 2 1 − e−3π2 /n2

i=1 ∞

j=0

2

2

−1 2 2 If ≥ n2 , 2 1 − e−3π /n < 1 and we conclude ||Q − u|| ≤ e

−π 2 2 n2

for n odd with ≥ n2 .

The result shows that a multiple of n2 steps suﬃce to drive the distance to zero exponentially fast. While not developed here, this result is sharp–for small with respect to n2 the distance to uniformity is close to its maximum value of 1 (See [8] pg. 29). It is instructive to view the bounding process in the light of representation theory. The sum is dominated by the representations close 1 to the trivial representation. 2π 2 = 1 − . This must be raised to the + O Thus when h = 1, Q(h) = cos 2π n n2 n4

126

DIACONIS

power of n2 or more to make it small. For more general groups, it also seems to hold that the representations close to the trivial representation control the rate of convergence. Problem 5 in Section 4 has more on this. The next example is central to present developments. Example 1.4 (Random transpositions) Let Sn be the symmetric group. The probability measure Q(w) deﬁned at (0.2) is invariant under conjugation Q(w) = = Q(ρ). For irreQ(v −1 wv). Thus its Fourier transform satisﬁes ρ(v −1 )Q(ρ)ρ(v) ducible ρ, Schurs lemma implies that Q(ρ) is a constant multiple of the identity: χρ (τ ) Q(ρ) = cI. Take the trace of both sides to see that c = n1 + n−1 n dρ with χρ (τ ) the character of ρ at a transposition and dρ the dimension. Thus the upper bound is 1 n − 1 χρ (τ ) 2k k 2 d2ρ . + 4||Q − u|| ≤ n n dρ ρ=1

To make further progress, we must get our hands dirty and come to terms with the character ratio. Fortunately, Frobenius did most of the work. The irreducible representations of Sn are indexed by partitions λ of n. If λ = (λ1 , λ2 , , . . . , λr ) with λ1 ≥ λ2 ≥ . . . ≥ λr > 0 and λ1 + . . . + λr = n, Frobenius showed that r

1 χλ (τ ) = λ2i − (2i − 1) λi . dλ n(n − 1) i=1

To see what is involved in the bounding, take the representation closest to the trivial representation—the n − 1 dimensional ‘permutation’ representation. This corresponds to (n − 1, 1) and the term to be bounded is 2 2k . (n − 1)2 1 − n Using 1 − x ≤ e−x we see that this is smaller than e−2c for k = 12 n log n + c n. This lead term dominates and determines the rate of convergence stated in Theorem A of the Introduction. The details lean on years of work by group theorists and combinatorialists (Tableaux Combinatorics). See [8] pg. 36-47 for a detailed proof which is a streamlined version of the original. Other walks and conjugacy classes There have been a number of further careful studies of random walks on groups which are constant on conjugacy classes. One of the earliest is random walk on the hypercube C2n . This can be recast as a successful analysis of the Ehrenfest Urn model of statistical physics [8] pg. 19. Of course, any walk on an abelian group is constant on conjugacy classes. Hildebrand [39] gave a careful complete analysis of a random transvections walk on SLn (Fq ) using character theory. Roughly, he showed that for ﬁxed q and n large, n + c steps are necessary and suﬃcient. A sweeping generalization was make by Gluck [34]. He studied random walks on ﬁnite groups of Lie type with

RANDOM WALKS ON GROUPS: CHARACTERS AND GEOMETRY

127

probability distribution Q supported on small generating conjugacy classes. In close to complete generality, he proves that order rank (G) steps are necessary and suﬃce. Gluck uses the upper bound lemma and must thus bound character ratios of the form χ(c)/χ(id). As opposed to earlier eﬀorts which used hairy but explicit formulae for character ratios, Gluck proved that the ratios are uniformly bounded θ above by terms of form q −χ(id) for e.g. θ = 1/10. These proofs work by induction and make careful use of the structure of Lie type groups. The argument is so powerful that it is worthwhile trying to abstract it; Gluck (personal communication) reports that this did not seem easy to do for the symmetric group. In a diﬀerent development, Gluck [35] applied his character ratio bounds to get sharp results on ﬁrst return times for these random walks. Character ratios ﬁgure in the analysis of many other applications of group theory. Some of these are explained in Gluck [34]. His results described above omit some groups “in the corners” and have some restrictions on the size of the generating conjugacy class. An elegant rounding out and natural completion of Gluck’s results was carried out by Liebeck-Shalev [52, 51]. As will appear later, these constant on conjugacy class walks are a backbone of the approach outlined here to studying general walks. We thus have a very solid base to build on—a comprehensive theory for the ﬁnite simple groups of Lie type. There is one striking open problem here. Hildebrand’s work proves a sharp cutoﬀ (n+c steps necessary and suﬃce, the transition to randomness happens as c varies). The work on other Lie type walks results in statements such as ‘fewer than rank (G) steps are not enough and there is a constant A > 1 so that A · rank (G) + c steps suﬃce’. One thus does not have a sharp cutoﬀ. There are good reasons to conjecture that the lead terms in the upper and lower bounds of any walk constant on conjugacy classes match up for a sequence of Lie type groups of growing rank. This is carefully explained in [9] or [65] which give surveys of this cutoﬀ phenomena. Turning to walks on the symmetric group, Rousell [64] has proved sharp bounds on random walk generated by small conjugacy classes such as three-cycles or products of two transpositions. Her results are of the following form: let c be a conjugacy n+θ) class in Sn with f (c) ﬁxed points. Then n(log n−f (c) steps are necessary and suﬃce to achieve randomness (as θ varies). At the other end, Lulov-Pak [56] treat walks generated by a single large conjugacy class. Here is one of their results. Let Q be a probability supported on the conjugacy class of an -cycle, with > n/2. Then the walk generated by Q has a cut oﬀ at k = log n/ log(n/(n − )). This is closely related to earlier work of Roichman [61] who derived useful bounds on character ratios for large conjugacy classes in Sn . Alas his work has not been pushed through to give sharp bounds in probability problems. This is a hard but potentially fruitful area of study. One striking result along these lines has been proved by Lulov [55] and Fomin-Lulov [32]. Fix h and take n a multiple of h. Consider the walk generated by the conjugacy class which is a product of n/h h-cycles. This random walk gets random after two steps (and one step will not do). Lulov and Pak [56] give a more comprehensive survey, several conjectures and many further results. A nice development of the character theoretic approach to random walk on the hyperoctahedral group Bn appears in Schoolﬁeld [67]. He gives careful bounds for

128

DIACONIS

the walk on a small generating conjugacy class and applies the result to a problem that arose in gene shuﬄing. I ﬁnd it easiest to state their ﬁnal result in the language of playing cards. Imagine a row of face down playing cards in order on the table. To start, card one is at left, . . ., card n is at the right. Pick a random block of k cards and turn them over, both end for end and face up. How many times should this be repeated to mix both the order and face up—face down pattern? Character theory methods can also be applied to compact groups such as the orthogonal group On or SLn (Zp ). Following work of Rosenthal [62] and Porod [60]. I have worked on such problems in [25]. This last, joint work with Laurent SaloﬀCoste, attacks a 50 year old math/physics problem posed by Mark Kac. This may be rephrased as an analysis of the walk on On generated by picking a random pair of coordinates and then rotating by a random angle in that 2-space (random Givens rotations). It has been wonderfully developed and completed in work of Carlin et al. [7], Janvresse [47] and Maslin [57]. The natural conjecture: “order n log n steps are necessary and suﬃce and there is a cutoﬀ in total variation” is still open. Random walks that are constant on conjugacy classes can be seen as a special case of bi-invariant walks on a Gelfand pair. This is speciﬁed by a subgroup H ⊂ G and a probability Q so that Q(y) = Q(h1 yh2 ). Then the tools of spherical functions can be usefully employed. Surveys of random walks on Gelfand pairs appear in Letac [48, 49], Diaconis [8] and Belsley [4]. Taking G ⊂ G × G, the spherical functions are the characters of G. I have not found this connection particularly illuminating. The two languages have their own rhythm and classes of examples. In particular, I have never been able to use any of the classical Gelfand pairs as a useful comparison for less symmetric walks. Other techniques To conclude this section let me mention that there are many other ways of studying random walks on groups that do not involve character theory. A survey of analytical approaches can be found in Diaconis and Saloﬀ-Coste [20]. These included Poincar´e, Nash, Sobolev, and Log Sobolev inequalities. There are techniques for lifting walks to covering walks, volume growth considerations and much else. One striking result may be mentioned here as an example of what a comprehensive theory might give. Let G be a ﬁnite p-group of bounded derived length and Frattini rank. For any random walk based on a symmetric minimal generating set, the squared diameter of the group in this generating set is necessary and suﬃcient to achieve randomness. For the easiest example, the simple ±1 random walk on the integers (mod p) has diameter of order p in these generators and the example above shows that order p2 steps are necessary and suﬃcient to achieve uniformity. For the next simplest example, consider one of the extra special groups of order p3 . Any set of two generators has diameter of order p and again order p2 steps are necessary and suﬃce. There are three proofs of the general (diam)2 result and many applications; in [19] it is proved using moderate geometric growth. In [22] it is proved using Nash inequalities. In [20] it is proved using coverings and Harnack inequalities.

RANDOM WALKS ON GROUPS: CHARACTERS AND GEOMETRY

129

In addition to the analytic techniques above, there are purely probabilistic techniques—coupling and strong stationary times which give eloquent deﬁnitive results in special cases. The forthcoming book of Aldous and Fill [3] treats these. I have given a treatment in [8] and in joint work with Fill [12]. Igor Pak [58], has developed strong stationary times in marvelous ways. There are also a variety of mixing schemes studied by methods not captured above. Chief among these are walks on Sn associated to riﬄe shuﬄing cards. After the original work of Bayer-Diaconis showing seven ordinary riﬄe shuﬄes suﬃce, Bidigare-Hanlon-Rockmore followed by Diaconis and Brown extended things to walks on the chambers of a hyperplane arrangement. This includes all ﬁnite reﬂexion groups. Then, Ken Brown extended things to walks on idempotent semi groups. This includes walks on spherical buildings. These shuﬄing walks have many connections with group theory and symmetric function theory. I have written a recent survey of these developments [10]. Finally, marvelous results which may be interpreted in the language of random walk on ﬁnite groups have been proved in studying expander graphs. These make deep contact with modern mathematics, drawing on work of Deligne and Selberg. I will not try to paint this picture here but refer to Lubotzky [54] for extensive pointers to the literature.

2

Analytic geometry

This section sets out the analytic tools concerning eigenvalues that allow geometric methods (paths between elements, diameter, covering numbers, volume growth) to be used to bound rates of convergence of random walk. Throughout, G is a ﬁnite group, Q(s) = Q(s−1 ) a symmetric probability distribution. Often Q(s) = 1/|S| or zero as s ∈ S or not with S = S −1 a symmetric set of generators. Associated to Q is a graph with vertex set G and an edge from s to t if Q(st−1 ) > 0. Geometry refers to the geometry of this graph. It is also helpful to associate a |G| × |G| matrix, the transition matrix with Mst = Q(ts−1 ). This gives the chance of going from s to t in one step of the walk. The matrix M is symmetric and doubly stochastic. It thus has real eigenvalues πi and the Perron-Frobenius theorem says that 1 = π0 ≥ π1 . . . ≥ π|G|−1 ≥ −1. Just to warm up, here is an easy (and useful) bound for the smallest eigenvalue. Lemma 2.1 For G and Q above, the smallest eigenvalue satisfies π|G|−1 ≥ −1 + 2Q(id). Proof If Q(id) = 0 this is certainly true. If Q(id) > 0, let Q(s) = Q(s)/(1 − Q(id)) for s = id, Q(id) = 0. This is symmetric with eigenvalues π i ≥ −1. Thus 1 − Q(id) ≥ −1. This gives the result. ✷ π |G|−1 1−Q(s) Remark It is not necessary to have Q(id) > 0 to bound negative eigenvalues (see Diaconis and Saloﬀ-Coste [22]). This is the most commonly occurring special case. The main bound on convergence to uniformity using eigenvalues is

130

DIACONIS

Lemma 2.2 For G and Q as above, |G|−1 1 4||Qk − u||2 ≤ |G| ||Qk − u||22 = |G| Q2k (id) − = πi2k . |G| i=1

Proof From the deﬁnition of total variation distance in the Introduction, 2 2 k Qk (s) − 1 Q (s) − 1 ≤ |G| 4||Qk − u||2 = |G| |G| 2 2 1 = |G| + = |G|Q2k (id) − 1. Qk (s) − |G| |G| s |G|−1 2k From the matrix interpretation, |G|Q2k (id) = tr M 2k = πi . This comi=0

pletes the proof.

✷

Corollary 2.3 For G and Q as above 4||Qk − u||2 ≤ |G|π2k ,

π = max π1 , π|G|−1 .

Remark The bound in Lemma 2.2 has been shown to be quite sharp in many examples. There is usually a reasonably close matching lower bound. The bound in the corollary is looser and usually oﬀ by factors of log |G|. It is sometimes all that is available. How to bound eigenvalues Deﬁne an inner product on real valued functions f1 (s)f2 (s). The linear space of all functions (the group on G by f1 |f2 = s

(G). A symmetric probability Q deﬁnes a Laplace operator algebra) is denoted L2 (I − Q)f (s) = f (s) − f (t)Q(ts−1 ). This has eigenvalues 1 − πi . The associated quadratic form is called the Dirichlet Form in probabilistic circles: E(f |f ) = (I − Q)f, f =

1 (f (s) − f (st))2 Q(t). 2 s,t

As usual, we may bound eigenvalues by the minimax principle: let V be a real linear space, T a symmetric linear map on V with eigenvalues q0 ≤ q1 ≤ . . .. For a subspace W , set M (W ) = max T v, v /||v||2 , v ∈ W , v = 0. Then qi = min {M (W ) : dim(W ) = i + 1}. See Horn and Johnson [40] for a nice development. The minimax characterization implies be symmetric probabilities on G with eigenvalues πi , π i Corollary 2.4 Let Q, Q If for some constant A > 0, E ≤ AE then for all i, and associated forms E, E. πi ≤ 1 − (1 − πi )/A. ˜ is Several applications of this result are given later in this section. Usually, Q a walk about which we know everything and Q is a walk we want to study. The main use of Corollary 2 is the following basic upper bound.

RANDOM WALKS ON GROUPS: CHARACTERS AND GEOMETRY

131

be symmetric probabilities on G. If E ≤ AE then Theorem 2.5 Let Q, Q 2k k/2A − u||22 . ||Qk − u||22 ≤ πmin + e−k/A + ||Q

Proof From Lemma 2, ||Qk − u||22 =

1 |G|

|G|−1 i=1 e−x ,

2k + πi2k ≤ πmin

1 |G|

i:πi >0

πi2k .

1 − x > e−2x for 0 < x ≤ 12 . Thus Now use the calculus bounds 1 − x ≤ 2k −(1−πi )2k k/A A πi > 0 gives πi2k ≤ 1 − (1−Aπi ) ≤e ≤ πi . ✷ Theorem 1 gives bounds on a probability of interest in terms of a known probability in the presence of a comparison between forms. How to compare forms (and ﬁrst examples). If S is a symmetric generating set of G let |t| = min : t = s1 . . . sk . |id| = 0. k

For such a representation of t, let N (s, t) = # (times s appears). Fix a minimal representation for each t. be symmetric probabilities on G. Let S be a symmetric Theorem 2.6 Let Q, Q generating set with Q(s) > 0 for s ∈ S. Then E ≤ AE for A = max s∈S

1 |t|N (s, t)Q(t). Q(s) t∈G

Proof For x, t in G, write t = s1 . . . sk as above. Then, for any function f , f (x)−f (xt) = (f (x) − f (xs1 ))+f (xs1 )−f (xs1 s2 )+. . .+(f (xs1 . . . sk−1 ) − f (xt)) . Squaring both sides and using the Cauchy-Schwarz inequality gives (f (x) − f (xt))2 ≤ |t| (f (x) − f (xs1 ))2 + . . . + (f (xs1 . . . sk−1 ) − f (xt))2 . Summing in x x

(f (x) − f (xt))2 ≤ |t|

(f (x) − f (xt))2 N (s, t).

x∈G

s∈S

|f ). The right side Multiply both sides by Q(t)/2 and sum in t. The left side is E(f is Q(s) 1 ≤ AE(f |f ). (f (x) − f (xs)) |t|N (s, t)Q(t) 2 x∈G Q(s) t s∈S

✷ The ﬁrst time one looks at Theorem 2.6, things look hopeless. As will emerge, it is often possible to get useful bounds on A. This is illustrated in a series of examples below. First examples We begin with a very basic example which ‘works’ for all walks on all groups. This is then specialized to the walk on the symmetric group based on the generating set of a transposition and an n-cycle.

132

DIACONIS

Example 2.7 Let Q be a symmetric probability on G with support (Q) ⊃ S for a generating set S. Then the 2nd eigenvalue of Q satisﬁes π1 (Q) ≤ 1 − η/γ 2 with η = min Q(s), s∈S

γ = diameter of G for S.

Proof Use the comparison measure Q(t) ≡ 1/|G|. This has π 1 = 0. In Theorem 2.6, bound 1/Q(s) ≤ 1/η and |t|, N (s, t) ≤ γ. Thus A ≤ γ 2 /η. The result now follows from Corollary 2.4. ✷ Remark Example 2.7 coupled with Lemma 2.1 and Corollary 2.2 shows 4||Qk − u||2 ≤ |G|π2k with π ≤ max(1 − η/γ 2 , |1 − 2Q(id)|). This is often a useful, if not good, bound. Example 2.8 Let G = Sn , S = id, (1, 2), c, c−1 with c = (1, 2, . . . , n) an n-cycle. Take Q(t) = 1/4 or 0 as t is in S or not. In the language of cards, this walk either does nothing, switches the top two cards, puts the top to bottom or puts bottom to top, each with probability 1/4. Here it is easy to show that γ = diam ≤ 32 n2 , η = 14 , 1 − 2Q(id) = 12 . Thus 1 2k k 2 4||Q − u|| ≤ n! 1 − 4 ≤ exp{(log n!) − 2k/9n4 }. 9n Using log n! ∼ n log n we see that k of order n5 log n steps suﬃce to achieve randomness. Example 2.9 With G and Q as above, we may improve the bound on π1 by ˜ as in (2) of the Introduction. comparing with random transpositions; thus take Q 2 From character theory we know that π 1 = 1 − n . To use Theorem 2, write A≤4 |s|2 Q(s) s

where the sum is over transpositions s. Any transposition s can be written with at most 3n generators, so A ≤ 36n. Using π 1 = 1 − n2 in Corollary 2.4 gives 3 π1 ≤ 1 − 1/18n . Now, the argument of Example 2.8 shows order n4 log n steps suﬃce. Example 2.10 With G and Q as above, we will knock oﬀ another power of n thus getting to order n3 log n by using the full force of Theorem 2.5. Using A from Example 2.9, we have 1 2k k 2 k 2 −k/36n2 k/2A 2 +e + ||Q − u||2 . 4||Q − u|| ≤ |G| ||Q − u||2 ≤ |G| 2 Using the careful analysis of random transpositions we have the right hand side is small for k of order n3 log n. Here, it is possible to give a lower bound showing that order n3 logn steps are needed. See e.g. Wilson [75]. This series of examples should be extensible to a wide variety of groups and walks. The next section expands on this.

RANDOM WALKS ON GROUPS: CHARACTERS AND GEOMETRY

3

133

Other appearances of random transpositions

When Shahshahani and I completed our analysis of random transpositions it seemed very delicate; the method of proof breaks down if the generating set is not a union of conjugacy classes. The last example of section 3 shows that transpositions plus comparison can handle examples with no relation to conjugacy. In joint work with Saloﬀ-Coste [17] we treated many further examples: consider a connected graph on {1, 2, . . . , n}, take a transposition for each edge. This gives a generating set. If the graph is a path, we get the usual Coxeter generators. If the graph is a star, the walk becomes ‘transpose random with one’. Of course, for the complete graph, we get random transpositions. The comparison theory is easy and we derived reasonably sharp answers for this general set of problems. For a path the answer is order n3 log n. This had been done earlier by Aldous [1] using coupling. The lower bound for this case was only done recently by Wilson [75]. It introduces a powerful lower bound technique that seems very useful. For a star, n(log n + c) steps are necessary and suﬃce. Again, the example had been done earlier by Flatto-Odlyzko-Wales using the fact that the restriction of an irreducible representation of Sn restricts Sn−1 in a multiplicity free way. All other graphs give new examples, handled by a uniform method. The paper with Saloﬀ-Coste treats overhand shuﬄes, an open problem of Borel-Levy (choose a random packet from the center of the deck and cut it to the top.). While I will not expand on it here, the comparison approach is not restricted to symmetric random walk (Q(w) = Q(w−1 )). This is developed in great detail in Diaconis and Saloﬀ-Coste [22]. In particular, for n odd, the generating set {(1, 2), (1, 2, . . . , n)} (no identity, no inverses) yields a random walk that equilibriates in n3 log n steps. Another technical improvement developed in joint work with Saloﬀ-Coste [22] uses averages over random paths. I’ve been very pleased to see two applications in biology: DNA sequences evolve by a variety of transmutations (substitutions, insertions, deletions). There are also ‘translocations’, exchanging genetic material between chromosomes. This can be studied by picturing a row of symbols, picking a pair (i, j) from some distribution, and reversing the order i ↔ j i + 1 ↔ j − 1, . . .. Fill and Schoolﬁeld studied this (keeping track of face up and face down symbols which have a biological meaning) using the analog of random transpositions on the hyperoctahedral group Bn . Durrett [31] studied the process on the symmetric group and includes a serious comparison between data and model. A very diﬀerent application, to phylogenetic trees, is also based on random transpositions; see Diaconis and Holmes [15] which also contains pointers to coagulation processes in chemistry. All of this development leans on Frobenius’s neat character formula of example 1.4. I was surprised to see an earlier application of this formula in work of RothausThompson [63]. They were studying perfect codes on the symmetric group. This is a set of permutations w1 , w2 , . . . , wk such that the balls of radius h in the metric based on transpositions exactly partition the group. The problem can be phrased as convolution with the random transpositions measure and the character formula (and some clever number theory) yields some partial results. There is much left

134

DIACONIS

open as well. The earliest Application A most surprising discovery was made by Basil Gordon (personal communication). The Basic random transposition model appeared in work of Hurwitz [43, 44]. Hurwitz was counting the number of n-sheeted covers of the Riemann sphere with d-simple branch points. Riemanns existence theorem says that such covers are in one to one correspondence with ways of writing the identity permutation in Sn as a product of d transpositions. Here the individual transpositions specify how to glue the covering sheets over the various branch points. Transpositions occur because of the (assumed) square root singularities. d Up to multiplying by a factor of n2 this number of ways equals Qd (id). Using the Fourier inversion formula χλ (τ ) d Qd (id) = d2λ dλ λ n

Now Frobenius’s formula gives an explicit result. In fact, Hurwitz wanted to count irreducible covers. In group theory terms this means that he requires that the transpositions that appear must generate Sn . He originally solved this problem by a form of exclusion-inclusion (if the transpositions don’t generate Sn they generate a subgroup). His version of this is one of the earliest appearances of what is now called the exponential formula of enumerative combinatorics (see Stanley [70], Chapter 6)). By looking at examples computed in this way for small n, Hurwitz guessed that all the mess clears away and the ﬁnal answer is simple. Hurwitz made the conjecture: Let σ have cycle lengths k1 , k2 , . . . , kρ . The number of factorizations τ1 . . . τn+ρ−2 = σ of the permutation σ into a product of transpositions τi , where τ1 , . . . , τn+ρ−2 = Sn is ρ k ki +1 . (n + ρ − 2)! nρ−3 Π i i=1 ki ! The full result was proved by Goulden and Jackson [36] . In fact, Hurwitz outlined a proof which was completed by Strehl [72]. The enumerative theory of surfaces and their covers and triangulations has a rich development. Useful surveys are given by Jones [46], Zvonkin [76] and in the article by Condor that appears in this volume. There are amazing recent connections to modern theoretical physics through Gromov-Witten invariants. See the article by Goulden-Jackson-Vakil [37] for pointers. I have no doubt that some of these developments can be turned into shuﬄing theorems but this lies in the future. Deformations My most recent encounter with random transpositions comes through joint work with Arun Ram on analysis of ‘Systemic scan Metropolis algorithms’. This involves a deformation into the Iwahori-Hecke algebra. In the end, the familiar analysis based on Frobenius’s character formula allowed sharp estimates. I think there is much further work to be done here and will include a high level overview.

RANDOM WALKS ON GROUPS: CHARACTERS AND GEOMETRY

135

The Metropolis algorithm is a mainstay of scientiﬁc computing. It is used in physics, chemistry, biology, statistics and business applications. Billera and Diaconis [6] and Diaconis and Saloﬀ-Coste [24] are surveys with extensive pointers to the literature. The present example involves comparing two variants (random versus systematic scan). The problem is to draw repeated random samples from a non-uniform distribution on Sn . Fix 0 < θ < 1. Let π(w) = θ−(w) /P (θ−1 ) where P (θ−1 ) = θ−(w) w

is the normalizing constant. Here (w) is the length of the permutation w in the usual generating set si = (i, i + 1), 1 ≤ i ≤ n − 1. Of course, for θ = 1 this becomes the uniform distribution but, for e.g. θ = 12 , this concentrates on permutations with larger lengths. These non-uniform distributions are known as Mallows models and are widely applied. A standard way to sample from π(w) involves a random walk on Sn which may be called the Random Scan Metropolis Algorithm. It is simple to state. The walk starts at some ﬁxed permutation (say id) and proceeds by making random pairwise adjacent transpositions according to the following scheme. Suppose the walk is currently at w. Choose i uniformly in 1 ≤ i ≤ n − 1. If (si w) > (w) the walk moves to si w. If (si w) < (w), ﬂip a coin with probability of heads θ. If this coin comes up heads, move to si w. If the coin comes up tails, the walks stays at w. This generates a random sequence w0 = id, w1 , w2 , . . .. Simple theory shows that the probability of (wk = w) → π(w) as k → ∞. In our second variant, the walk proceeds as above but instead of choosing the proposal transpositions at random, things proceed systematically; ﬁrst try s1 then s2 , . . . , then sn−1 then sn−1 , sn−2 , . . . , then s1 . At each stage one compares the length and makes an auxiliary coin toss if needed. Call the result of one pass through (based on 2(n − 1) steps) the systematic scan Metropolis algorithm. Such systematic scans are widely employed in applications to Ising-like models in physics and image analysis. Again, simple theory shows that they converge to π(w). It is natural to ask how long each of the algorithms takes to converge and which or when one is better. The problems are largely open for the usual applications of the Metropolis algorithm. For the special permutation case discussed here, the analysis can be pushed through. Roughly stated, the systematic scan procedure takes order n passes (and so order n2 steps). This is stated more carefully below. In very recent work, Benjamini, et al. [5] have shown that the random scan version also takes order n2 steps to converge. This seems surprising, since the systematic procedure builds in some extra structure. This is the ﬁrst and only example where such a comparison has been made. The paper with Ram [16] carries this out for general ﬁnite reﬂection groups with similar ﬁndings. The reason for mentioning the subject here is that the analysis rests on a novel probabilistic interpretation of multiplication in the Hecke algebra. Let W be a ﬁnite Coxeter group generated by simple reﬂections s1 , . . . , sn . These deﬁne a length function with (id) = 0, (si ) = 1 and (si w) = (w) ± 1. The IwahoriHecke algebra H corresponding to W is the vector space with basis Tw for w ∈ W

136

DIACONIS

and multiplication given by

Tsi w Ti Tw = (q − 1)Tw + qTsi w

if (si w) = (w) + 1 if (si w) = (w) − 1

where Ti = Tsi . We have Ti2 = (q − 1)Ti + q or equivalently (Ti − q)(Ti + 1) = 0. I have always found this multiplication intriguing and tried to ﬁnd a stochastic interpretation. In the work with Ram we proved Theorem 3.1 Let W be a finite Coxeter group as above. Set q = θ−1 ,

Ti = Ti /q,

Tw = q −(w) Tw for w ∈ W.

Then the systematic scan Metropolis chain (symbolically K1 K2 . . . Kn Kn . . . K1 ) has the same transition matrix as multiplication by T1 T2 . . . Tn Tn . . . T1 in the IwahoriHecke algebra with basis Ti . Central to our work is the fact due to Breiskorn and Deligne that the long systematic scan (K1 K2 . . . Kn Kn . . . K1 ) · · · (K1 K2 K2 K1 )(K1 K1 ) corresponds to multiplying by an element in the center of H. This is the ‘q analog’, or deformation of the fact that the sum of all transpositions is in the center of the group algebra. The action by this element can be explicitly diagonalized. The eigenvalues are closely related to Frobenius’s formula and it was possible to push through a successful analysis. Let me end this sea of exposition with a clear mathematical result from my work with Ram. Theorem 3.2 Let K be the transition matrix for one pass of the short systematic scan algorithm on Sn . For = n/2 − (log n/ log θ) + c with c > 0, for all n 4||K1 − π||2 ≤ (eθ

2c+1

− 1) + n!θn

2 /8−n log n/ log θ+n(c+1/4)

.

Conversely, for < n/4, for fixed θ, ||K1 − π|| tends to 1 as n → ∞. I must not leave this part of the world without mentioning that Diaconis and Hanlon [14] analyzed the Metropolis deformation of the random transpositions chain on Sn with stationary distribution σ(w) = Z −1 θ|w| where |w| is the length if all transpositions are used. This deformation led to the Jack symmetric functions. It is an intriguing problem to see if the two parameter Macdonald polynomials can be obtained in this way.

RANDOM WALKS ON GROUPS: CHARACTERS AND GEOMETRY

4

137

Some open problems

There are an inﬁnite variety of open problems. These range from speciﬁc to fairly general. Speciﬁc problems can be helpful in pointing to the need for new tools and understanding. 1. It is well known (Suzuki [73]) that for p a prime, SLn (Fp ) is generated by elementary row operations, adding or subtracting one row from another. Let Eij be the n × n matrix with ones down the diagonal and a one in position (i, j) (for ﬁxed i = j). Then these elementary transvections generate. It is natural to ask for the speed of convergence. This problem appears in a variety of guises; Diaconis and Saloﬀ-Coste [23] relate it to a particle system and to a special case of the product replacement algorithm. Using the method outlined in section three they were able to show that order n4 (log p)2 steps suﬃce. The best lower bound available is of order n2 log p. This is a natural enough problem that an answer should be sought. The approach taken in [23] is to use comparison with the walk generated by random transvections (the whole conjugacy class). Hildebrand [39] analyzed this walk by character theory. Kai Magaard showed us that any transvection can be written as a product of at most 5n elementary transvections so we were oﬀ and running. Igor Pak has obtained some improvements and extensions but the general problem of determining the right rate (and perhaps showing there is a cut oﬀ) remains open. 2. Steinberg [71] showed that any ﬁnite group of Lie type can be generated by two generators. This oﬀers a list of problems: pick your favorite group (or one you’d like to get to know), ﬁgure out what Steinberg’s generators are and get to work. To make things easier, you might begin with p = 2. Glucks bounds and some basic geometry should suﬃce to show that order a small polynomial in rank(G) steps suﬃce. Even this would require honest work. Finding the right answer would be a major achievement. Here is a speciﬁc case which I ﬁnd interesting. Take SLn (Fp ). A Singer cycle is an element of maximal order. It can be explicitly constructed as the n × n companion matrix of a primitive polynomial. Let A be a Singer cycle and B = E12 (elementary transvection). These two matrices generate SLn (Fp ). They are the appropriate analogs of n-cycle and transposition in Sn . Bound the rate of convergence of this random walk. 3. Towards generality; all ﬁnite simple groups are generated by almost all pairs of elements. For the alternating group, this is a theorem of Dixon. It has been improved extended and reﬁned by a generation of group theorists. See Shalev [68] for the latest results. This suggests a class of problems: pick two elements of such a group at random and study the expected relaxation time for the random walk. I conjecture that it is bounded by a small polynomial in Rank(G). In the case of the alternating group, I conjecture it is bounded √ by n3 log n. The best that is known rigorously is order ec n (Babai). Sticking to An , it is possible that there are a bounded number of generators such that the relaxation time is of order n log n; a tantalizing conjecture of Lubotzky

138

DIACONIS says this can’t happen see Gamburd-Pak [33] for more on this.

4. The question of understanding the cutoﬀ phenomenon is perhaps more abstract. If an , bn are two sequences tending to inﬁnity with bn /an tending to zero and kn = an + cbn then a sequence of probabilities Qn on groups Gn satisﬁes a cut oﬀ if there are real functions f (c), g(c) ≥ 0 such that g(c) → 0, f (c) → 1 and n f (c) ≤ Q∗k n − u ≤ g(c).

A variety of our examples above satisfy this; for random transpositions, one may take an = n2 log n, bn = n. The fact that the phenomena was discovered at all suggest that it is generic. There may be some soft way to prove this along the lines of concentration phenomena in modern combinatorics. Failing this, one can try to establish it for sets of examples along the lines of Rousell [64], Lulov-Pak [56]. In [19] we show that Nilpotent groups of low derived length with small generating sets do not satisfy cutoﬀs. Random walks on groups like Cpn for n large do satisfy cutoﬀs. What’s going on? It is tempting to try to relate these cutoﬀs to other types of phase transitions as in Dubois et al. [30]. 5. In many examples there is a natural ordering on the irreducible representa tions ρ such that for “simple” probability measures Q, Q(ρ) is monotone decreasing as ρ moves away from the trivial representation. For example, for 2πh G = Cn with n odd and Q(1) = Q(−1) = 1/2, |Q(h)| = |cos n | this is one when h = 0 and decreases for increasing h, 0 ≤ h ≤ n/2. For random xλ (j) 1 I. In [8] the character ratio was shown transpositions Q(ρλ ) = n + dλ to be monotone in the usual partial order on partitions. This question can be asked in purely group theoretic terms. To be speciﬁc, let G be a ﬁnite simple group. Let c be a conjugacy class not in the center. Show that the character ratio |xρ (c)|/xρ (id) is monotone decreasing in the dimension xρ (id). Put this precisely the conjecture may be false but since many special cases have been found, some mild weakening must hold. The Riemann-Lebesgue lemma is actually an asymptotic version of this: let G be a compact group. Let f : G → C be in L1 then, f (ρ) tends to zero as dim(ρ) tends to inﬁnity. 6. One of the exciting developments of group theory is the modern theory of pgroups. At long last a theory of these monsters is beginning to emerge. This is summarized in recent books by Dixon et al. [29]. The theory focuses on groups of size pn with large class. One may study random walk on these. For example, the groups of maximal class are all generated by two generators. Pick a group and a generating set and start working. To be speciﬁc, the Nottingham group may be represented as polynomials f (x) = x+a2 x2 +. . .+ an xn , with ai in Fp , taken mod xn+1 . These form a group under composition. A generating set is x, x + x2 . What is the rate of convergence as a function of p and n? Here is one further speciﬁc example. Let G = Cp wrCp . This is a group of order pp+1 . It appeared early in Philip Hall’s study of regular pgroups (G is a “smallest” example of a non-regular group) if G is represented

RANDOM WALKS ON GROUPS: CHARACTERS AND GEOMETRY

139

as Cp acting on Cpp by cyclic shift then a generator of Cp and any non-zero vector in Cpp generate. Uyemura-Reyes [74] gives upper and lower bounds of order p3 and p3 log p respectively but the right answer is unknown. Reyes shows how random walk on such wreath products arise in card shuﬄing and relates these questions to recent work on the lamplighter group along with work of Grigorchuk et al. [38] who used the eigenvalues of such walks to disprove a question of Atiyah. 7. It is very natural to study Markov Chains on ﬁnite rings. The meat ax falls into this class. Here is one simple, completely open example. In Fp , consider the walk which moves from x to x + 1 or x2 with probability 1/2. I’m morally certain that this takes order log p steps. At present, I don’t even have a rough description of the stationary distribution (it is certainly not uniform). Further questions are in [24]. I am pleased to report that almost all the questions I posed in my book [8] have been usefully settled. The most annoying open one is Thorp’s model of shuﬄing cards [8] pg. 90. Added in proof. A very recent application of random transpositions occurs in Diaconis, P., Mayer-Wolf, E., Zeitouni, O., and Zerner. M. (2002). Uniqueness of invariant measures for split-merge transformations and the Poisson-Dirichlet Law. To appear, Ann. Probab. They prove a conjecture of Vershik on the stationary distribution of a coagulation-fragmentation process occurring in chemistry by showing that it reduces to following the cycle structure under random transpositions. References [1] Aldous, D. (1983). Random walk on ﬁnite groups and rapidly mixing Markov Chains. In Seminaire de Probabilit´es XVII, 243-297. Springer Lecture notes in Math. 986. [2] Aldous, D. and Diaconis, P. (1986). Shuﬄing cards and stopping times, Amer. Math. Month. 43, 333-348. [3] Aldous, D. and Fill, J. (2002). Reversible Markov Chains and random walk on graphs. Forthcoming book. [4] Belsley, E. (1998). Rates of convergence of random walk on distance regular graphs, Prob. Th. Related Fields 112, 493-533. [5] Benjamini, I., Beger, W., Hoﬀman, Z. Mossell, E. (2002). Mixing time for biased card shuﬄing. Preprint, Dept. of Mathematics, University of Washington. [6] Billera, L. and Diaconis, P. (2001). A geometric interpretation of the MetropolisHastings algorithm, Statistical Science 16, 335-339. [7] Carlin, E., Carvolho, M. and Loss, M. (2002). Determination of the spectral gap for Kac’s master equation. To appear Acta Math. [8] Diaconis, P. (1988). Group representations in Probability and Statistics, Institute of Mathematical Statistics, Hayward, CA. [9] Diaconis, P. (1996). The cutoﬀ phenomenon in ﬁnite Markov Chains, Proc. Nat. Acad. Sci. 43, 1659-1664. [10] Diaconis, P. (2002). Mathematical developments from the analysis of riﬄe shuﬄing. To appear, M. Liebeck (ed.). Proc. Durham Conference on Groups. [11] Diaconis, P. and Brown, K. (1998). Random walk and hyperplane arrangements, Ann. Probab. 26, 1813-1854. [12] Diaconis, P. and Fill, J (1990). Strong stationary times via a new form of duality, Ann. Prob. 18, 1483-1522.

140

DIACONIS

[13] Diaconis, P. and Graham, R. (1981). The analysis of sequential experiments with feedback to subjects, Ann. Statist. 9, 3-23. [14] Diaconis, P. and Hanlon, P. (1992). Eigen-analysis for some examples of the Metropolis algorithm, Contemp. Math. 138, 99-117. [15] Diaconis, P. and Holmes, S. (2002). Random walk on trees and matchings. In Electronic Jour. Probab. [16] Diaconis, P. and Ram, A. (2000). Analysis of systematic scan Metropolis algorithms using Iwahori-Hecke algebra techniques, Mich. Math. Jour. 48, 157-190. [17] Diaconis, P. and Saloﬀ-Coste, L. (1993). Comparison techniques for random walk on ﬁnite groups, Ann. Prob. 21, 2131-2156. [18] Diaconis, P. and Saloﬀ-Coste, L. (1993). Comparison theorems for reversible Markov Chains, Ann. Appl. Prob. 3, 696-730. [19] Diaconis, P. and Saloﬀ-Coste, L. (1994). Moderate growth and random walk on ﬁnite groups, GAFA 4, 1-36. [20] Diaconis, P. and Saloﬀ-Coste, L. (1995). An application to Harnack inequalities to random walk on nilpotent quotients, Jour. Four. Anal. Applications, Special Kahane Issue, 189-207. [21] Diaconis, P. and Saloﬀ-Coste, L. (1995). Random walks on ﬁnite groups: A survey of analytic techniques. In Probability Measures on Groups XI (H. Heyer ed.). World Scientific, Singapore, 44-77. [22] Diaconis, P. and Saloﬀ-Coste, L. (1996). Nash inequalities for ﬁnite Markov Chains, Jour. Th. Probab. 9, 459-510. [23] Diaconis, P. and Saloﬀ-Coste, L. (1996). Walks on generating sets of Abelian groups, Prob. Th. Related Fields 105, 393-421. [24] Diaconis, P. and Saloﬀ-Coste, L. (1998). What do we know about the Metropolis algorithm? Jour. Comp. System Sci. 5, 20-36. [25] Diaconis, P. and Saloﬀ-Coste, L. (2000). Bounds for Kac’s master equation, Commun. Math. Phys. 209, 729-755. [26] Diaconis, P. and Shahshahani, M. (1981). Generating a random permutation with random transpositions, Z. Wahr. Verw. Gebete 57, 159-179. [27] Diaconis, P. and Shahshahani, M. (1987). The subgroup algorithm for generating uniform random variables, Prob. Eng. Infosci. 1, 15-32. [28] Diaconis, P. and Shahshahani, M. (1987). Time to reach stationality in the BernoulliLaplace diﬀusion model, SIAM Jour. Math. Analysis 18, 208-218. [29] Dixon, J., Dusautoy, M., Mann, A., Segal, D. (1999). Analytic pro-P groups, 2nd ed., Cambridge University Press, Cambridge. [30] Dubois, O., Monassor, R., Selman, B. and Zecclina (eds.). Special issue on phase transitions in combinatorial problems, Theoretical Computer Science 265. [31] Durrett, R. (2002). Shuﬄing Chromosomes, Technical Report, Dept. of Mathematics, Cornell University. [32] Fomin, S. and Lulov, N. (1995). On the number of rim hook tableaux, Zap. Nauchn. Sem. Pomi. 223, 218-226. (Also, http://www.math.lsa.umich.edu/ Fomin/papers/). [33] Gambaurd, A. and Pak, I. (2002). Expansion of product replacement groups, S.O.D.A. 2002. [34] Gluck, D. (1994). Characters and random walk on ﬁnite classical groups, Adv. Math. 129, 46-72. [35] Gluck, D. (1999). First hitting times for some random walks on ﬁnite groups, Jour. Th. Probab. 12, 739-756. [36] Goulden, I. and Jackson, D. (1997). Transitive factorization into transpositions and holomorphic mappings on the sphere, Proc. Amer. Math. Soc. 125, 51-60. [37] Goulden, I. Jackson, D. and Vakil, R. (1999). The Gromov-Witten potential of a

RANDOM WALKS ON GROUPS: CHARACTERS AND GEOMETRY

141

point, Hurwitz numbers, and Hodge integrals. To appear Proc. Lond. Math. Soc. [38] Grigorchuk, R. Linell, P., Schick, T. and Zuk, A. (2000). On a conjecture of Atiyah, C.R. Acad. Sci. Paris. t331, Serie I, 663-668. [39] Hildebrand, M. (1992). Generating random elements in SLn (Fq ) by random transvections, J. Alg. Combin. 1, 133-150. [40] Horn, R. and Johnson, C. (1985). Matrix analysis, Cambridge Press. Cambridge. [41] Hughes, B. (1995). Random walks and random environments, Vol. 1, Oxford Press, Oxford. [42] Hughes, B. (1996). Random walks and random environments, Vol. 2, Oxford Press, Oxford. ¨ [43] Hurwitz, A. (1891). Uber Riemann’ sche Fl¨achen mit Gegebenen Verzweigungs Punkten, Math. Ann. 39, 1-66. ¨ [44] Hurwitz, A. (1902). Uber Die Anzahl Der Riemann’schen Fl¨achen Mit Gegebenen Veazweigungspunkten, Math. Analen. 55, 55-66. [45] Ingram, R. (1950). Some characters of the symmetric group, Proc. Amer. Math. Soc. 1, 358-369. [46] Jones, G. (1995). Enumeration of homomorphisms and surface-coverings, Quarter. J. Math. 46, 485-507. [47] Janvresse, E. (2002). Spectral gap for Kac’s model of Boltzmann equation, Ann. Probab. [48] Letac, G. (1981). Problems classiques de probabilitie sur un couple de Gelfand. In Springer Lecture Notes in Math 861. [49] Letac, G. (1982). Les Functions spheriques d’un couple de Gelfand symetrique et les chaines de Markov, Adv. Appl. Probab. 14, 272-294. [50] Liebeck, M. and Shalev, A. (2001). Diameters of ﬁnite simple groups: Sharp bounds and applications, Ann. Math. 154, 383-406. [51] Liebeck, M. and Shalev, A. (2001). Random r − s generation of ﬁnite classical groups, Bull. Lond. Math. Soc. 34, 185-188. [52] Liu, J. (2001). Monte Carlo Strategies in Scientific Computing, Springer, N.Y. [53] Lov´ asz, L. and Winkler, P. (1998). Mixing times, Ams. Dimacs. Series 14, 189-204. [54] Lubotzky, A. (1995). Cayley graphs: eigenvalues, expanders and random walks, LMS Lecture Note Series 218, Cambridge University Press, Cambridge. [55] Lulov, N. (1996). Random walk on the symmetric group generated by conjugacy classes, Ph.D. Dissertation, Dept. of Math., Harvard University. [56] Lulov, N. and Pak, I. (2002). Rapidly mixing random walks and bounds on characters of the symmetric group. Technical Report, Dept. of Mathematics, M.I.T. [57] Maslin, D. (2002). On the eigenvalues of Kac’s mastser equation. To appear, Composito Math. [58] Pak, I. (1997). Random walks on permutations: Strong uniform time approach. Ph.D. Dissertation, Dept. of Mathematics, Harvard University. [59] Pak, I. (2000). What do we know about the product replacement algorithm. In groups and computation III, 301-347, De Gruyter, Berlin. [60] Porod, U. (1996). The cut-oﬀ phenomenon for random reﬂections, Ann. Probab. 24, 74-99. [61] Roichman, Y. (1996). Upper bound on characters of the symmetric groups, Inven. Math. 125, 451-485. [62] Rosenthal, J. (1994). Random rotations: Characters and random walks on S0(n), Ann. Probab. 22, 398-423. [63] Rothaus, O. and Thompson, J. (1966). A combinatorial problem in the Symmetric Group, Pacific Jour. Math. 18, 175-178. [64] Roussel, S. (2000). Phenomene de cutoﬀ pour certaines marches aleatoires sur le

142

DIACONIS

groupe symetrique, Coll. Math. 86, 111-135. [65] Saloﬀ-Coste, L. (1997). Lectures on ﬁnite Markov Chains, Lecture Notes in Math. 1665, Springer, Berlin. [66] Saloﬀ-Coste, L. (2001). Probability on groups: Random walks and invariant diﬀusions, Notices of Amer. Math. Soc. 48, 968-977. [67] Schoolﬁeld, C. (1998). Random walks on wreath products of groups and Markov Chains on related Homogeneous spaces. Ph.D. Dissertation, Johns Hopkins University. [68] Shalev, A. (2000). Asymptotic group theory, Notices Amer. Math. Soc. 48, 383-389. [69] Silver, J. (1996). Weighted Poincare and exhaustive approximation techniques for scaled Metropolis-Hasings and spectral total variation convergence bounds in inﬁnite commutable Markov chain theory. Ph.D. Dissertation, Dept. of Mathematics, Harvard University. [70] Stanley, R. (1999). Enumerative Combinatorics, Vol. II, Cambridge University Press, Cambridge. [71] Steinberg, R. (1962). Generators for simple groups, Canadian Jour. Math. 14, 277283. [72] Strehl, V. (1996). Minimal transitive products of transpositions – The Reconstruction of a proof by A. Hurwitz. Seminar Lothaire Combin. 37 Art 537C. [73] Suzuki, M. (1982). Group Theory, I., Springer, New York. [74] Uyemura-Reyes (2002). Random walks, perfect shuﬄes, and BHR shuﬄes, Ph.D. Thesis, Dept. of Mathematics, Stanford University. [75] Wilson, D. (1997). Mixing times of Lozenge tiling and card shuﬄing Markov Chains. Preprint, Microsoft Research. [76] Zvonkin, A. (1997). Matrix integrals and map enumeration: An accessible introduction, Math. Comput. Modeling 26, 281-304.

ON DISTANCES OF 2-GROUPS AND 3-GROUPS ˇ DRAPAL ´ ALES

1

Department of Mathematics, Charles University, Sokolovsk´ a 83, 186 75 Prague, Czech Rep.

Abstract This paper is concerned with ﬁnite groups G(◦) and G(∗) of order n that are not isomorphic, and where the size of {(u, v) ∈ G × G; u ◦ v = u ∗ v} is the least possible (with respect to the given n). It surveys the case of 2-groups, discusses the possible generalization of the known results to p-groups, p an odd prime, and establishes the least possible distance in the case when G(∗) is an elementary abelian 3-group. Let G(◦) and G(∗) be ﬁnite groups of order n. Consider the set {(u, v) ∈ G × G; u ◦ v = u ∗ v} and denote its size by d(◦, ∗). The number d(◦, ∗) is called the (Hamming) distance of ◦ and ∗. If d(◦, ∗) < n2 /4 and n is a power of two, then G(◦) ∼ = G(∗), by [4]. Section 1 lists further results about distances of 2groups, while Section 2 discusses the associated proof machinery, and its possible generalization to p-groups, p an odd prime. In Section 2 there are also presented non-isomorphic p-groups G(◦) and G(∗), |G| = n > p, for which d(◦, ∗) = n2 (p2 − 1)/(4p2 ). This result is the best possible, when G(◦) is an elementary abelian 3-group—in Section 3 we shall show that in such a case d(◦, ∗) < 2n2 /9 implies G(◦) ∼ = G(∗). If H ≤ G(◦), then the set of all left (or right) cosets of H in G(◦) is denoted by L◦ (H) and R◦ (H), respectively. If A ⊆ G and B ⊆ G, then the size of {(u, v) ∈ A × B; u ◦ v = u ∗ v} will be denoted by d(A, B).

1

Survey of results

If G(◦) and G(∗) are groups of ﬁnite order n with d(◦, ∗) ≤ n2 /9, then there must be G(◦) ∼ = G(∗). This was proved in [2] by a combinatorial argument that uses the associativity of ◦ and ∗ in many ways, but never refers to the subgroup structure of any of the both groups. It was conjectured in [2] that G(◦) ∼ = G(∗) follows already from d(◦, ∗) < n2 /4, and Donovan, Praeger and Oates-Williams constructed in their common paper [1] a number of 2-groups with d(◦, ∗) = n2 /4. Several such examples can be found in [2] as well. All published cases share the following conditions (1) and (2): (1) There exists S < G such that x ◦ h = x ∗ h and h ◦ x = h ∗ x, for all h ∈ S and x ∈ G. 1 Work supported by institutional grant MSM 113200007, and by Grant Agency of Charles University, grant 269/2001/B-MAT/MFF.

144

´ DRAPAL

(2) There exists H ≤ G(◦), S < H and |H : S| = 2, where for all (α, β) ∈ L◦ (H) × R◦ (H) one can ﬁnd (α0 , β0 ) ∈ L◦ (S) × R◦ (S) such that α0 ⊆ α and β0 ⊆ β, and for every (u, v) ∈ α×β, u◦v = u∗v if and only if (u, v) ∈ α0 ×β0 . Note that (1) and (2) imply a weaker condition (3) There exists H ≤ G(◦), S < H and |H : S| = 2, such that d(α, β) = |S|2 for all (α, β) ∈ L◦ (H) × R◦ (H). Theorem 1.1 Suppose that G(◦) and G(∗) are ﬁnite groups which satisfy (1) and (2). Then S and H can be chosen in such a way that S is normal in both G(◦) and G(∗), and that G(◦)/S ∼ = G(∗)/S is either cyclic or dihedral (the elementary abelian group of order four is to be regarded as a dihedral group here). This theorem is the main result of [5]. It was discovered in the context of an ongoing research of 2-groups that can be placed at quarter distance. Such a phrase will be used for two groups G1 and G2 of order n, for which there exist groups G(◦) ∼ = G2 with d(◦, ∗) = n2 /4. = G1 and G(∗) ∼ Lemma 1.2 Suppose that groups G(◦) and G(∗) have the common unit. Then both U = {u ∈ G; u ◦ x = u ∗ x for all x ∈ G} and V = {v ∈ G; x ◦ v = x ∗ v for all x ∈ G} are subgroups of G(◦) and G(∗). Proof If u1 , u2 ∈ U and x ∈ G, then u1 ◦ u2 = u1 ∗ u2 , and (u1 ◦ u2 ) ◦ x = ✷ u1 ∗ (u2 ◦ x) = u1 ∗ u2 ∗ x = (u1 ◦ u2 ) ∗ x. The rest is easy. If U and V are of index 2, then either U = V , or S = U ∩ V is of index 4 in G. If U = V , then G/U is cyclic of order 2, and if U = V , then G/S is elementary abelian of order 4. If G is ﬁnite of order n and d(◦, ∗) = n2 /4, then these situations correspond to the most simple situations described by Theorem 1.1. One can also start from a group G = G(·), from its subgroups U and V of index 2, and from an element h ∈ G that satisﬁes some additional conditions, and deﬁne an operation ∗ as follows: If U = V and h ∈ Z(G) ∩ U , put uv if u ∈ U or v ∈ U , u∗v = uvh if u ∈ G \ U and v ∈ G \ U ; if U = V , h ∈ Z(U ∩V ), huh = u for all u ∈ U \V and hvh = v for all v ∈ V \U , put  if u ∈ U or v ∈ V ,  uv uvh if u ∈ / U and v ∈ U \ V , u∗v =  / U and v ∈ G \ (U ∪ V ). uvh−1 if u ∈ Proposition 1.3 The operation ∗ is a group operation in both these constructions. The above statement can be directly veriﬁed. As an example, let us consider the case when U = V and x, y, z ∈ G \ (U ∪ V ). This set is one of the four cosets

ON DISTANCES OF 2-GROUPS AND 3-GROUPS

145

of G/S, S = U ∩ V , and hence its every element, say z, can be expressed as uv, where u ∈ U \ V and v ∈ V \ U . Thus hz = huv = uh−1 v = uvh = zh. There is x ∗ y = xyh−1 ∈ S and y ∗ z = yzh−1 ∈ S, by the deﬁnition of ∗. Therefore x ∗ (y ∗ z) = x(y ∗ z) = xyzh−1 = xyh−1 z = (xyh−1 ) ∗ z = (x ∗ y) ∗ z. Properties of the groups thus constructed are discussed in [8]. If U and V are deﬁned as in Lemma 1.2, and if they are of index 2, with d(◦, ∗) = n2 /4 and n = |G|, then G(∗) can be always derived from G(◦) by one of the above constructions (a fact, which is not diﬃcult to prove). Let us now consider, for n a power of two, a graph with vertices corresponding to all isomorphism types of n-element groups, and with edges connecting those groups that can be placed at quarter distance. Computations done by Natalia Zhukavets and Martin B´ alek show that this graph is connected when n ≤ 32 (and it is conjectured that it is connected for all powers of two). The constructions of Proposition 1.3 establish the connectivity for n ≤ 16, and in the case n = 32 they suﬃce to interconnect all groups but one. The exceptional group (cf. [8]) is made adjacent to the rest by a construction that reverses Theorem 1.1 in the case when G/S is cyclic of order 4. This method (which we do not describe here, but which is similar to methods of Proposition 1.3) can be generalized to the case when G/S is a cyclic group of order m ≥ 4, m a power of two, and then another method corresponds to the case when G/S is a dihedral group of order 2m. If G(◦) and G(∗) satisfy the assumptions of Theorem 1.1, and n = |G| is a power of two, then G(∗) can be obtained from G(◦) by one of the above mentioned methods [5]. Nevertheless, all these methods together do not suﬃce to show the connectivity of the quarter distance graph if n = 64. They yield two components, one of which has only 8 vertices, and thus there must exist also other constructions, if the conjecture about connectivity holds. The search for such a construction can be guided by the following result [6]:

Theorem 1.4 Suppose that G(◦) and G(∗) are ﬁnite 2-groups of order n with d(◦, ∗) ≤ n2 /4. If G(◦) and G(∗) are not isomorphic, then there exist subgroups S < H ≤ G(◦) that satisfy (1) and (3). One can easily prove that (1) and (3) imply (2), if S G(◦). This suggests that any further constructions of the quarter distance are extensions of a construction that corresponds to Theorem 1.1. We shall ﬁnish this section by pointing out another aspect of the case d(◦, ∗) < n2 /4, where G(◦) and G(∗) are 2-groups of order n. The proof of isomorphism published in [4] guarantees the existence of τ : G(◦) ∼ = G(∗), but does not say anything about the number of x ∈ G √ with τ (x) = x. It turns out [7] that this number has to be greater than (3 + 1/ 3)n/4 ≈ 0.89n. This means that for a given 2-group G(◦) one obtains all groups G(∗) with d(◦, ∗) < n2 /4 just by permuting a small subset of G.

´ DRAPAL

146

2

Inequalities and examples

The main results of [4] and [6] are based on a complex induction process that builds an isomorphism G(◦) ∼ = G(∗) by coupling (through an isomorphism) each 2-subgroup of G(◦) with a unique 2-subgroup of G(∗). This coupling is ﬁrst done for subgroups of order 2, and then extended, step by step, to subgroups of the double order. Each step is divided, in fact, into two substeps. If there is given σ:S∼ = T , where S < G(◦) and T < G(∗), and if there is given H ≤ G(◦), where S < H and |H : S| = 2, then one ﬁrst determines a (unique) K ≤ G(∗) with T < K and |K : T | = 2, and then constructs a (unique) isomorphism τ : H ∼ = K. The uniqueness is to be understood with respect to a system of inequalities which are devised in such a way that the object (i.e., a subgroup or an isomorphism) which is being selected is the only one which fulﬁlls the respective inequalities. We shall say more about inequalities that govern the choice of the isomorphism. For an isomorphism σ : S ∼ = T , where S ≤ G(◦) and T ≤ G(∗), write u ≡σ v if v = u ◦ h = u ∗ σ(h) for some h ∈ S, and u σ ≡ v if v = h ◦ u = σ(h) ∗ u for some h ∈ S. These relations are equivalences [4, Proposition 4.2], and each block of ≡σ is clearly contained in α ∩ α , for uniquely determined α ∈ L◦ (S) and α ∈ L∗ (T ). The importance of σ ≡ and ≡σ follows from [4, Proposition 4.4], which is here rephrased in the following way: Lemma 2.1 Let ui , vi ∈ G, i ∈ {1, 2}, be such that u1 ◦ v1 = u2 ◦ v2 = u1 ∗ v1 . (i) If u1 ≡σ u2 , then u2 ◦ v2 = u2 ∗ v2 if and only if v1 σ≡ v2 ; and (ii) if v1 σ≡ v2 , then u2 ◦ v2 = u2 ∗ v2 if and only if u1 ≡σ u2 . For α ∈ L◦ (S) consider the greatest block A ⊆ α of ≡σ , and put ϕσ (α) = |A| − s/2. Similarly, ψσ (β) = |B| − s/2, where B ⊆ β is a block of σ ≡ of the maximal size, β ∈ R◦ (S). If |S| = s is a power of two, then d(α, β) ≥ s(ϕσ (α) + ψσ (β)) − 4ϕσ (α)ψσ (β),

(∗)

by [4, Proposition 4.11]. This inequality does not have to hold, if S is not a 2-group. If S isa 2-group, then this inequality is an important tool◦ for a selection of σ such ψσ (β) > n/4, β ∈ R (S). that ϕσ (α)n/4, α ∈ L◦ (S), and One can ask why one does not work directly with sizes |A| and |B|, and why they are adjusted by the subtraction of s/2. There is no reason other than the fact that inequality (∗) can be manipulated more easily when the adjusted values are used, and that its interaction with other necessary inequalities is then more transparent. There are reasons to believe that (p) (p) (p) d(α, β) ≥ s(ϕ(p) σ (α) + ψσ (β)) − 2πϕσ (α)ψσ (β)

(†)

holds in every p-group, p a prime. Here π = p/(p − 1), α ∈ L◦ (S), β ∈ R◦ (S), σ : (p) (p) S∼ = T , S ≤ G(◦), T ≤ G(∗), s = |S|, ϕσ (α) = |A| − s/p and ψσ (β) = |B| − s/p, where A ⊆ α and B ⊆ β are blocks of the maximal size. Inequality (†) coincides with (∗), if p = 2, and will be veriﬁed in Section 3 for the case s = 3. For other cases it is to be regarded as a conjecture. The potential proof

ON DISTANCES OF 2-GROUPS AND 3-GROUPS

147

is complicated by the need to exclude conﬁgurations that point to the existence of a q-element subgroup, q a prime smaller than p. No such conﬁgurations interfere when p = 2, and that is why one can prove (∗) just by arguments that involve only basic Latin square properties and a straightforward induction. If G(+) = Zp ×Zp , and G(·) is a cyclic group on Zp ×Zp , where (1, 0)i+pj = (i, j), then d(+, ·) = p3 (p − 1)/2 = p4 /(2π). This was observed in [2, Example 7.3], and was an inspiration to the conjecture that d(◦, ∗) < n2 /(2π) implies G(◦) ∼ = G(∗) when p is the least prime dividing n = |G| (see also [3, Conjecture 3.1]). We shall now show that this conjecture is false when p ≥ 3. The construction is embarrassingly simple. Proposition 2.2 Let p ≥ 3 be an odd integer, and put C = {(−p+1)/2, . . . , −1, 0, 1, . . . , (p−1)/2}, and consider it as an abelian group with addition modulo p. Deﬁne on C ×C, in addition to the additive structure, a multiplicative structure of a cyclic group by setting (1, 0)i+pj = (i, j) for all i, j ∈ C. Then d(+, ·) = p2 (p2 − 1)/4. Proof If u = (0, j) for some j ∈ C, then u · v = u + v for all v ∈ C × C. Suppose 1 ≤ i ≤ (p − 1)/2 and 1 ≤ x ≤ (p − 1)/2, and let us have j, y ∈ C. Then (i, j) · (−x, y) = (1, 0)(i−x)+p(j+y) , where j + y ∈ C is computed modulo p. We obtain (i, j) · (−x, y) = (i, j) + (−x, y), as i − x ∈ C. Furthermore, (i, j) · (x, y) = (1, 0)(i+x)+p(j+y) , and we see that (i, j) · (x, y) = (i, j) + (x, y) just when i + x ≥ (p + 1)/2. This occurs in (p2 − 1)/8 cases, and the same number of inequalities is obtained from the products of (−i, j) and (−x, y). There are p2 choices of (j, y), ✷ and hence d(+, ·) = p2 (p2 − 1)/4 as required. Corollary 2.3 Let p be an odd prime, n ≥ 1 an integer, and suppose that p2 divides n. Then there exist nonisomorphic groups G(◦) and G(∗), n = |G|, such that d(◦, ∗) = n2 (p2 − 1)/(4p2 ). Proof Consider groups from Proposition 2.1, and their product with a common ✷ group of order n/p2 .

3

Computations

The purpose of this section is to provide starting steps for treatments of distances of 3-groups, to about the same extent as [3] initiated the study of distances of 2-groups. Lemma 3.1 Let G(◦) and G(∗) be groups, and let S ≤ G(◦) be a subgroup of order 3. Then d(α, β) = 1 for all α ∈ L◦ (S) and β ∈ R◦ (S). Proof Start from the contrary, and let w ∈ α ◦ β, ui ∈ α and vi ∈ β, 1 ≤ i ≤ 3, be such that ui ◦ vi = w = u1 ∗ v1 = u2 ∗ v2 = u3 ∗ v3 . Then u2 ∗ v1 = u3 ∗ v2 , u1 ∗ v2 = u2 ∗ v3 , and the quadrangle criterion gives u3 ∗ v3 = (u3 ∗ v2 ) ∗ (u2 ∗ v2 )∗ ∗ (u2 ∗ v3 ) = (u2 ∗ v1 ) ∗ (u1 ∗ v1 )∗ ∗ (u1 ∗ v2 ) = u2 ∗ v2 = w, a contradiction (inverses in G(∗) are denoted by x∗ ). ✷

´ DRAPAL

148

Lemma 3.2 Let G(◦) and G(∗) be groups of an odd order, and let σ : S ∼ = T be an isomorphism, where S ≤ G(◦) and T ≤ G(∗) are subgroups of order 3. Then (3) (3) (3) d(α, β) ≥ 3(ϕ(3) σ (α) + ψσ (β) − ϕσ (α)ψσ (β))

for all (α, β) ∈ L◦ (S) × R◦ (S). (3)

(3)

Proof Put a = ϕσ (α) and b = ψσ (β), and recall that there exist A ⊆ α and B ⊆ β that are blocks of ≡σ and σ≡, respectively, |A| = a + 1 and |B| = b + 1. By [4, Lemma 4.6], d(α, β) ≥ 3(b − a), and hence also d(α, β) ≥ 3(a − b). This means that d(α, β) ≥ 3(a + b − ab) holds if a = 0 or b = 0, and we can assume a, b ∈ {1, 2}. Let us ﬁrst have a = 2. Then only the case b = 1 requires our attention. We shall show that for all w ∈ α ◦ β there exists (x, y) ∈ α × β with w = x ◦ y = x ∗ y. Consider y1 ∈ B and y2 ∈ β \ B, and choose x1 , x2 ∈ α with w = x1 ◦ y1 = x2 ◦ y2 . There cannot be both w = x1 ∗ y1 and w = x2 ∗ y2 , by Lemma 2.1 The case b = 2 is similar, and hence we can assume a = b = 1. Deﬁne x ∈ α and y ∈ β as the only elements out of A and B, respectively, and put w = x ◦ y. If there exists (u, v) ∈ α × β with w = u ◦ v = u ∗ v, then one obtains d(α, β) ≥ 3 by the same argument as in the preceding paragraph. Let us assume u ◦ v = u ∗ v for all (u, v) ∈ α × β with u ◦ v = w. If there exists (u, v) ∈ A × B and w ∈ α ◦ β with u ◦ v = u ∗ v = w and w = w, then there exists (u , v ) ∈ A × B with u ◦ y = w = x ◦ v . In such a case Lemma 2.1 implies u ◦ y = u ∗ y and x ◦ v = x ∗ v , and we get d(α, β) ≥ 3 again. Hence we can assume that for (u, v) ∈ α × β there is u ◦ v = u ∗ v if and only if (u, v) ∈ A × B and u ◦ v = w. Determine α , α ∈ L∗ (T ) by A ⊆ α and x ∈ α , and determine β , β ∈ R∗ (T ) by B ⊆ β and y ∈ β . Note that a = 1 = b imply α = α and β = β . The set (α ∗ β ) ∩ (α ∗ β ) contains the set x ∗ B = x ◦ B = A ◦ y = A ∗ y, and hence it has at least two elements. Its size has to divide the order of T (cf. [4, Lemma 1.2]), and thus α ∗ β = α ∗ β . There is w ∈ (α ∗ β ) ∩ (α ∗ β ), and therefore there exists, by [4, Lemma 1.3], a 6-element subgroup K ≤ G(∗) with α ∪ α ∈ L∗ (K) and β ∪ β ∈ R∗ (K), a contradiction to the assumption that G is of an odd order. ✷ Lemma 3.3 Consider isomorphisms σi : S ∼ = Ti , i ∈ {1, 2}, where S ≤ G(◦) and (3) (3) Ti ≤ G(∗) are 3-element subgroups. If σ1 = σ2 , then ϕσ1 (α) + ϕσ2 (α) ≤ 2. Proof Let Ai ⊆ α be the corresponding blocks of maximal size, i ∈ {1, 2}. If |A1 ∩A2 | ≥ 2, then there necessarily T1 = T2 , and also σ1 = σ2 . Hence |A1 |+|A2 | ≤ 4, and the rest is clear. ✷ Proposition 3.4 Let G(◦) and G(∗) be groups of an odd order n, and assume d(◦, ∗) < 2n2 /9. Then for every S ≤ G(◦), |S| = 3, there exists a unique T ≤ G(∗), (3) for which one can ﬁnd a (unique) isomorphism σ : S ∼ ϕσ (α) > n/3, = T with α ∈ L◦ (S).

ON DISTANCES OF 2-GROUPS AND 3-GROUPS

149

Proof We assume d(◦, ∗) < (n/3)(2n/3), and therefore there has to exist β ∈ R◦ (S) with d(G, β) < 2n/3. This means that d(α, β) < 2 for some α ∈ L◦ (S), and thus d(α, β) = 0, by Lemma 3.1. However, d(α, β) = 0 clearly implies the existence of T ≤ G(∗) with (α, β) ∈ L∗ (T ) × R∗ (T ), and the existence of such a σ : S ∼ = T that (3) (3) (3) ϕσ (α) = 2 = ψσ (β). From Lemma 3.2 we get 2n/3 > d(G, β) ≥ 3 (2−ϕσ (α)), (3) (3) ϕσ (α) > α ∈ L◦ (S), and the latter value equals 2n − 3( ϕσ (α)). We obtain 4n/9. If σ : S ∼ = T , where T ≤ G(∗) and (σ , T ) = (σ, T ), then Lemma 3.3 gives (3) (3) (ϕσ (α) + ϕσ (α)) ≤ 2n/3, and hence S determines exactly one pair (σ, T ) that satisﬁes requirements of the proposition. ✷ Proposition 3.4 yields a bijection between 3-element subgroups of G(◦) and G(∗), as it can be applied in both directions. Elementary abelian 3-groups have more subgroups of order 3 than every other group of the same order, and we can thus state: Corollary 3.5 Let G(◦) and G(∗) be groups of order 3k , k ≥ 1. If G(◦) is elementary abelian and d(◦, ∗) < 2 · 32k−2 , then G(∗) is elementary abelian as well. References [1] D. Donovan, S. Oates-Williams and C. E. Praeger, On the distance between distinct group Latin squares, J. Comb. Des., 5(1997), 235-248. [2] A.Dr´ apal, How far apart can the group multiplication tables be?, Europ. J. Combinatorics 13(1992), 335–343. [3] A. Dr´ apal, On distances of multiplication tables of groups, Proceedings of Groups St. Andrews 1997 in Bath, C. M. Campbell and E. F. Robertson (eds.), Cambridge University Press, 1999, pp.248–252. [4] A.Dr´ apal, Non-isomorphic 2-groups coincide at most in three quarters of their multiplication tables, Europ. J. Combinatorics 21(2000), 301–321. [5] A. Dr´ apal, On groups that diﬀer in one of four subsquares (submitted). [6] A. Dr´ apal, Diﬀerence set of 2-groups at quarter distance is distributed uniformly, manuscript. [7] A. Dr´ apal, Multiplication tables of 2-groups which diﬀer in less than a quarter of places yield an isomorphism with many ﬁxed points (submitted). [8] N. Zhukavets, On small distances of small 2-groups, Comment. Math. Univ. Carolinae 42(2001), 247–257.

ZETA FUNCTIONS OF GROUPS: THE QUEST FOR ORDER VERSUS THE FLIGHT FROM ENNUI MARCUS DU SAUTOY Mathematical Institute, 24-29 St Giles, Oxford OX1 3LB, UK E-mail: [email protected] URL http://www.maths.ox.ac.uk/~dusautoy

“Restate my assumptions. 1. Mathematics is the language of nature. 2. Everything around us can be represented and understood through numbers. 3. If you graph these numbers, patterns emerge. Therefore: there are patterns everywhere in nature.” Max Cohen in the ﬁlm Π Mathematics is about the search for patterns, to see order where others see chaos. We are very lucky to ﬁnd ourselves studying a subject which is neither so rigid that the patterns are easy, yet not too complicated lest our brains fail to master its complexities. John Cawelti sums up this interplay perfectly in a book not about mathematics but about mystery and romance [1]: “if we seek order and security, the result is likely to be boredom and sameness. But rejecting order for the sake of change and novelty brings danger and uncertainty. . . the history of culture can be interpreted as a dynamic tension between these two basic impulses. . . between the quest for order and the ﬂight from ennui.” There are many objects in the mathematical world that look wild and untamable - the primes, the decimal expansion of π. In this survey I want to look at one of the wild corners of our own subject of group theory. Despite having a classiﬁcation of the building blocks of all ﬁnite groups, we are a long way from understanding the chemicals we can build from these atoms. The challenge begins with trying to understand the groups built out of the simplest of the simple groups, a cyclic group of order p. Trying to classify groups of prime power order has long been considered a diﬃcult and wild problem. A major theme throughout the last two hundred and ﬁfty years of mathematics is the power of an analytic function called a zeta function to bridge that divide between dangerous uncertainty and the quest for order.

1 1.1

Breaking the test ban treaty on zeta functions Euler

Leonhard Euler was probably the ﬁrst person to realise the scope of a zeta function to capture information and reveal patterns where previously there were none. The Bernoulli family had saved Euler from a life in the church and within time Euler joined Daniel Bernoulli in St. Petersburg. No doubt they talked about Daniel’s

ZETA FUNCTIONS OF GROUPS

151

interest in trying to calculate a precise value for the sum of the squares of the harmonic series: 1 + 1/4 + 1/9 + 1/16 + ... + 1/n2 + ... Daniel Bernoulli had estimated its value to be something in the region of 8/5 in a letter to Goldbach in 1728. As Euler wrote in 1735: “So much work has been done on the series that it seems hardly likely that anything new about them may still turn up. . . I, too, in spite of repeated eﬀort, could achieve nothing more than approximate values for their sums. Now, however, quite unexpectedly, I have found an elegant formula depending upon the quadrature of the circle”, in modern parlance the number π. Whereas the decimal expansion of π is a chaotic scramble of numbers, Euler hit the scientiﬁc headlines by identifying the following value for the sum of the squares of the harmonic series: 1 + 1/4 + 1/9 + 1/16 + ... + 1/n2 + ... = π 2 /6. This was the ﬁrst inkling that the zeta function ζ(s) = 1 + 2−s + 3−s + ... + n−s + ... would prove a powerful tool in ﬁnding order where previously there seemed just randomness. Euler went on to identify the values of the zeta function at all even numbers in terms of even powers of π : ζ(2n) =

22n−1 |B2n | 2n π . (2n)!

In recognition of the support of the Bernoulli family which had brought him to St. Petersburg, he even managed to identify the constants B2n involved in his evaluation as the Bernoulli numbers discovered by Daniel’s father Jacob. No one to this day has identiﬁed a comparable expression for the zeta function at odd integers. Where the zeta function proved most powerful was in understanding another chaotic sequence of numbers - the primes. Again it was Euler who understood why there was a connection between the zeta function and the primes. The Euler product rewrites the zeta function as an inﬁnite product over primes p of very simple rational functions: ζ(s) =

p prime

1 . (1 − p−s )

This expression captured in one formula the Greek discovery of the Fundamental Theorem of Arithmetic. But Euler could also use the zeta function to express another Greek discovery. The fact that the harmonic series 1 + 1/2 + 1/3 + ... + 1/n + ... diverged meant that there must be inﬁnitely many primes.

152 1.2

DU SAUTOY Dirichlet

Euler did not realise the full signiﬁcance of his product to release the secrets of the primes. It was Dirichlet in 1837 who made the next step. Dirichlet used to sleep with Gauss’ Disquisitiones Arithmeticae under his pillow in the hope that inspiration would come in the night. Gauss had explained in his treatise the conjecture due to Fermat that if r and N were coprime then there were inﬁnitely many primes p with p = r mod N. Whether inspired by a dream prompted by the book under his pillow or not, Dirichlet began the proliferation of zeta functions by deﬁning a variant of the classical zeta function. He deﬁned for each primitive residue class character χ, a character of (Z/N Z)∗ for some N , what is now called the Dirichlet L-function: L(s, χ) =

∞

χ(n)n−s .

n=1

By proving that the L-function is non-zero at s = 1, Dirichlet successfully proved that the sequence of integer r, r +N, r +2N, r +3N contained inﬁnitely many prime numbers as Fermat had guessed. −s where a is an inﬁnite sequence of numbers Any function of the form ∞ n n=1 an n is called a Dirichlet series, in honour of Dirichlet who ﬁrst started the perturbation of the classical zeta function. Dirichlet succeeded Gauss to the professorship in G¨ ottingen where he probably talked to Riemann about the power of these functions. It was of course Riemann who realised the full potential of the zeta function to tell you information about prime numbers. Riemann had been greatly inspired by Cauchy’s work emerging from the Paris academies about functions of complex variables. By considering the zeta function as a function on complex variables Riemann found that he had access to one of the ultimate goals of mathematics. He discovered an explicit formula for the number of primes less than any integer N in terms of the zeros of the zeta function. To see the zeros, he had to analytically continue the function beyond the region of convergence Re(s) > 1. His observation that the zeros seemed to all have real part equal to 1/2 will earn anyone who can prove his Hypothesis a million dollars thanks to the Clay foundation. It would also explain why the primes look like they were distributed as if they were chosen using prime number dice. 1.3

Dedekind

Dedekind, who was a contemporary of Riemann in G¨ ottingen, was perhaps the ﬁrst to use these zeta functions to try to understand properties of algebraic structures. He deﬁned for any number ﬁeld K extending the rational numbers Q, a function which now goes by the name of the Dedekind zeta function: N (a)−s ζK (s) = a

=

a

|ϑK : a|−s .

ZETA FUNCTIONS OF GROUPS

153

The sum is taken over all the ideals in the ring of integers ϑK and N (a) denotes the norm of the ideal which is equal to the index of the ideal in ϑK as an additive subgroup. In the middle of the nineteenth century it could be seen that the analytic behaviour of this function at the pole at s = 1 gave one algebraic information. For example, the residue at this pole contains information about the class number of K which measures how badly the ﬁeld deviates from uniquely factorizing. This was ﬁrst proved by Dirichlet for quadratic forms which is equivalent to quadratic ﬁelds. Subsequently Dedekind generalized Dirichlet’s ideas to the case of a general number ﬁeld. 1.4

Artin, Hasse, Weil

In 1924 Artin extended Dedekind’s approach by looking at zeta functions attached to ﬁnite extensions of global ﬁelds of characteristic p. For example, one can consider analysing the ideal structure of the ﬁeld K got by extending the ﬁeld Fp (x) by √ x3 − x. Let D be the integral closure of Fp [x] in K. In his thesis, Artin studied the zeta function: |D : a|−s ζD (s) = a

where the sum is taken over all ideals in the ring D. Artin discovered that this zeta function encoded information about the number of points in projective space on the associated elliptic curve E : y 2 = x3 − x which was at the heart of building the ﬁeld extension K. In particular, set Npm = |E(Fpm )| = (a, b) ∈ F2pm : b2 = a3 − a + 1 where the addition of the extra point corresponds to the point at inﬁnity in projective space. Artin proved that ζD (s) = (1 − p−s )ζ(Ep , s) where

ζ(Ep , s) = exp

∞ m=1

p−ms N pm m

.

So Artin’s zeta functions are intimately connected to the behaviour of the number of points on the variety used to deﬁne the ﬁeld extension. Artin succeeded to prove that ζ(Ep , s) is a rational function in p−s for a range of elliptic curves. In particular, he showed that the zeta function can be expressed in the following form: 1 − (p + 1 − Np )p−s + p1−2s (1 − p−s )(1 − p1−s ) (1 + πp p−s )(1 + πp p−s ) = ; (1 − p−s )(1 − p1−s )

ζ(Ep , s) =

for a certain complex number πp where πp denotes the complex conjugate.

154

DU SAUTOY

In the case of around 40 elliptic curves Artin was able to prove what is known as the Riemann Hypothesis for curves, namely that |πp | = |πp | = p1/2 . Hasse than extended the argument to prove the Riemann Hypothesis for all elliptic curves. The major breakthrough came during the Second World War. Weil deduced a more general expression for the zeta function of a higher genus curve where the degree of the numerator is twice the genus. Weil’s great success was to prove, whilst he was in prison in France during the war for desertion, that the zeros of these rational functions all had absolute value p1/2 , the Riemann Hypothesis for function ﬁelds. The Weil conjectures concern generalizations of the properties of these functions when one counts points on more general varieties than just curves. These conjectures were spectacularly conﬁrmed in 1973 thanks to the genius of Deligne who exploited the great ediﬁce constructed by Grothendieck. 1.5

Birch Swinnerton-Dyer

Since solutions to equations over the rational integers would reduce mod p to solutions in ﬁnite ﬁelds, was there some way to piece these zeta functions together to give information about rational solutions to diophantine equations? Hasse proposed that one should view the zeta functions counting points over Fpn as local zeta functions of a global object. By using the idea of the Euler product, Hasse proposed taking the Euler product of these local objects as the deﬁnition of the global object. The global zeta function of the elliptic curve is essentially (there are ﬁnitely many bad primes where a variation should be used but we ignore this) deﬁned to be: ζ(Ep , s) ζ(E, s) = p

=

(1 − (p + 1 − Np )p−s + p1−2s ) p

(1 − p−s )(1 − p1−s )

This zeta function converges for Re(s) > 3/2. This follows from the following standard facts: (1) n>0 (1 + an ) converges absolutely if and only n>0 |an | converges. (2) p prime p−s converges for Re(s) > 1. The Birch Swinnerton-Dyer Conjecture asserts (in the simplest case) that an elliptic E will have inﬁnitely many solutions if and only if the global zeta function takes the value zero at s = 1. The full conjecture is analogous in some ways to Dirichlet’s class number formula as the residue at the zero encodes delicate arithmetic information about the elliptic curve. They made their conjecture despite the fact that unlike the Riemann zeta function, it was not known whether the zeta function of an elliptic curve actually had analytic continuation to s = 1. Thanks to the complete proof of the Taniyama-Shimura conjectures by Christophe Breuil,

ZETA FUNCTIONS OF GROUPS

155

Brian Conrad, Fred Diamond and Richard Taylor, this analytic continuation follows from the fact that every elliptic curve is modular. Note that this connection too is based on the idea of the zeta function. It says that the coeﬃcients in the Dirichlet series for the zeta function of the elliptic curve can be reproduced using some associated modular form. 1.6

Borevich, Shafarevich, and Igusa

Following the success of counting solutions to equations in towers of ﬁelds over Fp , some number theorists wondered what would happen if you encoded solutions in a diﬀerent direction. The ﬁeld Fp can be interpreted as the ﬁrst layer in an alternative tower, namely the system of rings Z/pm Z. Let f (x1 , ..., xn ) be a polynomial with coeﬃcients in Z. Deﬁne A(p, m) = |{(x1 , ..., xn ) ∈ (Z/pm Z)n : f (x1 , ..., xn ) = 0

mod pm }|.

Borevich and Shafarevich proposed to encode this information in a local zeta function. But in contrast to the zeta functions of Artin-Hasse-Weil, this would just be a simple power series without the need to exponentiate: Pf (p, t) =

∞

A(p, m)tm .

m=0

This exponentiation in the Artin-Hasse-Weil arises in part from the historical connection with zeta functions counting ideals in function ﬁelds. Just as the number of solutions in a tower of ﬁnite ﬁelds is determined by a ﬁnite part of the tower, it was conjectured that knowledge about the number of solutions mod pn should be dictated by knowledge of a ﬁnite number of layers. In technical terms, it was conjectured that the zeta function of Borevich and Shafarevich should be a rational function in t. The breakthrough in proving this rationality came in Igusa’s recognition that p-adic integrals could be used to express the zeta function. These integrals will be a crucial tool in the application of zeta functions in group theory, so it will be good to understand how this series can be captured by a p-adic integral. The additive Haar measure µ on the ring of p-adic integers Zp is a way to measure the size of subsets. It has the following properties: (1) µ(Zp ) = 1; (2) The measure of a subset is unchanged by additive translation, i.e. µ(S) = µ(a + S) (3) Using (1) and (2) we can calculate the measure of an additive subgroup or its coset: µ(pn Zp ) = µ(a + pn Zp ) = p−n . This follows because Zp is a disjoint union of pn cosets of pn Zp all of which must have the same size by (2). (4) Because the additive cosets are a base of neighbourhoods for the topology in Zp every open subset is a Boolean combination of these cosets. Therefore (3) allows us to calculate the measure of any open subset. For example, what is the

156

DU SAUTOY

measure of the subset pn Z∗p , where Z∗p denotes the units of Zp ? µ(pn Z∗p ) = µ(pn Zp \pn+1 Zp ) = µ(pn Zp ) − µ(pn+1 Zp ) = p−n (1 − p−1 ). This measure extends to an additive measure on Znp . Having deﬁned the measure, we can illustrate what a p-adic integral is with respect to this measure. How does one calculate

|f (x1 , ..., xn )|s dµ I(s) = Zn p

where |.| denotes the p-adic valuation? If the integrand is constant then calculating an integral simply involves calculating the measure of the subset being integrated over. So we rewrite the integral as a sum of integrals where the integrand is constant: ∞

|f (x1 , ..., xn )|s dµ I(s) = m=0 V (m)

where V (m) = {(x1 , ..., xn ) ∈ Znp : v(f (x1 , ..., xn )) = m}. Then I(s) =

∞

p−ms µ(V (m)).

m=0

Now V (m) = W (m)\W (m + 1) where W (m) = {(x1 , ..., xn ) ∈ Znp : v(f (x1 , ..., xn )) ≥ m} = {(x1 , ..., xn ) ∈ Znp : f (x1 , ..., xn ) = 0 = (a1 , ..., an ) + pm Znp

mod pm }

(a1 ,...,an )∈A(m)

and A(m) = { (a1 , ..., an ) ∈ (Z/pm Z)n : f (a1 , ..., an ) = 0 mod pm }. This is a disjoint union of A(p, m) additive cosets each of measure p−nm . Hence I(s) =

∞

p−ms A(p, m)p−nm − A(p, m + 1)p−n(m+1) .

m=0

But Pf (p, p−s ) =

∞

A(p, m)p−ms .

m=0

Hence I(s) = Pf (p, p−n−s ) − ps (Pf (p, p−n−s ) − 1).

ZETA FUNCTIONS OF GROUPS

157

Rearranging this we get: Pf (p, p−n−s ) =

1 − p−s I(s) . 1 − p−s

These p-adic integrals could be calculated easily if the polynomial was nonsingular because Hensel’s lemma provided a way to lift solutions mod p to solutions modulo higher powers of p. The complexity of these functions arose from the singular points. To evaluate integrals where the polynomial was singular, Igusa employed a technique due to Hironaka called resolution of singularities. 1.7

Non-commutative zeta functions

With this continued expansion of the subject of zeta functions throughout the twentieth century, Selberg called for a test ban treaty to halt the further proliferation of zeta functions. Selberg himself had not helped his cause by discovering a natural zeta function during his investigations of negatively curved spaces which some say provides one of the best insights as to why Riemann’s Hypothesis might be true. Despite Selberg’s desire for a halt to any more members of the zeta function club, the group theorists have jumped upon the band-wagon. The ﬁrst example of zeta functions to capture information about groups was the use of the classical Riemann zeta function to express something about the classiﬁcation of ﬁnite abelian groups. Let d be a natural number. Let A(d) be the set of abelian groups, up to isomorphism, generated by at most d elements. Deﬁne a zeta function |G|−s . ζ(A(d), s) = G∈A(d)

Theorem 1.1

ζ(A(d), s) = ζ(s)...ζ(ds).

Proof This follows simply from the classiﬁcation of ﬁnite abelian groups: A(d) = {Z/n1 Z ⊕ · · · ⊕ Z/nd Z : n1 |n2 · · · |nd } = Z/m1 Z ⊕ · · · ⊕ Z/m1 · · · md Z : (m1 , . . . , md ) ∈ Nd>0 . This translates into the following calculation of the zeta function counting abelian groups: −s m−s . . . (m1 · · · md )−s ζ(A(d), s) = 1 (m1 m2 ) (m1 ,...,md )∈Nd>0

= ζ(s)...ζ(ds). The idea of this survey will be to further push the boundaries of zeta functions to tell us something about groups we understand slightly less than the abelian groups.

158

DU SAUTOY

What we shall discover is that these group theoretic zeta functions bring out many of the features of the number theoretic zeta functions that have been forerunners of these ideas. (For a good introduction to the range of classical zeta functions we have explored we refer the reader to the excellent book [22]. We also refer the reader to two other sources [13] and [24] which complement the ensuing survey of zeta functions of groups.)

2

Using p-adic integrals to capture finite p-groups

We may understand the classiﬁcation of ﬁnite abelian groups. But the next step up is a mystery. Trying to classify ﬁnite nilpotent groups is a diﬃcult task. Nilpotent groups decompose as direct products of their Sylow p-subgroups. So it suﬃces to provide a classiﬁcation of ﬁnite p-groups. It is not clear that a sensible classiﬁcation is possible of all groups of order pn . One of the ﬁrst attempts to get to grips with such a classiﬁcation was to attempt at least to count how many groups of order pn there are up to isomorphism. Definition 2.1 Let p be a prime, n an integer and denote by f (n, p) = the number of groups (up to isomorphism) of order pn . The following table shows the majority of our current knowledge of the explicit values of f (n, p) based on computer computation: f (1, p) = 1 f (2, p) = 2 f (3, p) = 5 f (4, 2) = 14 f (4, p) = 15 for p odd f (5, 2) = 51 f (5, 3) = 67 f (5, p) = 2p + 2 gcd(p − 1, 3) + gcd(p − 1, 4) + 61 for p ≥ 5 f (6, p) is given by a quadratic polynomial in p whose coeﬃcients depend on p mod 60. Higman and Sims (see [19] and [28]) gave an asymptotic formula for the behaviour of this function as n grows (and p is ﬁxed): 3

f (n, p) = p(2/27+o(1))n as n → ∞.

(2.1)

However in a second paper [20], Higman predicted a more subtle behaviour of this function as you vary the prime p and ﬁx n which starts to reveal itself in the above evidence for n = 5 and 6. This is encapsulated in what has become known as Higman’s PORC conjecture: Conjecture 2.1 (PORC) For ﬁxed n there is an integer N and polynomials Pn,i (X) for 0 ≤ i ≤ N − 1 so that if p ≡ i mod N then f (n, p) = Pn,i (p).

ZETA FUNCTIONS OF GROUPS

159

PORC stands for Polynomial On Residue Classes. Higman’s PORC conjecture has withstood any attack since Higman’s own contribution in [20] in which he proved that counting class 2 elementary abelian p by elementary abelian p-groups was PORC. What I want to explain is how zeta functions can be used to understand certain regularities in these arithmetic functions. I will make a slight reﬁnement of this counting function: Definition 2.2 Let p be a prime. For integers n, c, d deﬁne P(c, d, p) to be the set of ﬁnite p-groups (up to isomorphism) of class at most c generated by at most d generators and put f (n, p, c, d) = card {G ∈ P(c, d, p) : |G| = pn } . Although we don’t have a classiﬁcation for anything beyond c = 1, the abelian case, it will still turn out to be possible to use zeta functions to understand this sequence of numbers. We encode the arithmetic data using the following deﬁnition: Definition 2.3 Deﬁne the zeta function of class c, d-generator p-groups to be: ζc,d,p (s) =

∞ n=0

=

f (n, p, c, d)p−ns

|G|−s .

G∈P(c,d,p)

Using the fact that ﬁnite nilpotent groups are direct products of their Sylow p-subgroups we can see these zeta functions counting p-groups as local factors of a global object counting ﬁnite nilpotent groups. Definition 2.4 For integers n, c, d deﬁne N (c, d) to be the set of ﬁnite nilpotent groups (up to isomorphism) of class at most c generated by at most d generators and put g(n, c, d) = card {G ∈ N (c, d) : |G| = n} . Deﬁne the zeta function of class c, d-generator nilpotent groups to be: ζc,d (s) =

∞

g(n, c, d)n−s

n=1

=

|G|−s .

G∈N (c,d)

Euler’s observation that the fundamental theorem of arithmetic can be expressed in an identity between zeta functions extends to a group theoretic version: Theorem 2.2 iFor integers c and d there exists an Euler product ζc,d (s) = ζc,d,p (s). p prime

160

DU SAUTOY

Example 2.1 ζ1,d (s) = ζ(s)...ζ(ds) ζ1,d,p (s) = ζp (s)...ζp (ds) =

1 (1 − p−s )...(1 − p−ds )

Actually the ﬁnite groups are already captured in a single object. If I take the free nilpotent group Fc,d of class c on d generators then its ﬁnite quotients of order pn are a complete list of the p-groups we are hoping to count. So we would like to understand how to count these ﬁnite images up to isomorphism. This connects the subject to a topic that has been a running theme at the St. Andrews meetings ever since Dan Segal raised the interesting question of subgroup growth in Groups St. Andrews 1985 [27]. Definition 2.5 Let G be a ﬁnitely generated nilpotent group. Deﬁne an (G) = |{H ≤ G : |G : H| = n} an (G) = |{H G : |G : H| = n}. In a seminal paper [18] published in 1988 by Grunewald, Segal and Smith, zeta functions were proposed as potentially powerful tool to understand the ﬁner arithmetic of these sequences of numbers. Definition 2.6 Let G be a ﬁnitely generated nilpotent group. Deﬁne the zeta function of G and the normal zeta function of G to be respectively: an (G)n−s = |G : H|−s ζG (s) = ζG (s) =

H≤G

an (G)n−s =

|G : H|−s .

HG

These Dirichlet series look like a very natural non-commutative generalization of Dedekind’s zeta functions attached to number ﬁelds. The decomposition of nilpotent groups into Sylow p-subgroups translates here into a corresponding Euler product: Theorem 2.3 Let G be a ﬁnitely generated nilpotent group. Then ζG (s) = ζG,p (s) p prime

where ζG,p (s) = apn (G)p−ns . There is a corresponding Euler product for the normal zeta function. Example 2.2

ζZd (s) = ζZd (s) = ζ(s)...ζ(s − d + 1).

ZETA FUNCTIONS OF GROUPS

161

Proof Let M be a non-singular d × d matrix with entries from Z. Then Zd · M deﬁnes an additive subgroup of index | det(M )|. Every subgroup that we are counting arises in this way. Two matrices M and N deﬁne the same subgroup if and only if GLd (Z)M = GLd (Z)N. A unique set of coset representatives for the action of GLd (Z) on the left is given by triangular matrices: 

a11 a12  0 a22     

a1,d−1 a2,d−1 . . .. . . ad−1,d−1 0



a1,d a2,d .. . ad−1,d ad,d

     

where 0 ≤ ai,j < aj,j . These matrices are sometimes referred to as matrices in Hermite normal form. Let A be the set of all these matrices and A(a11 , ..., add ) be the subset of matrices with ﬁxed diagonal elements a11 , ..., add . We can use these matrices to calculate the zeta function of Zd : ζZd (s) = |Zd : A · Zd |−s A∈A

=

A∈A

=

−s a−s 11 ...add

−s a−s 11 ...add

a11 ,...,add ∈N>0 A∈A(a11 ,...,add )

=

−s a−s 11 ...add |A(a11 , ..., add )|

a11 ,...,add ∈N>0

=

d−1−s a−s 11 ...add

a11 ,...,add ∈N>0

= ζ(s)...ζ(s − d + 1).

It is instructive to compare the two calculations we have made for zeta functions relating to abelian groups. ζ1,d (s) = ζ(s)...ζ(ds) ζZd (s) = ζ(s)...ζ(s − d + 1) The second zeta function counts all the normal subgroups in the free abelian group. The quotients deﬁned by these normal subgroups deﬁne all the ﬁnite abelian groups being counted in the ﬁrst zeta function. But there is a certain amount of overcounting because several normal subgroups deﬁne the same isomorphic quotient. Indeed two normal subgroups Zd · M and Zd · N deﬁne the same quotient if and only if there is an automorphism of Zd mapping Zd · M to Zd · N. In other words if M GLd (Z) = N GLd (Z). Hence a unique set of matrix representatives for the ﬁnite images of Zd up to isomorphism is given by a set of double coset representatives

162

DU SAUTOY

for the action of GLd (Z) on both the right and left. This is precisely the diagonal matrices diag(a11 , ..., add ) where aii |ajj if i < j. These matrices are sometimes referred to as matrices in Smith normal form. But there is nothing particularly special here about sticking to the free abelian group. We show now how we can extend this philosophy to understand the zeta functions counting ﬁnite images up to isomorphism of free nilpotent groups. However as yet we don’t have a well-developed theory of normal forms of matrices to understand these ﬁnite images. Instead we are going to use the vehicle of p-adic integrals to try to understand something of the groups we want to count. Let us start by showing how to use p-adic integrals to count the normal subgroups in the free class two nilpotent group on two generators. The Mal’cev correspondence between nilpotent groups and Lie algebras sets up a one-to-one index preserving correspondence between normal subgroups of ﬁnite p-power index in F2,2 = x, y, z : [x, y] = z

and ideals in the nilpotent Lie algebra: Lp = {Zp x ⊕ Zp y ⊕ Zp z : (x, y) = z} . Proposition 2.4 ζF2,2 ,p (s) =

|Lp : H|−s = ζLp (s).

HLp

For more general nilpotent groups, the correspondence works for almost all primes (see section 4 of [18]). In the ﬁrst instance, an ideal H of Lp is an additive subgroup of Z3p . Therefore there exists a matrix M such that H = Z3p · M. But instead of looking for a unique representative matrix, we consider the set of all matrices which generate the ideal H. It turns out to be easiest to restrict M to be a triangular matrix, but it isn’t necessary to make this restriction (see for example [11]). So deﬁne M(H) = M ∈ Tr3 (Zp ) : H = Z3p · M . Lemma 2.1 Let µ be the additive Haar choice of matrix  a11 a12 a22 A= 0 0 0

measure on Tr3 (Zp ) = Z6p . Then for any  a13 a23  ∈ M(H) a33

as a representative for this subset, we get that the measure of the matrices generating H is given by µ(M(H)) = (1 − p−1 )3 |a11 |1 |a22 |2 |a33 |3 . Proof This generalizes the one-dimensional setting where if H is an additive subgroup a11 Zp then all the other generators for this subgroup are M(H) = a11 Z∗p .

ZETA FUNCTIONS OF GROUPS

163

We calculated the measure of this subset in the ﬁrst lecture as |a11 |(1 − p−1 ). The higher powers of |a22 | and |a33 | in this 3-dimensional example come from the chance to perturb the matrix generating H by adding Zp -combinations of the second and third row to the ﬁrst and second row. Let me work backwards from the expression for the zeta function to a p-adic integral: |Lp : H|−s ζLp (s) = HLp

=

HLp

A∈M(H)

= (1 − p−1 )−3

−1 |a11 |s |a22 |s |a33 |s (1 − p−1 )3 |a11 |1 |a22 |2 |a33 |3 dµ

A∈M

|a11 |s−1 |a22 |s−2 |a33 |s−3 dµ

where

M=

M(H).

HLp

In the case of counting all abelian subgroups of Z3p where we ignore any Lie structure, this set would of course be the complete set of triangular matrices since every triangular matrix generates some additive subgroup. But not all these additive subgroups are ideals when we introduce the Lie structure into consideration. So when is a matrix generating an ideal?   a11 a12 a13 a22 a23  then H = Z3p · A is an ideal in Lp if and Lemma 2.2 Let A =  0 0 0 a33 only if v(a33 ) ≤ v(a11 ), v(a22 ), v(a12 ). (v(a) denotes the p-adic valuation of a.) Proof Let a1 = a11 x + a12 y + a13 z, a2 = a22 y + a23 z, a3 = a33 z then H is the Zp -span of a1 , a2 , a3 . It is an ideal if and only if taking the Lie bracket of each ai with a generator x, y, z we can re-express the result in terms of the linear span of a1 , a2 , a3 . So for example (x, a1 ) = a12 z and this is the linear span of a1 , a2 , a3 if and only if v(a33 ) ≤ v(a12 ). We can piece all these properties together to get the following theorem: Theorem 2.5 ζF2,2 ,p (s) = (1 − p−1 )−3

A∈M

|a11 |s−1 |a22 |s−2 |a33 |s−3 dµ

where M = {(aij ) ∈ Z6p : v(a33 ) ≤ v(a11 ), v(a22 ), v(a12 )}. This integral can be calculated by hand to get the following:

164

DU SAUTOY

Theorem 2.6 ζF2,2 ,p (s) = ζp (s)ζp (s − 1)ζp (3s − 2). Proof Putting nij = v(aij ) and using our calculation of µ(aZ∗p ) = p−v(a) (1−p−1 ) we get

|a11 |s−1 |a22 |s−2 |a33 |s−3 dµ I(s) = A∈M

=

pn11 (1−s) pn22 (2−s) pn33 (3−s) µ(pn11 Z∗p × pn12 Z∗p × pn22 Z∗p × pn33 Z∗p )

n11 ,n12 ,n22 ≥n33

= (1 − p−1 )4

p−n11 s pn22 (1−s) pn33 (2−s) p−n12

n11 ,n12 ,n22 ≥n33

Changing variables so that n11 = m11 + n33 and similarly for n12 and n22 we get

I(s) = (1 − p−1 )4 =

p−m11 s pm22 (1−s) pn33 (2−3s) p−m12

m11 ,m12 ,m22 ,n33 (1 − p−1 )4

(1 − p−s )(1 − p1−s )(1 − p2−3s )(1 − p−1 )

.

How do we adapt this now to count only ﬁnite images up to isomorphism? Two normal subgroups N and M deﬁne the same isomorphic ﬁnite quotient in the free nilpotent group F2,2 if and only if there is an automorphism φ of F2,2 with the property that N = φM. So the number of normal subgroups deﬁning the same ﬁnite quotient as a subgroup H is just the size of the orbit of the action of the automorphism group G of F2,2 acting on the lattice of normal subgroups. The size of an orbit is the same as the index of the stabilizer of the automorphism group G acting on H, i.e. |G : StabG(H)|. So we can now relate the zeta functions counting normal subgroups in F2,2 and ﬁnite quotients of F2,2 up to isomorphism: Theorem 2.7 ζ2,2 (s) =

|F2,2 : H|−s |G : StabG(H)|−1 .

HF2,2

We can linearize the problem again so that the local factors can be expressed in terms of counting ideals in Lp up to the action of the automorphism group G(Lp ). Theorem 2.8 ζ2,2,p (s) =

HLp

|Lp : H|−s |G(Lp ) : StabG(Lp ) (H)|−1 .

ZETA FUNCTIONS OF GROUPS

165

The automorphism group is a subgroup of GL3 (Zp ), the automorphism group of just the additive structure of Lp . It has the following description:     λ11 λ12 µ1   : (λij ) ∈ GL2 (Zp ), µi ∈ Zp . G(Lp ) =  λ21 λ22 µ2   0 0 λ11 λ22 − λ12 λ21 If A ∈ M(H) then we can give a description of the stabilizer subgroup: Lemma 2.3

StabG(Zp ) (H) = G(Zp ) ∩ A−1 GL3 (Zp )A.

Proof φ ∈ StabG(Zp ) (H) if and only if Z3p · Aφ = Z3p · A, i.e. AφA−1 ∈ GL3 (Zp ). Can we extend our integral counting normal subgroups to take account of this new piece in the zeta function? The key is to use the natural Haar measure ν that exists on the automorphism group G(Lp ). This Haar measure has similar properties to the additive Haar measure that we have already introduced. It measures the size of subsets of G(Lp ). In particular (1) ν(G(Lp )) = 1 (2) It is constant on cosets of subgroups. This has the eﬀect that if H is a subgroup of G(Lp ) then since there are |G(Lp ) : H| disjoint subsets which cover G(Lp ) they each must have measure |G(Lp ) : H|−1 . This means we can write: |Lp

H|−s |G(Zp ) : StabG(Zp ) (H)|−1

−1 −3 = (1 − p ) |a11 |s−1 |a22 |s−2 |a33 |s−3 ν G(Zp ) ∩ A−1 GL3 (Zp )A dµ. :

A∈M(H)

Deﬁne the subset N ⊂ Tr3 (Zp ) × G(Zp ) by N = (A, K) : A ∈ M, K ∈ G(Zp ) ∩ A−1 GL3 (Zp )A then: Theorem 2.9 ζ2,2,p (s) = (1 − p−1 )−3

N

|a11 |s−1 |a22 |s−2 |a33 |s−3 dµdν.

We have therefore transformed our zeta functions into p-adic integrals. The above analysis can be carried out to count ﬁnite quotients up to isomorphism in any free nilpotent group. This allows us to apply the techniques of Igusa and Denef which imply the rationality of certain classes of so-called deﬁnable integrals. In particular we can prove the ﬁrst regularity in the arithmetic of counting ﬁnite p-groups. The complete proofs can be found in [3] and [6]. Theorem 2.10 For a ﬁxed prime p and integers c and d, the function ζc,d,p (s) is a rational function in p−s .

166

DU SAUTOY

Corollary 2.11 For a ﬁxed prime p and integers c and d the function f (n) = f (n, p, c, d) satisﬁes a linear recurrence relation with constant coeﬃcients. So what is the rational function for the class two, two generator p-groups? My Ph.D. student Christopher Voll has calculated it to be: Example 2.3

ζ2,2 (s) = ζ(s)ζ(2s)ζ(3s)2 ζ(4s).

His calculation required taking triangular matrices for ideals in Hermite normal form. Then one needs to look for a normal form, like the Smith normal form for abelian groups, using the action of G(Zp ) rather than the full GL3 (Zp ). Details are contained in Voll’s thesis [30]. Problem 1 Calculate ζc,d (s) for more c and d. In particular, the zeta function counting normal subgroups of ﬁnite index in the free class 2 nilpotent group on 3 generators was previously calculated in [18]. It takes the following form: ζF2,3 (s) = ζ(s)ζ(s − 1)ζ(s − 2)ζ(3s − 5)ζ(5s − 8)ζ(6s − 9)

W (p, p−s )

p

where W (X, Y ) = 1 + X 3 Y 4 + X 4 Y 3 + X 6 Y 5 + X 7 Y 5 + X 10 Y 8 . The analysis of this calculation should contribute towards determining an expression for ζ2,3 (s). To help any brave soul on their way, here are some explicit computations of the ﬁrst few coeﬃcients provided courtesy of Eamon O’Brien. We record in the following table the values of f (n, p, 2, 3)− f (n, p, 2, 2) for various small values of p and n: f (n, p, 2, 3)− f (n, p, 2, 2) n = 1 n = 2 n = 3 n = 4 n = 5 n = 6 p=2 a0 0 0 3 16 46 p=3 0 0 0 3 18 56 p=5 0 0 0 3 20 64 The rationality of these local zeta functions expresses a regularity in the variation of these numbers when we ﬁx the prime p and we look at p-groups of increasing order. In the next section we consider the variation in the other direction: namely ﬁx n and vary p. This is of course relevant to Higman’s PORC conjecture.

3

Uniformity

Having proved that for each prime p the zeta functions ζc,d,p (s) are rational functions, to analyse PORC we need to understand how these rational functions vary as we vary the prime p. We make the following: Conjecture 3.1 For each c and d there exist ﬁnitely many rational functions Wi (X, Y ) ∈ Q(X, Y ) (i = 1, ..., N ) such that if p ≡ i mod N then ζc,d,p (s) = Wi (p, p−s ).

ZETA FUNCTIONS OF GROUPS

167

The rational functions Wi (X, Y ) have the form Wi (X, Y ) =

(1 −

Pi (X, Y ) a b i1 X Y i1 )...(1 −

X aidi Y bidi )

.

The nature of the denominator of Wi (X, Y ) will be a result of the fact that we are summing geometric progressions. Note in particular that this conjecture implies the following: Corollary 3.2 Suppose Conjecture 3.1 is true. Then for n ∈ N and i = 1, ..., N there exist polynomials rn,i (X) ∈ Q[X] such that if p ≡ i mod N then f (n, p, c, d) = rn,i (p) i.e. the function f (n, p, c, d) is PORC in p. Since f (n, p, n − 1, n) = f (n, p) this includes Higman’s PORC conjecture as a special case. The proof that the conjecture implies PORC for the coeﬃcients f (n, p, c, d) of ζc,d,p (s) follows by expanding the rational function Wi (X, Y ) as ∞ di

k aij bij Pi (X, Y ) X Y j=1

k=0

and reading oﬀ the coeﬃcient of Y n . We have explained how the zeta functions counting ﬁnite p-groups are variations on zeta functions counting normal subgroups in free nilpotent groups. The question of how the zeta functions counting normal subgroups varies with the prime p was one of the principal questions raised in Grunewald, Segal and Smith’s paper [18] where these functions were ﬁrst introduced. Motivated by the analogy with the Dedekind zeta function of a number ﬁeld, they speculated whether these zeta functions might satisfy some Chebotaryov density type theorem, a generalization of Dirichlet’s Theorem on primes in arithmetic progression. This speculation was supported by an example of the Heisenberg group extended to a quadratic number ﬁeld (details of which are presented in section 5.2). They stated that it was “plausible” that the following question might have a positive answer: Question Let G be a finitely generated nilpotent group and ∗ ∈ {≤, }. Do there exist finitely many rational functions W1 (X, Y ), . . . , Wr (X, Y ) ∈ Q(X, Y ) such that for each prime p there is an i for which ∗ ζG,p (s) = Wi (p, p−s )?

If the answer to this question is “yes” we say the zeta function is ﬁnitely uniform. If also r = 1, we say that the zeta function is uniform. Note that we won’t necessarily expect the decomposition to be PORC since the behaviour of primes in number ﬁelds of degree > 2 is not simply determined by residue classes. In [18] this question was elevated to the status of a conjecture in the case that the nilpotent group is a free group:

168

DU SAUTOY

Conjecture 3.3 Let F be a ﬁnitely generated free nilpotent group of class c on d generators and ∗ ∈ {≤, }. Then there exists a rational function W (X, Y ) ∈ Q(X, Y ) such that for almost all primes p ∗ (s) = W (p, p−s ). ζF,p

In other words, the zeta functions of free nilpotent groups are expected to be uniform. Since the free nilpotent group is the group which is relevant to PORC, a proof of Conjecture 3.3 for ∗ = will probably form an essential stepping stone to a proof of Conjecture 3.1 and hence Higman’s PORC conjecture. This connection gives a much greater signiﬁcance to the original Conjecture 3.3 than was ﬁrst supposed. In the original paper [18] where Conjecture 3.3 was made, a proof is given for class 2 free nilpotent groups. In joint work with Grunewald [10] we have established this conjecture for 2-generator free nilpotent groups of arbitrary class. We collect together our present knowledge then in the following: Proposition 3.4 Conjecture 3.3 is true for (a) class 2 free nilpotent groups [18]; and (b) 2-generator free nilpotent groups [10]. It is a simple observation to see that a ﬁnite number of exceptional primes are covered by residue classes by taking N in Conjecture 3.1 to be divisible by these primes. The same argument implies that a function which is PORC on almost all primes is PORC on all primes. The evidence that we documented in section 2 certainly means that we are going to expect some genuine PORC behaviour in Conjecture 3.1 and Corollary 3.2. In other words, there won’t be just one rational function or polynomial in general that will work for almost all primes. Since Conjecture 3.3 implies that the behaviour of normal subgroups in free nilpotent groups will not give rise to a genuine PORC behaviour, we are expecting the analysis that will come from the action of the algebraic automorphism above to provide this genuine PORC behaviour. Although zeta functions of number ﬁelds looked like an appropriate model for zeta functions of groups, recent analysis in [9] of the p-adic integral used to express these zeta functions has signiﬁcantly shifted the perspective that one should take on these zeta functions. It turns out that it is the zeta functions counting points on varieties considered by Artin, Hasse and Weil (see section 1.4) which oﬀers the better analogy for zeta functions counting groups, rather than the Dedekind zeta functions. All the integrals we have considered to date are examples of things we have called cone integrals: Definition 3.1 Cone integrals are deﬁned for a set of cone integral data consisting of polynomials D = {f0 (x), g0 (x), . . . , fl (x), gl (x)} by

|f0 (x)|s |g0 (x)| dµ ZD (s, p) = Vp (D)

ZETA FUNCTIONS OF GROUPS where

169

Vp (D) = x ∈ Zm p : v (fi (x)) ≤ v (gi (x)) for i = 1, . . . , l

and µ is the additive Haar measure on Zm p . We saw that the conditions on the matrices deﬁning ideals in the free class two, two generator nilpotent group were deﬁned by comparing valuations of entries in the matrix. In general, if the polynomials are simply monomials in entries of the matrices then the integrals are relatively straightforward to calculate. This is because v(x1 · x2 ) = v(x1 ) + v(x2 ). In particular because the conditions just reduce then to conditions on the valuations, the integrals simply become sums over lattice points contained in cones in Rm ≥0 deﬁned by the cone conditions. This guarantees the independence of p in the resulting rational functions. Proposition 3.5 Suppose ZD (s, p) is a cone integral in which each of the polynomials in the cone data are monomials. Then there exists a rational function P (X, Y ) such that for almost all primes p ZD (s, p) = P (p, p−s ). For example, the integral counting normal subgroups in F2,2, reduces to counting (n11 , n12 , n22 , n33 ) ∈ N in the cone bounded by n11 = n33 , n12 = n33 , n22 = n33 . The problem comes when these polynomials are not monomial. The valuation v(x + y) is not determined simply by knowing the valuations of the individual variables v(x), v(y). The trouble is that the polynomial xy(x + y) has a singularity with non-normal crossings at (0, 0). How would you calculate

|xy(x + y)|s dµ? Z2p

A blow-up at (0, 0) serves to sort this problem out. A blow-up is an example of a resolution of singularity, the technique employed by Igusa to resolve the conjecture of Borevich and Shafarevich detailed in section 1.6. The blow-up preserves the space outside the singularity at (0, 0) but replaces this point by a projective line P1 which is the set of pairs (x , y ) up to the equivalence deﬁned by scalar multiplication. What the blow-up does is to replace two-dimensional aﬃne space A2 by a subset B(0,0) of A2 × P1 deﬁned by the equation xy = x y inside A2 × P1 . The blow-up is then the map h : B(0,0) → A2 deﬁned by (x, y, x , y ) → (x, y). It is an isomorphism outside h−1 (0, 0). The point x = y = 0 has been replaced by P1 . In general a resolution of singularity is an isomorphism oﬀ a set of measure 0 which is why it can be used to rewrite the integral. There is just a Jacobian which adjusts for the scaling in the measure introduced by the transformation eﬀected by the resolution. We have two open subsets Bx = {(x, y, x , y ) ∈ B(0,0) : x = 0} and By = {(x, y, x , y ) ∈ B(0,0) : y = 0} which cover B(0,0) with some overlap. On Bx we can choose a representative for each projective point so that x = 1. Similarly for

170

DU SAUTOY

By . Each of these open subsets has a chart which identiﬁes them with A2 : φy

:

Bx → A2

(x, y, 1, y ) → (x, y ) and φx

:

By → A2

(x, y, x , 1) → (x , y). What does the polynomial xy(x+y) look like with respect to the variables in the chart for Bx ? Since y = xy this polynomial is transformed into the polynomial x xy x + xy = x3 y (1 + y ). Although the variety deﬁned by x3 y (1 + y ) = 0 still has singularities at (0, 0) and (0, 1) these are merely what are called normal crossing and are harmless to the evaluation of the integral. The point is that two lines crossing in two dimensional space produce normal crossings whilst three lines crossing at a point do not. The condition for normal crossings ensures that a local coordinate system can be chosen around the singularity using tangent vectors to the varieties. On the other chart the polynomial is transformed into y 3 x (1 + x ). How do we useall this to calculate our original integral? We break the space Z2p = (x, y) ∈ Z2p over which we are integrating into two disjoint pieces U1 = (x, y) ∈ Z2p : v(x) ≤ v(y) U2 = (x, y) ∈ Z2p : v(x) > v(y) . Then h−1 (U1 ) ⊂ Bx and φy ◦ h−1 (U1) = Z2p = (x, y ) ∈ Z2p . Whilst h−1 (U2 ) ⊂ By and φx ◦ h−1 (U2 ) = pZp × Zp = (x , y) ∈ Z2p : x ∈ pZp . The integral now becomes:

|xy(x + y)|s dµ = |xy(x + y)|s dµ Z2p

=

U1 ∪U2

(x,y )∈Z2p

|x3 y (1 + y )|s |x|dµ

+

(x ,y)∈pZp ×Zp

(3.2)

|y 3 x (1 + x )|s |y|dµ

The extra |x| and |y| are the Jacobians coming from the change in measure eﬀected by these charts. To calculate these integrals is much simpler because the transformation means that on cosets of pZp one of y or (1 + y ) is a unit and therefore contributes nothing to the integral. In the second integral we are only considering one coset of pZp so we get immediately:

|y 3 x (1 + x )|s |y|dµ (x ,y)∈pZp ×Zp

= |x |s |y|3s+1 dµ (x ,y)∈pZp ×Zp

= (1 − p−1 )2 p−(s+1) ζ(s + 1)ζ(3s + 2).

ZETA FUNCTIONS OF GROUPS

171

Considering the ﬁrst integral in (3.2) we get

|x3 y (1 + y )|s |x|dµ = |x|3s+1 dµ Z2p

Zp

Zp

|y (1 + y )|s dµ

= (1 − p−1 )ζ(3s + 2) and

Zp

|y (1 + y )|s dµ =

y ∈pZp

|y |s dµ +

1+y ∈pZp

Zp

|y (1 + y )|s dµ

|1 + y |s dµ +

= 2(1 − p−1 )p−(s+1) ζ(s + 1) + 1 − 2p−1 .

y ∈pZ / p ∪−1+pZp

dµ

Summing all the integrals we get the ﬁnal answer:

|xy(x + y)|s dµ = (1 − p−1 )(1 − 2p−1 + 2p−(s+1) − p−(s+2) )ζ(s + 1)ζ(3s + 2). Z2p

In general the resolution of singularities transforms the space ZN p that we are integrating over so that we have a new space Y , a variety over Q, which is crisscrossed with a system of varieties Ei , i ∈ T. Consider the map θ : Y (Zp ) → Y (Fp ). Then for every a ∈ Y (Fp ) deﬁne Ua = θ−1 (a). Then on each subset Ua the polynomials in the cone integrals become monomial. Essentially there is a variable Xi for each variety Ei criss-crossing the space Y (Fp ) and if a ∈ Ei (Fp ) then you switch on this variable in the monomial representation of the polynomials. If you are oﬀ Ei (Fp ) then that variable does not appear in the monomial expression. So the monomial description of the polynomials is invariant on the following sets: for each I Ei (Fp ) \ Ej (Fp ). EI = i∈I

j∈T \I

The integral breaks up into a ﬁnite sum of integrals. In each of these ﬁnite sums the integral is a cone integral on monomials and is therefore uniform. The dependence in p of these integrals is therefore entirely controlled by the size of these sets: Theorem 3.6 [9] Let ZD (s, p) be a cone integral. Then there exist varieties Ei , i ∈ T deﬁned over Q sitting in some scheme Q-scheme Y and rational functions PI (x, y) ∈ Q(x, y) for each I ⊂ T with the property that for almost all primes p |EI |PI (p, p−s ). ZD (s, p) = I⊂T

Since the integrals counting ﬁnite p-groups are cone integrals we get the following corollaries:

172

DU SAUTOY

Theorem 3.7 [6] For each c and d there exist ﬁnitely many subvarieties Ei,c,d (i ∈ T (c, d)) of a variety Yc,d deﬁned over Q and for each I ⊂ T (c, d) a rational function Pc,d,I (X, Y ) ∈ Q(X, Y ) such that for almost all primes p ζc,d,p (s) = ec,d,p,I Pc,d,I (p, p−s ) I⊂T (c,d)

where ec,d,p,I = card{a ∈ Yc,d (Fp ) : a ∈ Ei,c,d (Fp ) if and only if i ∈ I} . Here Y means reduction of the variety

mod p, which is deﬁned for almost all p.

Since f (n, p) = f (n, p, n − 1, n) this Theorem for ζn−1,n,p (s) tells us something about the behaviour of the number of p-groups of order pn as we vary p. Although this does not establish yet the strong uniformity predicted by Higman’s PORC conjecture, we have established the following uniform behaviour: Corollary 3.8 For each n there exist ﬁnitely many subvarieties Ei,n (i ∈ T (n)) of a variety Yn deﬁned over Q and for each I ⊂ T (n) a polynomial Hn,I (X) ∈ Q[X] such that for almost all primes p en,p,I Hn,I (p) f (n, p) = I⊂T (n)

where en,p,I = card{a ∈ Yn (Fp ) : a ∈ Ei,n (Fp ) if and only if i ∈ I} . So counting p-groups is given by the number of points on varieties mod p (or NOPOV, not quite as catchy as Higman’s culinary shorthand). Not only that, the varieties Ei,c,d are explicitly deﬁned and arise from the resolution of singularities of a polynomial we associate to each pair (c, d). This theorem therefore oﬀers some hope to approach Higman’s PORC conjecture by analysing the nature of these varieties. Since there is some genuine PORC behaviour, the varieties are going to be a little more exotic than just rational varieties. What is likely is that in the analysis of the algebraic group, some consideration will be needed about whether Fp contains nth roots of unity. This will cause consideration of what the residue of p is modulo n. So we might see varieties like xn = 1 cropping up. The consequence of the analysis of these types of cone integrals is that the zeta functions of Artin, Hasse and Weil which count the number of points mod p on varieties oﬀer a much better analogy for the zeta functions of groups than the zeta functions of Dedekind attached to number ﬁelds. This clearly had implications then for the uniformity conjecture proposed originally by Grunewald, Segal and Smith. After all, as the elliptic curve considered by Gauss in his diaries illustrates, the number of points mod p on Y 2 = X 3 − X when p = 1 mod 4 behaves quite wildly. In particular, it is proved in [7] that it is not PORC. However, it was far from clear whether the integrals attached to groups could encode anything as

ZETA FUNCTIONS OF GROUPS

173

exotic as an elliptic curve. After all, the integrals begin quite linearly. Could it be that the varieties that arise from zeta functions of groups are all rational varieties or varieties like {y = xn } whose solutions mod p are polynomial on some ﬁnite subdivision of the primes? The following example presented in [7] and [8] blew that prospect out of the water: Theorem 3.9 Let G(E) be the Hirsch length 9, class two nilpotent group given by the following presentation: x1 , x2 , x3 , x4 , x5 , x6 , y1 , y2 , y3 : [x1 , x4 ] = y3 , [x1 , x5 ] = y1 , [x1 , x6 ] = y2 G(E) = [x2 , x4 ] = y2 , [x2 , x6 ] = y1 , [x3 , x4 ] = y1 , [x3 , x5 ] = y3 where all other commutators are deﬁned to be 1. Let E be the elliptic curve Y 2 = X 3 − X. Then there exist two rational functions P1 (X, Y ) and P2 (X, Y ) ∈ Q(X, Y ) such that for almost all primes p : (s) = P1 (p, p−s ) + |E(Fp )|P2 (p, p−s ). ζG,p

To see where the elliptic curve is hidden in this presentation, take the determinant of the 3 × 3 matrix (aij ) with entries aij = [xi , xj+3 ] and you will get the projective version of E. Corollary 3.10 The question of Grunewald, Segal and Smith has a negative answer. So Theorem 3.9 indicates that a positive answer to Conjecture 3.3 and Higman’s PORC Conjecture will be something special about free nilpotent groups. This example reveals that the arithmetic of nilpotent groups is much richer than originally anticipated. It was known that the arithmetic of quadratic forms can be encoded in the theory of ﬁnitely generated nilpotent groups (see [17]). It was a striking realization that the arithmetic of elliptic curves can also be encoded in the subgroup structure. It opens up a whole new side to the subject. Can the arithmetic of any variety be encoded in the subgroup structure of an appropriate nilpotent group? It also sets up an interesting hierarchy in the theory of these groups. Groups can be indexed by the varieties that are needed to describe their zeta functions. That this is canonically deﬁned requires the theory of motivic integration developed by Kontsevich, Denef and Loeser. This hierarchy is deﬁned in two papers [11] and [8]. At the bottom of the hierarchy are those groups whose zeta functions are ﬁnitely uniform. Grunewald, Segal and Smith conjectured then that the free nilpotent groups will be found here. It opens up a fascinating dialogue between groups and algebraic geometry. What can be said about any relationship between groups and varieties of a given genus for example? Stimulated by the above example, Christopher Voll [30] and Con Griﬃn, a PhD student of Ivan Fesenko, [16] have begun investigating other varieties that arise. The varieties are encoded in a similar manner by realising polynomials as determinants of matrices with linear polynomial entries and then using these determinants to deﬁne a corresponding presentation for a group.

174

DU SAUTOY

The method of resolution of singularities is not just of theoretical importance. It can be put to practical use in calculating the zeta functions of groups. In a paper [14] with my PhD student Gareth Taylor we used an analysis of the singularities involved in the integral counting subalgebras in sl2 (Zp ) to evaluate the integral directly. This substantially simpliﬁed the previous evaluation which had been made essentially by Ilani but required a huge case analysis for which a computer was employed (see [21], [4] and [5]). The philosophy in our paper is that the algebraic geometry motivates a much more judicious change of variable in the integral. Taylor subsequently observed that by even more cunning choices of blow-ups which are not so clearly motivated by the underlying algebraic geometry but by certain quirks of the cone conditions, he could simplify the calculation further. We reproduce Taylor’s calculation which appears in his PhD thesis [29]. Example 3.1 sl2 (Zp ) = e, f, h : (h, e) = 2e, (h, f ) = −2f, (e, f ) = h . Then for odd p ζsl2 (Zp ) (s) = ζp (s)ζp (s − 1)ζp (2s − 2)ζp (2s − 1)ζp (3s − 1)−1 . Proof This is a three-dimensional Lie algebra like the Heisenberg Lie algebra. We can therefore use 3 × 3 triangular matrices to record coordinates for a basis for a subalgebra H. The cone integral representing ζsl2 (Zp ) (s) is then as follows (see [4])

|a|s−1 |x|s−2 |z|s−3 |dµ| ζsl2 (Zp ) (s) = (1 − p−1 )−3 W

= (1 − p−1 )−3 I(s). where    v(x) ≤ v(4cy)  a c b  v(x) ≤ v(4cz) . W =  0 x y  ∈ Tr3 (Zp ) :   0 0 z v(xz) ≤ v(ax2 + 4bxy − 4cy 2 ) Since 4 is a unit for odd p, we can make a change of variable c = 4c. The ﬁrst condition ensures that c y/x is a p-adic integer. Therefore we can make the change of variable b = 4b − c y/x which transforms the integral into

I(s) = |a|s−1 |x|s−2 |z|s−3 |dµ| . x|c y x|c z z|(ax + b y) A blow-up at y = z = 0 serves to sort this integral out. Case (1) Consider v(z) ≤ v(y).

ZETA FUNCTIONS OF GROUPS

175

Let y = y/z ∈ Zp . Then the integral I(s) restricted to v(z) ≤ v(y) becomes:

I1 (s) =

=

x|c y z x|c z z|(ax + b y z) x|c z z|ax

|a|s−1 |x|s−2 |z|s−3 |z| |dµ|

|a|s−1 |x|s−2 |z|s−2 |dµ|

which is already monomial. Case (2) Consider v(z) > v(y). Let z = z/y ∈ pZp . The integral I(s) restricted to v(z) > v(y) becomes:

I2 (s) =

v(z ) ≥ 1 x|c y yz |(ax + b y)

s−3 |y| |dµ| . |a|s−1 |x|s−2 z y

The condition yz |(ax+b y) is equivalent to y|ax and z |b where b = b +ax/y ∈ Zp . Put b = b /z then this second integral just becomes:

I2 (s) =

v(z )

x|c y y|ax =

≥1

s−3 |y||z | |dµ| |a|s−1 |x|s−2 z y

(1 − p−1 )p1−s I1 (s) (1 − p1−s )

by integrating the z variable and observing that the resulting integral is the same as the integral in case (1) except that y appears in place of z. So let’s calculate the integral I1 (s). Since it is monomial we can calculate this integral by reducing it to summing over lattice points representing the valuations of the variables. It helps though to make a further division of the domain of integration into two pieces. Case (a) Suppose v(x) ≥ v(z). Put x = x/z ∈ Zp then the integral restricted to v(x) ≥ v(z) becomes:

s−2 2s−4 |a|s−1 x |z| |z| |dµ| I1a (s) = x |c

s−2 2s−3 |a|s−1 x |z| |x | |dµ| = Z4p

= (1 − p−1 )3

1 (1 − p−s )2 (1 − p2−2s )

where we made a transformation c = c /x .

176

DU SAUTOY

Case (b) Suppose that v(x) < v(z). Put z = z/x ∈ pZp then the integral restricted to v(x) < v(z) becomes:

s−2 |a|s−1 |x|2s−4 z |x| |dµ| I1b (s) = z |a z ∈ pZp

s−1 2s−3 2s−3 a z |x| |z | |dµ| = z ∈pZp

= (1 − p−1 )3

(1 −

p−s )(1

p1−2s − p2−2s )(1 − p1−2s )

where we made a transformation a = a/z . Hence I1 (s) = I1a (s) + I1b (s) = (1 − p−1 )3

(1 − p1−3s ) . (1 − p−s )2 (1 − p2−2s )(1 − p1−2s )

Therefore ζsl2 (Zp ) (s) = (1 − p−1 )−3 (I1 (s) + I2 (s)) " ! (1 − p−1 )p1−s = (1 − p−1 )−3 I1 (s) 1 + (1 − p1−s ) 1−3s ) (1 − p . = (1 − p−s )(1 − p1−s )(1 − p2−2s )(1 − p1−2s ) Taylor went on his thesis to make some monumental calculations motivated by the technique of resolutions of singularities. We reproduce the results in section 5 as they have helped provide further evidence for various conjectures about zeta functions of groups.

4

Subgroup growth and Euler products of cone integrals

Zeta functions can be used as analytic tools to provide subtle information about ∗ (s) converges on the subgroup growth of nilpotent groups. The zeta function ζG ∗ the right half complex plane {s ∈ C : Re(s) > αG } . The abscissa of convergence of ∗ (s) (where ∗ ∈ {≤, }) is equal to ζG log s∗n (G) log n n→∞

∗ αG = lim sup

where s∗n (G) = a∗1 (G) + · · · + a∗n (G). The abscissa of convergence determines something about the subgroup growth: ∗ = inf {α ≥ 0 : there exists c > 0 with s∗n (G) < cnα for all n} . αG

ZETA FUNCTIONS OF GROUPS

177

However to get ﬁner information about this growth, one would like to exploit the Tauberian Theorem. This requires though some knowledge about the possibility of analytically continuing the zeta function beyond its abscissa of convergence. We quote the following version from [25], p. 121: Proposition 4.1 (Tauberian Theorem) Suppose that for s in the half plane ∗ } , we have ζ ∗ (s) = g(s)(s−α∗ )−β +h(s) where g(α∗ ) = 0, β ∈ {s ∈ C : Re(s) > αG G G G ∗ }. N and g(s), h(s) are holomorphic in the closed half-plane {s ∈ C : Re(s) ≥ αG Then as x → ∞ ! ∗) " g(αG α∗G (log x)β−1 . s∗n (G) ∼ ∗ Γ(β) x αG For example, ζZ2 (s) = ζ(s)ζ(s−1) is a meromorphic function which converges on {s ∈ C : Re(s) > 2} . We can use the Tauberian Theorem to deduce the following subgroup growth for the number of subgroups of ﬁnite index in the rank two free abelian group: π2 2 n . sn (Z2 ) ∼ 12 Without such sophisticated analytic machinery it would be diﬃcult by elementary 2 means to derive the coeﬃcient π12 as the precise limit of the subgroup growth in the free abelian group. The occurrence of π 2 goes back to Euler’s calculation of the sum of the squares of the harmonic series which launched mathematicians’ interest in the zeta function. One has to be very careful about assuming that such analytic continuation is possible for the zeta function of a general nilpotent group. Even if the zeta function is uniform this is far from clear as the following example cautions: Proposition 4.2 The Euler product !

1+

p

p−1−s (1 − p−s )

" (4.3)

converges for {s ∈ C : (s) > 0} but has Re(s) = 0 as a natural boundary. In [9], Grunewald and I were able to use our analysis of the explicit expression for a cone integral in terms of counting points on varieties to be able to deduce the following theorem about the analytic behaviour of Euler products of cone integrals: Theorem 4.3 For cone integral data D the Euler product ZD (s) =

ZD (s, p)

p

has meromorphic continuation to {s ∈ C : Re(s) > αD − δ} where δ > 0 and αD is the abscissa of convergence of ZD (s).

178

DU SAUTOY

Note that the local zeta functions ZD (s, p) are rational functions in p−s and hence the global zeta function ZD (s) can be represented as a Dirichlet series which implies in turn that the abscissa of convergence αD is well deﬁned. The proof of this Theorem uses Artin L-functions to do the meromorphic continuation together with careful analysis of counting lattice points in cones to avoid pathologies like that in (4.3). The Artin L-function is deﬁned for each continuous ﬁnite dimensional complex representation ρ : G →GL(V ) of G, the absolute Galois group of Q, by the following Euler product: 1 . L(ρ, s) := −s det(1 − ρ(Frob p) · p ) p In our situation we look at the permutation representation of G as it acts on all the absolutely irreducible components of the varieties arising in the explicit expression for the cone integral. The deﬁnition of the Artin L-function arose out of attempts to generalize local class theory to non-abelian ﬁeld extensions. Local class ﬁeld theory identiﬁes the part of the ﬁeld which determines its abelian extensions. We are missing a corresponding invariant for the non-abelian extensions. The idea was to use the Artin L-function to provide some dialogue. After all the Artin L-function generalize the L-function of an abelian extension which corresponds to the case of one-dimensional representations, i.e. characters. One of the conjectures raised in Lubotzky’s lecture notes [23] on Counting Finite Index Subgroups for the Groups ’93 Galway/St. Andrews Conference was the following question about the abscissa of convergence of a nilpotent group: ∗ is rational. Conjecture 4.4 If G is a ﬁnitely generated nilpotent group then αG

Our analysis of cone integrals now answers this conjecture: Theorem 4.5 The abscissa of convergence αD of ZD (s) is a rational number. The proof depends on the Lang-Weil estimates for the number of points on varieties modulo p. Lubotzky declared following his conjecture that the abscissa of convergence of the Heisenberg group is 3/2. This is in fact the abscissa of convergence of the local factors. The global zeta function has abscissa of convergence 2. This raised the interesting question of whether there was an example of a nilpotent group whose abscissa of convergence was not an integer. Some of the calculations of zeta function of groups in section 5 provide examples where this is indeed the case. Returning to the setting of zeta functions counting ﬁnite p-groups and ﬁnite nilpotent groups, we can use the analysis of the growth of coeﬃcients in the Euler product of cone integrals to deduce the following: Corollary 4.6 There exist a rational number α(c, d), an integer β(c, d) ≥ 0 and γ(c, d) ∈ R such that the number of ﬁnite nilpotent group of size bounded by n, class bounded by c and generated by at most d generators is asymptotically given by γ(c, d)nα(c,d) (log n)β(c,d) .

ZETA FUNCTIONS OF GROUPS

179

Problem 2 Determine for given c and d values of α(c, d) and β(c, d). Problem 3 What is the relationship between α(c, d), β(c, d) and αF c,d and βF c,d where Fc,d is the free nilpotent group?

5

The future: speculation and conjecture

We take the time in this section to document some recent calculations that have been made of zeta functions of groups. By collecting these results together in one paper, we hope that new patterns might emerge from analysing the shape and structure of the examples. 5.1

Direct products of Heisenberg groups

In section 2 we showed how to use p-adic integrals to calculate the zeta function counting normal subgroups in the Heisenberg group H = x, y, z : [x, y] = z . Explicit calculations have been made of the zeta functions counting subgroups and normal subgroups in various direct products of the Heisenberg group. We reproduce the results in the following proposition: Proposition 5.1 (1) ≤ (s) = ζ(s)ζ(s − 1)ζ(2s − 2)ζ(2s − 3)ζ(3s − 3)−1 ζH ζH (s) = ζ(s)ζ(s − 1)ζ(2s − 3).

(2) ≤ ζH×H (s) = ζ(s)ζ(s − 1)ζ(s − 3)ζ(2s − 4)2 ζ(2s − 5)2 ζ(3s − 5) ≤ WH×H (p, p−s ) ×ζ(3s − 7)ζ(3s − 8) p

(s) ζH×H

= ζ(s)ζ(s − 1)ζ(s − 2)ζ(s − 3)ζ(5s − 5)ζ(3s − 4)2 ζ(5s − 4)−1 .

where ≤ WH×H (X, Y )

= 1 + X 2 Y + X 4 Y 2 − X 4 Y 3 − 3X 5 Y 3 + X 6 Y 3 − X 7 Y 3 + X 5 Y 4 −X 6 Y 4 − 3X 7 Y 4 + X 8 Y 4 − 2X 9 Y 4 + X 7 Y 5 − 2X 8 Y 5 + X 10 Y 5 −4X 11 Y 5 + X 9 Y 6 − X 10 Y 6 + 3X 11 Y 6 + 4X 12 Y 6 − 2X 13 Y 6 + X 14 Y 6 +X 11 Y 7 − X 12 Y 7 + 3X 13 Y 7 + 3X 14 Y 7 − X 15 Y 7 + X 16 Y 7 + X 13 Y 8 −2X 14 Y 8 + 4X 15 Y 8 + 3X 16 Y 8 − X 17 Y 8 + X 18 Y 8 − 4X 16 Y 9 + X 17 Y 9 −2X 19 Y 9 + X 20 Y 9 − 2X 18 Y 10 + X 19 Y 10 − 3X 20 Y 10 − X 21 Y 10 + X 22 Y 10 −X 20 Y 11 + X 21 Y 11 − 3X 22 Y 11 − X 23 Y 11 + X 23 Y 12 + X 25 Y 13 + X 27 Y 14 . (3) ζH×H×H (s)

= ζ(s)ζ(s − 1)ζ(s − 2)ζ(s − 3)ζ(s − 4)ζ(s − 5)ζ(5s − 7)ζ(7s − 8) ×ζ(8s − 14)ζ(3s − 6)3 WH×H×H (p, p−s ) p

180

DU SAUTOY

where (X, Y ) WH×H×H

= 1 − 3X 6 Y 5 + 2X 7 Y 5 + X 6 Y 7 − 2X 7 Y 7 + X 12 Y 8 − 2X 13 Y 8 + 2X 13 Y 12 −X 14 Y 12 + 2X 19 Y 13 − X 20 Y 13 − 2X 19 Y 15 + 3X 20 Y 15 − X 26 Y 20 . ≤ (s) and ζH×H×H (s) where made by Taylor in his thesis The calculations of ζH×H [29]. The other calculations were made in the original paper of Grunewald, Segal and Smith [18]. My PhD student, Christopher Voll, has proved in his thesis [30] a more general result about the zeta functions of direct products of Heisenberg groups. It concerns the uniformity in p of the local factors.

Theorem 5.2 Let H n denote the direct product of n copies of the Heisenberg group. Then there exist rational functions WH≤n (X, Y ) and WH n (X, Y ) with the property that for almost all primes p ≤ ≤ −s ) ζH n ,p (s) = WH n (p, p −s ζH ). n ,p (s) = WH n (p, p

5.2

Zeta functions of Heisenberg groups over number fields

The examples of the previous section occurred naturally in the context of zeta functions of Heisenberg groups over number ﬁelds. If K is an algebraic extension of Q of degree n with ring of integers ϑK then set   1 ϑ K ϑK ϑK  . H(ϑK ) =  0 1 0 0 1 When p is a prime which completely splits in K (s) = ζH ζH(ϑ n ,p (s). K ),p

In [18] the case of a quadratic number ﬁeld was calculated. The normal global zeta function of H(ϑK ) can be neatly expressed in terms of the zeta function of the number ﬁeld K. Proposition 5.3 Let K be a quadratic extension of Q. Then (s) = ζ(s)ζ(s − 1)ζ(s − 2)ζ(s − 3) ζH(ϑ K)

×ζ(5s − 4)ζ(5s − 5)ζK (3s − 4)ζK (5s − 4)−1 . Gareth Taylor’s calculation of the zeta function of three copies of the Heisenberg group completed the missing case of Grunewald, Segal and Smith’s calculation of the zeta function of the Heisenberg group over a cubic number ﬁeld. Grunewald, Segal and Smith successfully proved a uniformity result for Heisenberg groups over number ﬁelds which fuelled their speculation about the possibility of a more general uniformity result. In particular they proved:

ZETA FUNCTIONS OF GROUPS

181

Proposition 5.4 For each ﬁnite family f = (f1 , . . . , fd ) of positive integers, there is a rational function Wf (X, Y ) such that whenever p > 2 is unramiﬁed in K and decomposes in K into d primes of residue class degree f1 , . . . , fd ζH(ϑ (s) = Wf (p, p−s ). K ),p

My graduate student Christopher Voll has recently combined the analysis of the work of Grunewald and myself together with his perspective using the language of buildings to reﬁne this proposition. He speculates that the local zeta function will not depend on the complete knowledge of the decomposition of p into primes but will only depend on the number of prime factors with residue class degree 1. This subtlety was not apparent in the small examples considered because the primes were always of residue class degree 1 or else inert. Conjecture 5.5 If f and f are families of positive integers with the same number of 1 s appearing in each family then Wf (X, Y ) = Wf (X, Y ). Voll has conﬁrmed the conjecture so far in the case of cubic number ﬁelds. Still open is the conjecture raised in [18] to extend this result from the Heisenberg group to other free nilpotent groups of arbitrary class over number ﬁelds. Of course this includes Conjecture 3.3 as a very special case. Casting this in the language of Lie algebras we might ask: Conjecture 5.6 Let F be a ﬁnitely generated free Lie algebra over Z of class c. Then for each ﬁnite family f = (f1 , . . . , fd ) of positive integers, there is a rational function Wf (X, Y ) such that whenever p > c is unramiﬁed in K and decomposes in K into d primes of residue class degree f1 , . . . , fd then ζF ⊗ϑK ,p (s) = Wf (p, p−s ). 5.3

Class two quotients of Un (Z)

Let Un (Z) be the group of upper triangular unipotent matrices over Z. We can consider the maximal class two quotient Ln of this group which is given by the presentation Ln = x1 , . . . , xn , y1 , . . . , yn−1 : [xi , xi+1 ] = yi (i = 1, . . . , n − 1)

where all other commutators are the identity. The case of n = 2 is of course just the Heisenberg group once again. A student of Dan Segal’s, Dermot Grenham, calculated in his thesis [15] the case of n = 3. Taylor successfully applied his method of choosing appropriate blow-ups to calculate the zeta function counting normal subgroups in the case of n = 4. We record their results in the following: Proposition 5.7 (1) [15] For almost all primes p ζL≤3 ,p (s) =

WL≤3 (p, p−s ) (1 − p−s )(1 − p1−s )(1 − p2−s )(1 − p4−2s )(1 − p5−2s )(1 − p6−3s )

182

DU SAUTOY

where

WL≤3 (X, Y ) = 1 + X 3 Y 2 + X 4 Y 2 − X 4 Y 3 − X 5 Y 3 − X 8 Y 5 . ζL 3 ,p (s) =

1 + p3−3s . (1 − p−s )(1 − p1−s )(1 − p2−s )(1 − p4−3s )(1 − p6−5s )

(2) [29] For almost all primes p ζL 4 ,p (s) = ζZ4p (s)

(1 −

p5−3s )2 (1

−

WL4 (p, p−s ) − p8−5s )(1 − p10−6s )(1 − p12−7s )

p6−5s )(1

where WL4 (X, Y ) = 1 + X 4 Y 3 − X 5 Y 5 + X 8 Y 5 − X 8 Y 6 − X 9 Y 6 − X 10 Y 8 − X 12 Y 8 − X 13 Y 9 +X 13 Y 10 − 2X 14 Y 10 + X 14 Y 11 + X 15 Y 11 − X 16 Y 11 − X 17 Y 11 + 2X 17 Y 12 −X 18 Y 12 + X 18 Y 13 + X 19 Y 14 + X 21 Y 14 + X 22 Y 16 + X 23 Y 16 − X 23 Y 17 +X 26 Y 17 − X 27 Y 16 − X 31 Y 22 .

Conjecture 5.8 There exist rational functions WL≤n (X, Y ) and WLn (X, Y ) with the property that for almost all primes p ζL≤n ,p (s) = WL≤n (p, p−s )

ζL n ,p (s) = WLn (p, p−s ). 5.4

Grenham’s examples

Grenham’s calculation of the zeta function of L3 was actually as an example of a diﬀerent system of nilpotent groups of increasing Hirsch length: Gn = x1 , ..., xn , u1 , ..., un−1 |[xi , xn ] = ui , [xi , xj ] = [xk , uj ] = 1

where ( i, j, ≤ n − 1 and k ≤ n) The group Gn can be thought of as n − 1 copies of this Heisenberg group with one oﬀ diagonal entry identiﬁed in each copy. The previous section documented Grenham’s formula for the zeta function of G3 = L3 . Here we include the results of his calculations of the zeta functions of G4 and G5 made in [15]: Proposition 5.9 (1) ≤ ζG (s) = ζZ4p (s) 4 ,p

WG≤4 (p, p−s ) (1 − p5−2s )(1 − p6−2s )(1 − p7−2s )(1 − p10−3s )(1 − p12−4s )

where WG≤4 (X, Y ) = 1 + Y 2X 4 + Y 2X 5 + Y 2X 6 − Y 3X 5 − Y 3X 6 − Y 3X 7 + Y 3X 8 + Y 3X 9 −Y 4 X 9 − Y 4 X 10 − Y 4 X 11 − Y 6 X 14 − Y 6 X 15 − Y 6 X 16 + Y 7 X 16 + Y 7 X 17 −Y 7 X 18 − Y 7 X 19 − Y 7 X 20 + Y 8 X 19 + Y 8 X 20 + Y 8 X 21 + Y 10 X 25 .

ZETA FUNCTIONS OF GROUPS

183

(2) (s) = ζZ4p (s) ζG 4 ,p

where

(1 −

WG 4 (p, p−s ) − p10−5s )(1 − p12−7s )

p6−3s )(1

WG 4 (X, Y ) = 1 + Y 3 X 4 + Y 3 X 5 + Y 5 X 8 + Y 5 X 9 + Y 8 X 13 .

(3) ζG5 ,p (s) =

ζZ5p (s)

(1 − p2−s )WG≤5 (p, p−s ) (1 − p6−2s )(1 − p8−2s )(1 − p9−2s )(1 − p14−3s )(1 − p18−4s )(1 − p20−5s )

where WG≤5 (X, Y ) = 1 + Y X 2 + Y 2 X 4 + Y 2 X 5 + Y 2 X 6 + 2Y 2 X 7 + Y 2 X 8 + Y 3 X 9 + 2Y 3 X 10 +Y 3 X 11 + 2Y 3 X 12 + Y 3 X 13 + Y 4 X 12 + 2Y 4 X 14 + 2Y 4 X 15 + Y 4 X 16 +Y 4 X 17 + 2Y 5 X 17 + Y 5 X 18 + 2Y 5 X 19 + Y 5 X 20 − Y 6 X 18 − Y 6 X 20 +Y 6 X 21 + 2Y 6 X 22 + 2Y 6 X 23 + 2Y 6 X 24 + Y 6 X 25 − Y 7 X 22 − 2Y 7 X 23 −2Y 7 X 24 − 2Y 7 X 25 − Y 7 X 26 + Y 7 X 27 + Y 7 X 29 − Y 8 X 27 − 2Y 8 X 28 −Y 8 X 29 − 2Y 8 X 30 − Y 9 X 30 − Y 9 X 31 − 2Y 9 X 32 − 2Y 9 X 33 − Y 9 X 35 −Y 10 X 34 − 2Y 10 X 35 − Y 10 X 36 − 2Y 10 X 37 − Y 10 X 38 − Y 11 X 39 −2Y 11 X 40 − Y 11 X 41 − Y 11 X 42 − Y 11 X 43 − Y 12 X 45 − Y 13 X 47 . (4) (s) = ζZ5p (s) ζG 5 ,p

WG 5 (p, p−s ) (1 − p8−3s )(1 − p14−5s )(1 − p18−7s )(1 − p20−9s )

where WG 5 (X, Y ) = 1 + Y 3 X 5 + Y 3 X 6 + Y 3 X 7 + Y 5 X 10 + Y 5 X 11 + 2Y 5 X 12 + Y 5 X 13 +Y 7 X 15 + Y 7 X 16 + Y 7 X 17 + Y 8 X 17 + Y 8 X 18 + Y 8 X 19 + Y 10 X 21 +2Y 10 X 22 + Y 10 X 23 + Y 10 X 24 + Y 12 X 27 + Y 12 X 28 +Y 12 X 29 + Y 15 X 34 . Voll succeeded in proving the following: (s) are uniform in p. Theorem 5.10 The zeta functions ζG n ,p

The uniformity of the zeta functions counting all subgroups is still open: ≤ Conjecture 5.11 The zeta functions ζG (s) are uniform in p. n ,p

Explicit calculations for nilpotent groups or Lie algebras beyond class 2 had been non-existent until recently. However the following section documents some more of Taylor’s impressive explicit techniques.

184

DU SAUTOY

5.5

Maximal class nilpotent lie algebras

Some of the ﬁrst calculations of zeta functions of nilpotent groups of class greater than two have recently been achieved. Gareth Taylor has successfully employed his methods to calculate the integrals corresponding to some examples of Lie algebras of maximal class: Mn = z, x1 , . . . , xn : [z, xi ] = xi+1 , i = 1, . . . , n − 1 . The case n = 2 is the Heisenberg Lie algebra. We document here Taylor’s results made in [29]. By the correspondence between nilpotent Lie algebras and nilpotent groups, Taylor’s calculations translate into calculations of the local zeta functions for almost all primes of the corresponding nilpotent groups of maximal class. Proposition 5.12 (1) ≤ (s) = W3≤ (p, p−s )ζp (s)ζp (s − 1)ζp (2s − 3)ζp (3s − 5)ζp (4s − 6) ζM 3 ,p

where

W3≤ (X, Y ) = 1 + X 2 Y 2 + X 3 Y 2 − X 3 Y 3 + X 4 Y 3 − X 5 Y 4 +X 6 Y 4 − X 6 Y 5 − X 7 Y 5 − X 9 Y 7 .

(2) ζM (s) = 3 ,p

ζp (s)ζp (s − 1)ζp (3s − 2)ζp (4s − 2)ζp (5s − 3) . ζp (5s − 2)

(3) ≤ (s) = W4≤ (p, p−s )ζp (s)ζp (s − 1)ζp (2s − 3)ζp (2s − 4) ζM 4 ,p

ζp (3s − 6)ζp (4s − 7)ζp (4s − 8)ζp (7s − 12) where W4≤ (X, Y ) = 1 + X 2 Y 2 + X 3 Y 2 − X 3 Y 3 + X 4 Y 3 + 2X 5 Y 3 − 2X 5 Y 4 +X 7 Y 4 − 2X 7 Y 5 − X 8 Y 5 + 2X 9 Y 6 − 2X 10 Y 6 − X 11 Y 6 +X 10 Y 7 − 2X 12 Y 7 − X 13 Y 7 + X 13 Y 8 − X 14 Y 8 − X 16 Y 9 +X 15 Y 10 + X 17 Y 11 − X 18 Y 11 + X 18 Y 12 + 2X 19 Y 12 − X 21 Y 12 +X 20 Y 13 + 2X 21 Y 13 + 2X 22 Y 13 − X 22 Y 14 + X 23 Y 14 + 2X 24 Y 14 −X 24 Y 15 + 2X 26 Y 15 − 2X 26 Y 16 − X 27 Y 16 + X 28 Y 16 −X 28 Y 17 − X 29 Y 17 − X 31 Y 19 . (4) (s) = W4 (p, p−s )ζp (s)ζp (s − 1)ζp (3s − 2)ζp (5s − 2)ζp (7s − 4) ζM 4 ,p

ζp (8s − 5)ζp (9s − 6)ζp (11s − 6)ζp (12s − 17) · ζp (6s − 3)−1

ZETA FUNCTIONS OF GROUPS

185

where W4 (X, Y ) = 1 + X 2 Y 4 − X 2 Y 5 + X 3 Y 5 − X 2 Y 6 + 2X 3 Y 6 − X 3 Y 7 −X 5 Y 9 + X 6 Y 10 − 2X 5 Y 11 − X 7 Y 13 − X 8 Y 13 + X 7 Y 14 −X 8 Y 14 − X 8 Y 15 − X 9 Y 15 + X 9 Y 16 − X 9 Y 17 − X 10 Y 17 +2X Y 18 − X 10 Y 18 + X 10 Y 19 − 2X 11 Y 19 + X 10 Y 20 + X 11 Y 20 9

−X 11 Y 21 + X 11 Y 22 + X 12 Y 22 + X 12 Y 23 − X 13 Y 23 + X 12 Y 24 +X 13 Y 24 + 2X 15 Y 26 − X 14 Y 27 + X 15 Y 28 + X 17 Y 30 − 2X 17 Y 31 +X 18 Y 31 − X 17 Y 32 + X 18 Y 32 − X 18 Y 33 − X 20 Y 37 . ≤ (s) is interesting because the corresponding global zeta The formula for ζM 4 ,p function built out of the Euler product of these local factors has a non-integer abscissa of convergence: ≤ ≤ (s) = p ζM (s) has abscissa of convergence 5/2. Corollary 5.13 ζM 4 4 ,p

The next section provides an example of a class two nilpotent group whose abscissa of convergence counting all subgroups is not an integer. In [18] it was proved that the abscissa convergence of the zeta function counting normal subgroups in a class 2 nilpotent group is always an integer. In particular they proved the following: Proposition 5.14 Let G be a ﬁnitely generated nilpotent group of class 2. Then is equal to the torsion-free rank of G/[G, G], the abelianisation of G. αG Problem 4 Are there ﬁnitely generated nilpotent groups (necessarily of class is not an integer? greater than 2) for which αG 5.6

Free class two three generator group

The ﬁrst example calculated of a class two nilpotent group whose zeta function ≤ (s) has a non-integer abscissa of convergence is the group ζG F2,3 = x1 , x2 , x3 , y1 , y2 , y3 : [x1 , x2 ] = y1 , [x1 , x3 ] = y2 , [x2 , x3 ] = y3

where all other commutators are the identity. The group is the free class two three generator nilpotent group. This is another example of Gareth Taylor’s impressive computations. Proposition 5.15 For almost all primes p ζF≤2,3 ,p (s) = p−s )(1

p1−s )(1

(1 − − where W (X, Y ) =

−

p2−s )(1

(1 + p4−2s )W (p, p−s ) − p5−2s )(1 − p6−2s )(1 − p6−3s )(1 − p7−3s )(1 − p8−3s )

1 + X 3Y 2 + X 4Y 2 + X 5Y 2 − X 4Y 3 − X 5Y 3 − X 6Y 3 − X 7Y 4 − X 9Y 4 −X 10 Y 5 − X 11 Y 5 − X 12 Y 5 + X 11 Y 6 + X 12 Y 6 + X 13 Y 6 − X 16 Y 8 . The global zeta function ζF≤2,3 (s) has abscissa of convergence 7/2.

186 5.7

DU SAUTOY Functional equations

There is one rather remarkable feature of all these rational functions representing the local zeta functions of nilpotent groups: their numerators exhibit an extraordinary palindromic symmetry. This can be expressed as a local functional equation. By changing every occurrence of p to p−1 in the expression for the rational function one gets a new rational function. But by simply pulling out appropriate powers of p and p−s one can recover the original rational function. For example in the case of the free class two, three generator nilpotent group: ζF≤2,3 ,p (s)|p→p−1 = −p15−6s ζF≤2,3 ,p (s). Analysing all the uniform class two examples (which thanks to Gareth Taylor’s work are quite extensive by now) we ﬁnd that they all satisfy the following conjecture regarding the existence and shape of the functional equation. Conjecture 5.16 Let G be a class two nilpotent group of Hirsch length d whose abelianisation G/[G, G] has torsion-free rank m. Suppose that the zeta functions of G are uniform. Then for almost all primes p the local zeta functions satisfy the following local functional equations: d ≤ ≤ (s)|p→p−1 = (−1) p(2)−ds ζG,p (s) ζG,p d (s)|p→p−1 = (−1)d p(2)−(d+m)s ζG,p (s). ζG,p

where is some integer. (In all cases except F2,3 , can be taken to be d.) (This functional equation is a wonderful way to be sure that any explicit computation is correct or not. It is very diﬃcult to fake such an equation. Any examples in this paper which fail this functional equation are an indication that I’ve made a typing error!) Voll has conﬁrmed the functional equation for the normal zeta functions of all ﬁnitely generated class two nilpotent groups whose centres have torsion-free rank 2. This depends on the classiﬁcation of such groups made by Grunewald and Segal in [17]. Even Taylor’s examples of groups of maximal class satisfy the same functional equation for the zeta function counting all subgroups. However the functional equations for the zeta function counting normal subgroups take a slightly diﬀerent shape. Conjecture 5.17 Let Mn be the Lie algebra of maximal class of dimension d = n + 1. Then the local zeta functions satisfy the following local functional equations d ≤ ≤ (s)|p→p−1 = (−1)d p(2)−ds ζM (s) ζM n ,p n ,p d d+1 ζM (s)|p→p−1 = (−1)d p(2)−(( 2 )−1)s ζM (s). n ,p n ,p

ZETA FUNCTIONS OF GROUPS

187

During the lectures I gave in Oxford, I was sceptical whether these functional equations would survive once one passed to non-uniform examples like the group encoding the elliptic curve y 2 = x3 −x. On September 11 2001, my student Christopher Voll turned up to my house with an explicit calculation of the elliptic curve example. He made the calculation by exploiting new ideas he has for counting normal subgroups by using the language of buildings. His calculation identiﬁes the speciﬁc rational functions alluded to in Theorem 3.9 Proposition 5.18 Let E be the elliptic curve Y 2 = X 3 − X. Let G(E) be the Hirsch length 9, class two nilpotent group given in Theorem 3.9. Then for almost all primes p : (s) = ζZ6p (s) ζG(E),p

(1 −

W1 (p, p−s ) + |E(Fp )|W2 (p, p−s ) 7−5s p )(1 − p8−7s )(1 − p14−8s )(1 − p18−9s )

where W1 (X, Y ) =

1 + X 6 Y 7 + X 7 Y 7 + X 12 Y 8 + X 13 Y 8 + X 19 Y 15 W2 (X, Y ) = X 6 Y 5 1 − Y 2 1 + X 13 Y 8 .

1 − X 7Y 5

You will see that the polynomials W1 (X, Y ) and W2 (X, Y ) individually have functional equations but not quite of the same shape. W1 (X −1 , Y −1 ) = −X −26 Y −20 W1 (X, Y ) W2 (X −1 , Y −1 ) = −X −25 Y −20 W2 (X, Y ) However all was not lost. Along with the Riemann Hypothesis for curves, Weil also conjectured the existence of a functional equation satisﬁed by the zeta functions counting points. We can see this functional equation quite clearly in the case of the elliptic curve. If |E(Fpn )| denotes the number of solutions in projective space P 2 (Fpn ) of the equation y 2 z = x3 − xz 2 then |E(Fpn )| = 1 − (πpn + πp n ) + pn where πp is a complex number of absolute value p1/2 and πp πp = p. The function n → |E(Fpn )| extends naturally to a function over Z and satisﬁes the functional equation |E(Fp−n )| = p−n |E(Fpn )|. When we inserted this functional equation into the expression for the zeta function of the nilpotent group encoding the elliptic curve, everything slotted into place. The missing power of X in the functional equation for W2 (X, Y ) was supplied by the functional equation for the elliptic curve: Theorem 5.19 For almost all primes p, the normal zeta function of G(E) satisﬁes the following local functional equation (s)|p→p−1 = −p36−15s ζG(E),p (s). ζG(E),p

188

DU SAUTOY

The shape of this functional equation even satisﬁes Conjecture 5.16. Half an hour after making this miraculous discovery, Christopher and I switched on the television to see the twin towers crumbling to the ground. Voll’s calculation provides exciting evidence that this functional equation might truly be a general phenomenon for nilpotent groups. Since the functional equation conjectured by Weil is restricted to absolutely non-singular projective varieties, one might have to restrict one’s expectations of a general functional equation. This depends to some extent on how easy it is to encode arbitrary varieties into the presentation of a nilpotent group. In jointwork with Lubotzky [12] we did in fact establish a form of this functional equation for zeta functions counting subgroups isomorphic to the original group. The functional equation exploits symmetries in the associated building of the algebraic automorphism group of the nilpotent group. In a diﬀerent direction, Denef and Meuser [2] have proved a functional equation for a special class of Igusa zeta functions by exploiting the proof of Weil’s functional equation. 5.8

Class number formula

I believe that the subject of zeta functions of groups will have reached full maturity when we can give some interpretation of the residue of the zeta function at the pole located on the abscissa of convergence. As a wild guess it would be wonderful if it encoded for example something that we can call the class number of a nilpotent group G. This is deﬁned to be the number of nilpotent groups up to isomorphism with the same proﬁnite completion as G. All such groups will have the same zeta function. The class number is ﬁnite thanks to a Theorem of Pickel [26]. Dan Segal has suggested that this is more probably related to the zeta function counting subgroups whose proﬁnite completion is isomorphic to the original group. This zeta function is studied in [18] and [12]. To achieve such an analogue of the class number formula of the Dedekind zeta function of a number ﬁeld would certainly make number theorists wake up to the importance of these zeta functions of groups. References [1] J. Cawelti, Adventure, Mystery and Romance. University of Chicago Press, 1976. [2] J. Denef and D. Meuser, A functional equation of Igusa’s local zeta function, Amer. J. Math., 113 (1991), 1135-1152. [3] M.P.F. du Sautoy, Zeta functions and counting ﬁnite p-groups, Electronic Research Announcements of the American Math. Soc., 5 (1999), 112-122. [4] M.P.F. du Sautoy, The zeta function of sl2 (Z), Forum Mathematicum, 12 (2000), 197-221. [5] M.P.F. du Sautoy, Addendum to the paper The zeta function of sl2 (Z), Forum Mathematicum 12 (2000), 383. ´ [6] M.P.F. du Sautoy, Counting p-groups and nilpotent groups, Inst. Hautes Etudes Scientiﬁques, Publ. Math., 92 (2000), 63-112. [7] M.P.F. du Sautoy, A nilpotent group and its elliptic curve: non-uniformity of local zeta functions of groups, M.P.I. preprint 2000-85. Israel J. of Math. 126 (2001), 269-288. [8] M.P.F. du Sautoy, Counting subgroups in nilpotent groups and points on elliptic curves, M.P.I. preprint 2000-86. To appear in J. Reine Angew. Math.

ZETA FUNCTIONS OF GROUPS

189

[9] M.P.F. du Sautoy and F.J. Grunewald, Analytic properties of zeta functions and subgroup growth, Annals of Math, 152 (2000), 793-833. [10] M.P.F. du Sautoy and F.J. Grunewald, Uniformity for 2-generator free nilpotent groups, in preparation. [11] M.P.F. du Sautoy and F. Loeser, Motivic zeta functions of inﬁnite dimensional Lie ´ algebras, Ecole Polytechnique preprint series 2000-12. [12] M.P.F. du Sautoy and A. Lubotzky, Functional equations and uniformity for local zeta functions of nilpotent groups, Amer. J. Math., 118 (1996), 39-90. [13] M.P.F. du Sautoy and D. Segal, Zeta functions of groups, in New Horizons in pro-p Groups, edited by M.P.F. du Sautoy, D. Segal and Aner Shalev, Progress in Mathematics 184 Birkh¨ auser (2000), 249-286. [14] M.P.F. du Sautoy and G. Taylor, The zeta function of sl2 and resolution of singularities. Math. Proc. of the Cambridge Phil. Soc. 132 (2002), 57-73. [15] D. Grenham, Some topics in nilpotent group theory, DPhil thesis 1988, Oxford. [16] C. Griﬃn, Ph.D. thesis 2002, Nottingham. [17] F.J. Grunewald and D. Segal, Reﬂections on the classiﬁcation of torsion-free nilpotent groups, in Group Theory: Essays for Philip Hall edited by K.W. Gruenberg and J.E. Roseblade, Academic Press, London (1984), 121-158. [18] F.J. Grunewald, D. Segal and G.C. Smith, Subgroups of ﬁnite index in nilpotent groups, Invent. Math. 93 (1988), 185-223. [19] G. Higman, Enumerating p-groups, I, Proc. London Math. Soc. 10 (1960), 24-30. [20] G. Higman, Enumerating p-groups, II, Proc. London Math. Soc. 10 (1960), 566-582. [21] I. Ilani, Zeta functions related to the group SL2 (Zp ), Israel J. of Math., 109 (1999), 157-172. [22] K. Ireland and M. Rosen, A classical introduction to modern number theory, 2d edition. Graduate texts in mathematics 84, Springer-Verlag, New York–Berlin–Heidelberg, 1993. [23] A. Lubotzky, Counting ﬁnite index subgroups. In Groups ’93 Galway/St. Andrews, L.M.S. Lecture Note Series 212 C.U.P. (1995), 368-404. [24] A. Lubotzky and D. Segal, Subgroup Growth, to be published by Birkh¨ auser, Basel. [25] W. Narkiewicz, Number Theory, World Scientiﬁc Publishing Co, 1983. [26] P.F. Pickel, Finitely generated nilpotent groups with isomorphic ﬁnite quotients. Trans. Amer. Math. Soc. 160 (1971), 327-341. [27] D. Segal, Subgroups of ﬁnite index in soluble groups I. In Proceedings of Groups-St. Andrews 1985, L.M.S. Lecture Note Series 121 C.U.P (1986), 307-314. [28] C.C. Sims, Enumerating p-groups, Proc. London Math. Soc. 15 (1965), 151-166. [29] G. Taylor, Zeta functions of algebras and resolution of singularities, Ph.D. thesis 2001 Cambridge. [30] C. Voll, Zeta functions of algebras and enumeration in Bruhat-Tits buildings, Ph.D. thesis 2002 Cambridge.

SOME FACTORIZATIONS INVOLVING HYPERCENTRALLY EMBEDDED SUBGROUPS IN FINITE SOLUBLE GROUPS L. M. EZQUERRO

†

` and X. SOLER-ESCRIVA

‡ 1

†

Departamento de Matem´ atica e Inform´ atica, Universidad P´ ublica de Navarra, Campus de Arrosad´ıa. 31006 Pamplona, Spain ‡ Centre d’Investigaci´ o Operativa, Departament d’Estad´ıstica y Matem`atica Aplicada, Universitat Miguel Hern´ andez, Avinguda del Ferrocarril, s/n. 03202 Elx, Spain E-mail: [email protected]

1

Introduction

All groups considered in this note are ﬁnite. If H is a subgroup of a group G, we write and Recall that Subgroups which permute with all Sylow subgroups of the group, or S-permutable subgroups, were introduced by Kegel in his seminal paper [K 62]. P. Schmid, in [Sch 98], presented an extensive and elegant study of these subgroups. In that paper it is proved that for a core-free S-permutable subgroup T of a group G which also permutes with the normalizer of a Sylow subgroup N , then T ≤ N ([Sch 98]; Prop. D). Since the hypercenter Z∞ (G) of the group G, i.e. the last member of the ascending central series of G, is the intersection of the normalizers of all Sylow subgroups of G, we have that for a subgroup T of G the following are equivalent: (i) T is an S-permutable subgroup which permutes with the normalizers of all Sylow subgroups of G, and (ii) T G /TG ≤ Z∞ (G/TG ), for TG = g∈G T g , the core of T in G. Such a subgroup is said to be a hypercentrally embedded subgroup. We focus our attention on these special S-permutable subgroups. Hypercentrally embedded subgroups enjoy very good factorization properties. In fact, previously, Carocca and Maier, in [CM 98], had characterized hypercentrally embedded subgroups as those subgroups which permute with all pronormal subgroups. In this note we present some factorizations of hypercentrally embedded subgroups with some special types of subgroups which, in general, are not pronormal. We prove in Theorem 3.1 that in a ﬁnite group, these hypercentrally embedded subgroups permute with every F-normalizer, for F a saturated formation. In Theorem 4.2 we prove that hypercentrally embedded subgroups permute with subgroups of prefrattini type in ﬁnite soluble groups. These results cannot be reproduced for S-permutable subgroups as is shown in the ﬁnal Example 5.1. Moreover, we prove that hypercentrally embedded subgroups are subgroups with the cover and avoidance property; this property does not hold in general with S-permutable subgroups either. In the soluble universe, the subgroups formed by the above mentioned factorizations and the corresponding intersections are also subgroups with the cover and avoidance property. Makan in [M 70] proved that, in the soluble universe, normally embedded 1 The authors are partially supported by Proyecto ”Acciones y representaciones de grupos” of Gobierno de Navarra.

HYPERCENTRALLY EMBEDDED SUBGROUPS

191

subgroups also factorize with prefrattini subgroups and F-normalizers. Our results can be extended to factorizations involving subgroups whose Sylow subgroups are also Sylow subgroups of hypercentrally embedded subgroups. Makan’s proofs in [M 70] depend closely of the structure of normally embedded subgroups of ﬁnite soluble groups and in particular of the cover and avoidance properties. Thus, our proofs are essentially diﬀerent and, for instance, do not use cover and avoidance properties. As is pointed out above, it is remarkable that these factorizations are no longer valid if we use subgroups whose Sylow subgroups are also Sylow subgroups of S-permutable subgroups.

2

Hypercentrally embedded subgroups

Definition 2.1 A subgroup T of a ﬁnite group G is said to be (i) an S-permutable subgroup of G, if T permutes with every Sylow subgroup of G; (ii) a hypercentrally embedded subgroup of G, if every chief factor of G between TG and T G is central in G or, in other words, T G /TG ≤ Z∞ (G/TG ), where T G = T g : g ∈ G is the normal closure of T in G. Carocca and Maier, in [CM 98], characterized the hypercentrally embedded subgroups as those subgroups which permute with all pronormal subgroups. Later, Schmid in ([Sch 98]; Prop. D) proved that if T is a core-free S-permutable subgroup of a group G which also permutes with the normalizer of a Sylow subgroup N , then T ≤ N . With this result, we can simplify Carocca and Maier’s original proof. Proposition 2.2 (See [CM 98]) Let G be a group and T a subgroup of G. The following statements are equivalent: (i) T is an S-permutable subgroup which permutes with the normalizers of all Sylow subgroups of G; (ii) T G /TG ≤ Z∞ (G/TG ); (iii) T permutes with all pronormal subgroups of G. In particular, a core-free subgroup T of G is hypercentrally embedded in G if and only if T normalizes all pronormal subgroups of G. Proof: (i) ⇒ (ii). By ([Sch 98]; Prop. D), if T is a core-free S-permutable subgroup of a group G which also permutes with the normalizer of a Sylow subgroup N , then T ≤ N . Therefore if T is an S-permutable subgroup which permutes with the normalizers of all Sylow subgroups of G, then T /TG normalizes all Sylow subgroups of G/TG . In other words, T /TG ≤ Z∞ (G/TG ). (ii) ⇒ (iii). Notice that every subgroup H of G is normal in HZ(G) and then H is subnormal in HZ∞ (G). Therefore if T G /TG ≤ Z∞ (G/TG ) and H is pronormal in G, then T normalizes HTG . In particular, T permutes with H for every pronormal subgroup

192

` EZQUERRO, SOLER-ESCRIVA

H of G. Furthermore, if TG = 1, then T normalizes H, indeed. (iii) ⇒ (i). If T permutes with all pronormal subgroups of G, we have that, in particular, T is an S-permutable subgroup of G which permutes with the normalizers of all Sylow subgroups of G. Hypercentrally embedded subgroups are S-permutable subgroups. The converse does not hold in general as we will see in Example 5.1. Our ﬁrst result will be useful in proofs using induction arguments. The proof only involves routine checking. Lemma 2.3 Let G be a group, and T a hypercentrally embedded subgroup of G. Then, if K is a normal subgroup of G and M ≤ G, we have: (i) the subgroup T K is a hypercentrally embedded subgroup of G and T K/K is a hypercentrally embedded subgroup of G/K; (ii) if K ≤ M and M/K is a hypercentrally embedded subgroup of G/K, then M is a hypercentrally embedded subgroup of G. In the ﬁrst paragraph of Theorem 4 in [BBPA 96] it is proved that Y ∩ Z∞ (X) ≤ Z∞ (Y ), for any subgroup Y of a group X. Using these arguments it is not diﬃcult to prove the following. Lemma 2.4 Let G be a group, M a subgroup of G and T a hypercentrally embedded subgroup of G. Then (i) the subgroup M ∩ T is a hypercentrally embedded subgroup of M ; (ii) if T ≤ M , then T is a hypercentrally embedded subgroup ofM ; (iii) if moreover M is normal in G, then M ∩ T is a hypercentrally embedded subgroup of G. Recall that, if H/K is a chief factor of a group G, a subgroup T covers H/K if H = H ∩KT , and T avoids H/K if K = H ∩KT . A subgroup that covers or avoids any chief factor of the group is said to be a CAP-subgroup or a subgroup with the cover and avoidance property. A normal subgroup is a trivial example of a CAPsubgroup. On the other hand, in general S-permutable subgroups are subnormal subgroups which are not CAP-subgroups (see Example 5.1). For hypercentrally embedded subgroups we have the following. Theorem 2.5 Let G be a group and T a hypercentrally embedded subgroup of G. Then T is a subgroup with the cover and avoidance property in G.

3

Factorizations of hypercentrally embedded subgroups with Fnormalizers

In ([Sch 98]; Th. C) it is proved that in a soluble group G, an S-permutable subgroup T is hypercentrally embedded in G if and only if T permutes with some system normalizer of G. The system normalizers of a soluble group are the N normalizers, for N the saturated formation of the nilpotent groups. Our next result shows that in fact hypercentrally embedded subgroups of ﬁnite not-necessarily

HYPERCENTRALLY EMBEDDED SUBGROUPS

193

soluble groups permute with F-normalizers for any saturated formation F. For more details of F-normalizers of ﬁnite (not-necessarily soluble) groups see [BB 89]. It must be observed that F-normalizers are not pronormal subgroups in general. Theorem 3.1 Let F be a saturated formation and G a group. If T is a hypercentrally embedded subgroup of G and D is an F-normalizer subgroup of G, then T D = DT is a subgroup of G. In the soluble universe F-normalizers are CAP-subgroups which cover the F-central chief factors and avoid the F-eccentric ones. The factorization of Theorem 3.1 produces, in a soluble group, a new CAP-subgroup. Proposition 3.2 Let G be a soluble group and T a hypercentrally embedded subgroup of G. Let F be a saturated formation and D an F-normalizer of G. (i) the subgroup DT possesses the cover and avoidance property in G; more precisely, DT avoids the F-eccentric chief factors of G avoided by T and covers the rest. (ii) the subgroup D ∩ T possesses the cover and avoidance property in G; more precisely, D ∩ T covers the F-central chief factors of G covered by T and avoids the rest.

4

Factorizations of hypercentrally embedded subgroups with prefrattini subgroups

Definition 4.1 Let G be a soluble group. Consider a Hall system Σ of G. (i) [BBE 95] A set X of maximal subgroups of G is said to be weakly solid (or w-solid) if it satisﬁes the property that if M1 , M2 ∈ X with coreG (M1 ) = coreG (M2 ) and both complement a chief factor H/K of G, then M = (M1 ∩ M2 )H ∈ X. (ii) [BBE 95] Let X be a w-solid set of maximal subgroups of G. Consider the following subgroup W (G, X, Σ) = ∩{M ∈ X : Σ reduces into M }. We say that W (G, X, Σ) is the X-prefrattini subgroup of G associated with Σ. If {M ∈ X : Σ reduces into M } = ∅, we put W (G, X, Σ) = G. When X is the set of all maximal subgroups of the group, the X-prefrattini subgroups are just the classical prefrattini subgroups deﬁned by Gasch¨ utz. In fact, as was remarked in [BBE 95], this deﬁnition extends the classical ones due to Gasch¨ utz, Hawkes, F¨orster and Kurtzweil (see [DH 92] for details of these constructions). Our aim here is to obtain a factorization of X-prefrattini subgroups with hypercentrally embedded subgroups in a ﬁnite soluble group. In our proof we do not use the cover and avoidance property. The new subgroup that appears is an X T -prefrattini subgroup, where X T denotes the set of maximal subgroups, X T = {M ∈ X : T ≤ M },

` EZQUERRO, SOLER-ESCRIVA

194

which is always a w-solid of maximal subgroups of G, for any subgroup T of G. Theorem 4.2 Let G be a soluble group, Σ a Hall system of G and X a w-solid set of maximal subgroups of G. Let T be a hypercentrally embedded subgroup of G. Then W (G, X T , Σ) = T W (G, X, Σ) . Notice that, in a soluble group G, a prefrattini subgroup W (G, X, Σ) is a CAPsubgroup of G which avoids the X(Σ)-complemented chief factors of G and covers the rest, where X(Σ) = {M ∈ X : Σ reduces into M } (see [BBE 95]; Proposition p.274). Therefore W (G, X T , Σ) = T W (G, X, Σ) is a CAP-subgroup of G which avoids the X T (Σ)-complemented chief factors of G and covers the rest. However the X T (Σ)-complemented chief factors of G cannot be described in terms of the X(Σ)-complemented chief factors of G avoided by T . Example 4.3 Consider the group D = a : a4 = 1 ∼ = C4 , a cyclic group of order 4, and C = x : x2 = 1 ∼ = C2 , a cyclic group of order 2. Construct the direct product G = D × C, an abelian group of order 8. It is easy to see that Φ(G) = a2 and {Φ(G), C, a2 x} is the set of all minimal normal subgroups of G. Write S = a2 , x and M = ax. The set X = M ax(G) = {S, D, M } contains all maximal subgroups of G. Obviously X is a solid set of maximal subgroups of G. Notice that S ∩ D = S ∩ M = D ∩ M = Φ(G). Consider the subgroup T = a2 x, which is trivially a normal subgroup of G. Clearly, X T = {S}. Denote W = W (G, X, Σ), where Σ = {1, G} is a Hall system of G. Therefore W = Φ(G) and WT = S = W T . The minimal normal subgroup C of G is X-complemented (by D and M ) and it is avoided by T ; but C is an X T -Frattini chief factor of G. This implies that C is avoided by W and by T . However C is covered by WT . The above example shows that we cannot use a routine order argument as in Proposition 3.2 to prove that T ∩ W is also a CAP -subgroup. This fact needs a separate proof. Theorem 4.4 Let G be a soluble group, X a w-solid set of maximal subgroups of G and T a hypercentrally embedded subgroup of G. Let W = W (G, X, Σ) be the X-prefrattini subgroup of G associated to Σ. Then T ∩ W is a CAP -subgroup of G.

5

Final remarks

Example 5.1 The main results presented here are not true when using S-permutable subgroups instead of hypercentrally embedded subgroups. Let D = c, b : c7 = b2 = 1, cb = c6 be the dihedral group of order 14. Denote C = c ∼ = C7 . There exists an irreducible C-module W over GF (2) of dimension 3, such that the minimal polynomial of the action of c over W is x3 + x2 + 1. If {w1 , w2 , w3 } is a basis of W , then w1c = w2 , w2c = w3 , w3c = w1 + w3 . Consider the induced module V = W D . Then VC = W ⊕ W b . In the C-module W b the action of c 6 is described by (wb )c = wbc = wbcbb = (wc )b , for all w ∈ W . Since the minimal

HYPERCENTRALLY EMBEDDED SUBGROUPS

195

polynomial of the action of c6 over W is x3 +x+1, we have that W and W b are nonisomorphic irreducible C-modules. Therefore, the inertia subgroup is ID (W ) = C. This implies that V is an irreducible D-module over K, by ([HB-II 82];VII,9.6). Construct the semidirect product G = [V ]D. Then V = W × W b . If Q ∈ Syl7 (G), then Q = C v , for some element v ∈ V . Since W C is a subgroup of G, we have that (W C)v = W C v = W Q is a subgroup of G. Moreover W ≤ V = O2 (G) and then W is contained in all Sylow 2-subgroups of G. Hence W is an S-permutable subgroup of G. (i) The subgroup D is maximal in G. Therefore D is pronormal in G. However W does not permute with D. Thus the subgroup W is not hypercentrally embedded in G. Notice that WG = 1 and Z∞ (G) = 1. (ii) Since V is a minimal normal subgroup of G and 1 = W = W ∩ V = V , the subgroup W does not possess the cover and avoidance property in G. Hence Theorem 2.5 does not hold for S-permutable subgroups in general. (iii) The subgroup b is a system normalizer of G, that is an N -normalizer of G, for N the saturated formation of all nilpotent groups. The subgroup W does not permute with b. Hence Theorem 3.1 does not hold for S-permutable subgroups in general. (iv) Consider the w-solid set of maximal subgroups X := {D, P } of G, where P = V b, then X W = {P }. Let Σ be a Hall system of G such that P ∈ Σ. Then W (G, X, S) = b and W (G, X W , S) = P . Nevertheless, the product W b is not a subgroup of G and obviously W b = P . Hence Theorem 4.2 is not true for S-permutable subgroups. Makan in [M 70] proved that, in the soluble universe, normally embedded subgroups also factorize with (Gasch¨ utz-)prefrattini subgroups and F-normalizers. Recall that normally embedded subgroups are subgroups whose Sylow subgroups are also Sylow subgroups of normal subgroups. In [M 70] all proofs depend heavily on the structure of normally embedded subgroups of ﬁnite soluble groups and in particular on the cover and avoidance properties. Our results can be extended to factorizations involving subgroups whose Sylow subgroups are also Sylow subgroups of hypercentrally embedded subgroups. However our proofs are essentially diﬀerent to those of Makan and, for instance, do not use cover and avoidance properties. Remarks 5.2 The main results presented here remain true, with small changes, when we use subgroups whose Sylow subgroups are also Sylow subgroups of hypercentrally embedded subgroups. Let us say that a subgroup V of a group G is pro-permutably embedded in G if each Sylow subgroup of V is also a Sylow subgroup of a hypercentrally embedded subgroup of G. (1) A pro-permutably embedded subgroup of a ﬁnite group G covers or avoids every abelian chief factor of the group G. In particular in a soluble group all pro-permutably embedded subgroups are CAP-subgroups. (2) Given a soluble group G and a Hall system Σ of G, a pro-permutably embedded subgroup V of G, such that Σ reduces into V , permutes with the F-normalizer D of G associated with Σ, for any saturated formation F. The

` EZQUERRO, SOLER-ESCRIVA

196

product DV and the intersection D ∩V are CAP-subgroups of G as in Proposition 3.2. (3) If V is a pro-permutably embedded subgroup of a soluble group G, then the set X V = {M ∈ X : V g ≤ M for some g ∈ G}, is a w-solid set of maximal subgroups of G. The X-prefrattini subgroup of G associated with a Hall system Σ permutes with any pro-permutably embedded subgroup V of G such that Σ reduces into V and the product is the X V -prefrattini subgroup of G associated with Σ. The intersection is also a CAP-subgroup of G as in Theorem 4.4. As is pointed out above, it is remarkable that these factorizations are no longer valid if we use subgroups whose Sylow subgroups are also Sylow subgroups of Spermutable subgroups. References A. Ballester-Bolinches, H-normalizers and local definitions of saturated formations of finite groups. Israel J. Math. 67 (1989), 312-326. [BB 95] A. Ballester-Bolinches, Permutably embedded subgroups of finite soluble groups. Arch. Math. 65 (1995), 1-7. [BBE 91] A. Ballester-Bolinches and L. M. Ezquerro, On maximal subgroups of finite groups. Comm. Algebra, 19 (8) (1991), 2373-2394. [BBE 95] A. Ballester-Bolinches and L. M. Ezquerro, The Jordan-H¨ older theorem and prefrattini subgroups of finite groups. Glasgow Math. J. 37 (1995), 265-277. [BBPA 96] A. Ballester-Bolinches and M. C. Pedraza-Aguilera, On minimal subgroups of finite groups. Acta Math. Hungar 73 (4) (1996), 335-342. [CM 98] A. Carocca and R. Maier, Hypercentral embedding and pronormality. Arch. Math. 71 (1998), 433-436. [DH 92] K. Doerk and T. O. Hawkes, Finite soluble groups. De Gruyter (1992). [HB-II 82] B. Huppert and N. Blackburn, Finite groups II. Springer-Verlag (1982). [K 62] O. H. Kegel, Sylow-Gruppen und Subnormalteiler endlicher Gruppen. Math. Z. 78 (1962) 205-221. [M 70] A. Makan, Another characteristic conjugacy class of subgroups of finite soluble groups. J. Austral. Math. Soc. 11 (1970), 395-400. [Ma 73] A.Makan, On certain sublattices of the lattice of subgroups generated by the prefrattini subgroups, the injectors and the formation subgroups. Can. J. Math., Vol. XXV, No.4,(1973), 862-869. [MS 73] R. Maier and P. Schmid, The embedding of quasinormal subgroups in finite groups. Math. Z. 131 (1973), 269-272. [Sch 98] P. Schmid, Subgroups permutable with all Sylow subgroups. J. Algebra. 207 (1998), 285-293. [T 75] M. J. Tomkinson, Prefrattini subgroups and cover-avoidance properties in Ugroups. Canad. J. Math. 27 (1975), 837[BB 89]

ELEMENTARY THEORY OF GROUPS BENJAMIN FINE1 , ANTHONY M. GAGLIONE2 , ALEXEI MYASNIKOV3 , GERHARD ROSENBERGER4 and DENNIS SPELLMAN5 1 2 3 4 5

Department of Mathematics, Fairﬁeld University, Fairﬁeld, Connecticut 06430, USA Department of Mathematics, U.S. Naval Academy, Anapolis, Maryland 1402, USA Department of Mathematics, City College of CUNY, New York, New York 10031, USA Fachbereich Mathematik Universit¨ at, Dortmund, 44221 Dortmund, Germany Department of Mathematics, Temple University, Philadelphia, Pennsylvania 19132, USA

Abstract Remarkable ties between group theory, logic and algebraic geometry have come to light via the positive solution of the Tarski conjecture. This has led to further work on the universal theory of groups. In this paper we describe and survey this material a large body of which is not familiar to most group theorists

Contents 1. Introduction. 2. First Order Languages and Model Theory. 3. The Tarksi Problems. 4. Residually Free and Universally Free Groups. 5. Algebraic Geometry over Groups and Applications. 6. The Positive Solution to the Tarski Problems. 7. Discriminating, Co-discriminating and Squarelike Groups. 8. Open Questions.

1

Introduction

The elementary theory of a group G consists of all the ﬁrst-order or elementary sentences (see section 2) which are true in G. Although this is a concept which originated in formal logic, in particular model theory, it arises independently from the theory of equations within groups. Recall that an equation in a group G is a word W (x1 , ..., xn , g1 , ..., gk ) in free variables x1 , ..., xn and constants g1 , ..., gk which are elements of G. A solution consists of an n-tuple (h1 , ..., hn ) of elements from G which upon substitution for x1 , .., xn make the word trivial in G. Hence an equation is a ﬁrst-order sentence in the language L[G] consisting of the elementary language of group theory (again see section 2) augmented by allowing constants from the group G. 1 The research of the second author was partially supported by the Naval Academy Research Council.

198

FINE, GAGLIONE, MYASNIKOV, ROSENBERGER, SPELLMAN

Equations within groups play an important role in many areas of both algebra and formal logic (see [LS] or [G]) and the development of the theory of equations within groups has a long history. The greatest progress has been with the theory of equations within free groups. Fundamental work was done by Makanin [Mak1], [Mak2] and Razborov [Ra] in ﬁnding both an algorithm to determine if a system of equations in a free group has a solution and if it does to describe a solution. Another breakthrough result in this area was obtained by O. Kharlampovich and A. Myasnikov in the series of two papers [KhM1], [KhM2]. They proved that every ﬁnite system of equations S(X) = 1 over a free group can be eﬀectively transformed into a ﬁnite number of systems of a very particular type (so-called non-degenerate triangular quasi-quadratic systems, see Section 5), thus reducing the system S = 1 into ﬁnitely many quadratic equations over groups which are pretty close to being free (universally free groups). This work in part was motivated by several longstanding problems known as the Tarski Conjectures. The ﬁrst of these is that all non-abelian free groups have the same elementary theory while the second is that the elementary theory of these free groups is decidable (see section 3). While signiﬁcant work on these conjectures was done they resisted complete solution until the development (by Baumslag, Myasnikov and Remesslennikov [BMR1] [BMR2]) of an algebraic geometry over groups mirroring classical algebraic geometry over ﬁelds. Applying this algebraic geometry a positive solution to the Tarski problems was given by Kharlampovich and Myasnikov [KhM1], [KhM2], [KhM3],[KhM4], (An independent solution to the ﬁrst Tarski problem has also been proposed by Sela [Se]). Concurrently with the work on the Tarski conjectures there has also been a great deal of work tying logical concepts such as universal freeness (see section 3) with algebraic concepts such as free discrimination, commutative transitivity, equationally Noetherian property, theory of quasi-varieties, etc. This has led to further work in this area, including the development of discriminating and squarelike groups. Although this material has had a large impact on group theory it is not as well known among group theorists as it should be. The purpose of this paper is to introduce and survey this material in a format accessible to group theorists and to be as self-contained as possible. In sections 2 and 3 we describe the necessary material from model theory and ﬁrst-order logic and then introduce the Tarski problems and some of the earlier results on them; section 4 considers the theory of residually free groups; section 5 a brief description of algebraic geometry over groups and some applications; section 6 describes the techniques going into the solution of the Tarski problem and in section 7 we introduce discriminating, codiscriminating and squarelike groups.

2

First-order languages and model theory

The elementary theory of groups is tied to ﬁrst-order logic and to model theory. In this section we introduce the necessary material from these areas. We start with a ﬁrst-order language appropriate for group theory. This language which we denote L0 is the ﬁrst-order language with equality containing a binary operation symbol

ELEMENTARY THEORY OF GROUPS

199

. a unary operation symbol −1 and a constant symbol 1. A universal sentence of L0 is one of the form ∀x{φ(x)} where x is a tuple of distinct variables, φ(x) is a formula of L0 containing no quantiﬁers and containing at most the variables of x. Similarly an existential sentence is one of the form ∃x{φ(x)} where x and φ(x) are as above. A universal -existential sentence is one of the form ∀x∃y{φ(x, y)}. Similarly deﬁned is an existential-universal sentence. It is known that every sentence of L0 is logically equivalent to one of the form Q1 x1 ...Qn xn φ(x) where x = (x1 , ..., xn ) is a tuple of distinct variables, each Qi for i = 1, ..., n is a quantiﬁer, either ∀ or ∃, and φ(x) is a formula of L0 containing no quantiﬁers and containing free at most the variables x1 , ..., xn . Further vacuous quantiﬁcations are permitted. Finally a positive sentence is one logically equivalent to a sentence constructed using (at most) the connectives ∨, ∧, ∀, ∃. If G is a group then the universal theory of G consists of the set of all universal sentences of L0 true in G. Since any universal sentence is equivalent to the negation of an existential sentence it follows that two groups have the same universal theory if and only if they have the same existential theory. The set of all sentences of L0 true in G is called the ﬁrst-order theory or the elementary theory of G. We denote this by T h(G). We note that being ﬁrst-order or elementary means that in the intended interpretation of any formula or sentence all of the variables (free or bound) are assumed to take on as values only individual group elements - never, for example, subsets of nor functions on the group in which they are interpreted. Deﬁnition 2.1 Two groups G and H are elementarily equivalent (symbolically G ≡ H) if they have the same ﬁrst-order theory, that is T h(G) = T h(H). Group monomorphisms which preserve ﬁrst-order formulas are called elementary embeddings. Formally: Deﬁnition 2.2 If H and G are groups and f : H → G is a monomorphism then f is an elementary embedding provided whenever φ(x0 , ..., xn ) is a formula of L0 containing free at most the distinct variables x0 , ..., xn and (h0 , ..., hn ) ∈ H n+1 then φ(h0 , , ..., hn ) is true in H if and only if φ(f (h0 ), , ..., f (hn )) is true in G. If H is a subgroup of G and the inclusion map i : H → G is an elementary embedding then we say that G is an elementary extension of H. We note that the existence of an elementary embedding f : H → G is a suﬃcient condition for H and G to be elementarily equivalent. Further an isomorphism from G onto H is an elementary embedding. Another example of an elementary embedding is given by the following: Example 2.3 Let G be a group and let I be a nonempty set. Let δ : GI → G be the diagonal map, i.e., δ(g)(i) = g for all i ∈ I. Let D be an ultraﬁlter on I. Then the map d from G into the ultrapower GI /D given by g −→ [δ(g)]D is an elementary embedding. We note that two groups can be elementarily equivalent without one being an elementary extension of the other.

200

FINE, GAGLIONE, MYASNIKOV, ROSENBERGER, SPELLMAN

Example 2.4 Let F be a free group of countably inﬁnite rank with basis A = {an : n < ω}. Let F be the commutator subgroup of F . Then F and F are elementarily equivalent since they are isomorphic. None the less F is not an elementary extension of F . For example, the formula ∃x ([[a0 , , a1 ], x] = [[a0 , a1 ], a2 ]) is true in F but false in F . Here the commutator [x, y] = x−1 y −1 xy. A suﬃcient condition to prove that an inclusion map is an elementary embedding is provided by the next theorem. Theorem 2.5 Suppose H0 is a subgroup of H and suppose that to every ﬁnite subset {a1 , ..., an } of H0 and every element b ∈ H there exists an automorphism σ of H ﬁxing a1 , ..., an and mapping b into H0 . Then the inclusion map from H0 into H is an elementary embedding. We will see this used as a criteria in showing that two inﬁnite rank free groups are elementarily equivalent. In general, given two groups G and H, it is diﬃcult to determine whether or not G and H are elementarily equivalent. Sometimes the following general criterion, which is due to Keisler (see, [CK] ) is useful: two groups are elementary equivalent if and only if their ultrapowers (with respect to some non-principle ultraﬁlter) are isomorphic. Another general test for elementary equivalence is based on games (increasing chains of sets of ﬁnite partial isomorphisms) is due to Ehrenfeucht [Eh]. Szmielew [S] has completely characterized elementary equivalence of abelian groups. Recall that a group G has ﬁnite exponent if there is a positive integer n such that xn = 1 for all x ∈ G and G has inﬁnite exponent otherwise. Szmielew distinguishes between two types of linear independence in an abelian group A (which we shall write additively). If m is a positive integer and (ai )i∈I is a sequence of elements of A containing only ﬁnitely many nonzero terms, then (ai )i∈I is linearly independent modulo m provided that ni ai = 0 ⇒ ni ≡ 0( mod m) i∈I

for all i ∈ I. The sequence (ai )i∈I is linearly independent modulo m in the stronger sense provided that ni ai ∈ mA ⇒ ni ≡ 0( mod m) i∈I

for all i ∈ I. Szmielew then deﬁnes, for each prime p and each positive integer k, three quantities ρ(i) [p, k](A), i = 1, 2, 3, each of which is either a nonnegative integer or the symbol ∞, as follows: (1) ρ(1) [p, k](A) is the maximal number (if it exists) of elements of order pk and linearly independent modulo pk .

ELEMENTARY THEORY OF GROUPS

201

(2) ρ(2) [p, k](A) is the maximal number (if it exists) of elements linearly independent modulo pk in the stronger sense. (3) ρ(3) [p, k](A) is the maximal number (if it exists) of elements of order pk and linearly independent modulo pk in the stronger sense. Theorem 2.6 (Szmielew [S]) Let A and B be abelian groups. Then A and B are elementarily equivalent if and only if the following two conditions are satisﬁed 1. Either A and B both have ﬁnite exponent or they both have inﬁnite exponent. 2. For all primes p and positive integers k, one has ρ(i) [p, k](A) = ρ(i) [p, k](B) for i = 1, 2, 3. It is easy to see that the elementary theory of a given ﬁnitely generated abelian group is decidable - it follows readily from decidability of the elementary theory of Presburger arithmetic (non-negative integers with addition) (see [S]). A general description of abelian groups with decidable elementary theory is given also by Szmielew [S]: Theorem 2.7 (Szmielew [S]) 1. The elementary theory of the class of all abelian groups is decidable. 2. The elementary theory of an abelian group A is decidable if and only if the invariants ρ(i) [p, k](A) are computable (i.e., the functions (p, k) → ρ(i) [p, k](A) are computable for i = 1, 2, 3). A. I. Malcev solved the problems of elementary equivalence and decidability of elementary theories of the classical linaer groups [Mal]. It turns out that in this case everything depends on the ﬁeld of coeﬃcients. For example, for n, m ≥ 2 GL(n, K) ≡ GL(m, F ) if and only if k = m and K ≡ F . Similarly, T h(GL(n, F )) is decidable if and only if T h(F ) is decidable. The problem of decidability of ﬁnitely generated solvable groups was solved in three steps: by Y. Ershov (nilpotent case) [Er], N.Romanovskii (polycyclic case) [R], and by G.Noskov (general case). The net result says that a virtually abelian group G has decidable elementary theory if and only if it is virtually solvable. All these proofs reduce the problem of decidability of T h(G) to the corresponding problem for the ring of integers, which is, of course, undecidable. The problem of elementary equivalence of ﬁnitely generated nilpotent groups and unipotent groups (as well as the problem of decidability of their elementary theories) was solved in a series of papers by A. Myasnikov [M1], A. Myasnikov and V. Remeslennikov [MR1], [MR2], [MR3] and F. Oger [O]. The question of elementary equivalence for ﬁnitely generated solvable groups remains open. Before moving on to discuss ideas from model theory we mention two other results concerning elementary equivalence. Theorem 2.8 (Elementary Chain Theorem) If G0 → G1 → · · · → Gζ → . . .

202

FINE, GAGLIONE, MYASNIKOV, ROSENBERGER, SPELLMAN

is a chain of elementary embeddings then the direct union G is an elementary extension of each group in the chain. Theorem 2.9 (Downward Lowenheim-Skolem Theorem for Groups) If S is an inﬁnite subset of a group H then there is a subgroup H0 of H containing S such that |H0 | = |S| and the inclusion map H0 → H is an elementary embedding. We now introduce some basic ideas from model theory. Deﬁnition 2.10 If Φ is a consistent set of sentences of L0 , then the class of all groups G satisfying every sentence ϕ in Φ is the model class of Φ which we denote by M(Φ). We note that any model class is nonempty and is closed under isomorphism. Further the model class operator M reverses inclusions. That is, if Φ and Ψ are consistent sets of sentences of L and Φ ⊆ Ψ, then M(Ψ) ⊆ M(Φ). A class of groups which is the model class of some set of ﬁrst-order sentences is said to be axiomatic. Formally: Deﬁnition 2.11 If X is a nonempty class of groups closed under isomorphism, then X is axiomatic provided that there is at least one set Φ of sentences of L0 such that X = M(Φ). Important to model theory, relative to groups, are questions of completeness and decidability. Given a non-empty class of groups X closed under isomorphism then we say its ﬁrst-order theory is complete if given a sentence φ of L0 then either φ is true in every group in X or φ is false in every group in X . The ﬁrstorder theory of X is decidable if there exists a recursive algorithm which, given a sentence φ of L0 decides whether or not φ is true in every group in X . We will see these types of questions relative to the class of nonabelian free groups when we discuss the Tarski problems. The model theory of groups is also closely tied to the theory of group varieties. Recall that if F is a free group on x1 , ..., xn and w = w(x1 , ..., xn ) is a word in the generators then we say that the group G satisﬁes the law w(x1 , ..., xn ) = 1 if the substitution of any elements g1 , ..., gn ∈ G for x1 , ..., xn gives the identity in G. In the logical language we introduced this says that G satisﬁes the universal sentence ∀x1 , .., xn (w(x1 , ..., xn ) = 1). A group variety is the class of all groups satsifying each one of a given set of laws. Again in the language of model theory this can be described as the model class of the given set of laws. Group varieties can be described purely group theoretically as follows: Theorem 2.12 A class of groups X is a group variety if and only it is closed under subgroups, factor groups and unrestricted direct products.

ELEMENTARY THEORY OF GROUPS

203

If G is a group there is a minimal variety containing G. We call this the variety generated by G and denote it var(G). Similarly if Y is any class of groups there is a least variety var(Y) containing Y. Certain extensions of the varietal concept also play a role in the elementary theory of groups. Let X be a nonempty class of groups closed under isomorphism. Then X is a prevariety if X is closed under subgroups and X is closed under arbitrary direct products. Observe that since the trivial group 1 is a subgroup of any group G, every prevariety X must contain at least 1. Note also that the intersection of any family of prevarieties is again a prevariety; so, if Y is any class of groups there is a least prevariety pvar(Y) containing Y. This is the prevariety generated by Y. In the case that Y = {G} is a singleton, we write pvar(G) for the prevariety generated by G. When we discuss discrimination and separation properties in the ﬁnal section we will see another characterization of prevarieties. Finally, a universal sentence of the form (ui (x) = 1) → (w(x) = 1) ∀x i

is called a quasilaw or quasi-identity. Note that every identity ∀x(w(x) = 1) is equivalent to a quasilaw ∀x, y (y · y −1 = 1) → (w(x) = 1) . A quasivariety is the model class of a set of quasilaws. The quasivariety qvar(X ) generated by X is the model class of those quasilaws ϕ true in every group in X . If X is a nonempty class of groups then the universal closure ucl(X ) of X is the model class of the set of all universal sentences ϕ true in every group G in X .If X = {G} is a singleton, then we write ucl(G) for the universal closure of G and qvar(G) for the quasivariety generated by G. It is clear that both ucl(X ) and qvar(X ) are axiomatic. Moreover ucl(X ) is the least universally axiomatizable class containing X and qvar(X ) is the least quasivariety containing X . Further it is straightforward to verify that every quasivariety contains the trivial group 1, is closed under subgroups and is closed under direct products. A result of Mal’cev [M] shows that a prevariety is a quasivariety if and only if it is axiomatic. That is, every axiomatic prevariety is a quasivariety. However not every prevariety need be axiomatic and hence need not be a quasivariety. The following example shows this. Example 2.13 Call an abelian group reduced provided that it contains no nontrivial divisible subgroup. It is straightforward to verify that the class of all reduced abelian groups is a prevariety. One can produce an ultrapower (see [CK] ) of Z which contains a copy of Q. It follows that pvar(Z) is not axiomatic; hence, pvar(Z) is not a quasivariety.

3

The Tarksi problems

From a group theoretical point of view, interest in elementary theory has been motivated in large part by the Tarski problems or Tarski conjectures formulated

204

FINE, GAGLIONE, MYASNIKOV, ROSENBERGER, SPELLMAN

by A. Tarski around 1945. In a 1988 paper surveying combinatorial group theory [L], Roger Lyndon called the Tarski problems, which he described as folklore, among the outstanding open problems ( at that time) in the ﬁeld. Basically Tarksi conjectured that all nonabelian free groups have exactly the same elementary theory. We’ll state this in a more formal manner below. The genesis of this conjecture seems to be the following straightforward observations on free groups which show immediately that all nonabelian free groups have the same universal theory. Let α, β be ordinal numbers with α < β. Let Fα be the free group of rank α, and Fβ the free group of rank β. Then there is a natural embedding of Fα into Fβ as a proper free factor. Furthermore if ω is the ﬁrst limit ordinal then there are monomorphisms embedding Fω into F2 , the free group of rank 2, and hence into Fn for all 2 ≤ n < ω. However Fω cannot be a free factor in Fn for any ﬁnite n. It follows that if F and G are any two countable nonabelian free groups, then each is embeddable in the other. Hence the elementary theory of any countable nonabelian free group must strongly resemble the elementary theory of any other countable nonabelian free group. In particular if H is embedded into G then the universal theory of G is embedded in the universal theory of H, that is T h∀ (G) ⊂ T h∀ (H). Since any two nonabelian free groups are embeddable in each other this argument together with the Lowenheim-Skolem Theorem shows that any two nonabelian free groups have exactly the same universal theory. Theorem 3.1 Any two nonabelian free groups have exactly the same universal theory and hence exactly the same existential theory. On the other hand the universal theory of free groups is not complete in that a group not elementarily equivalent to any nonabelian free group can have the same universal theory as the class of nonabelian free groups. We call a group universally free if it has the same universal theory as the class of nonabelian free groups. The above statement on completeness is that there exist nonfree, universally free groups which can be distinguished from free groups by ﬁrst-order properties. In the next section we discuss this further and give a very beautiful group theoretical description of universally free groups. Observations of this type were the basis for the Tarski conjectures which we now make precise. Tarski Conjecture 1 Any two nonabelian free groups are elementarily equivalent. That is any two nonabelian free groups satisfy exactly the same ﬁrst-order theory. Tarski Conjecture 2 If the nonabelian free group H is a free factor in the free group G then the inclusion map H → G is an elementary embedding. Clearly the second conjecture is stronger than the ﬁrst and implies the ﬁrst. If true then the theory of the nonabelian free groups would be complete, that is given a sentence φ of L0 then either φ is true in every nonabelian free group or φ is false in every nonabelian free group. In addition to the completeness of the theory of the nonabelian free groups the question of its decidability also arises. Recall that this means the question of

ELEMENTARY THEORY OF GROUPS

205

whether there exists a recursive algorithm which, given a sentence φ of L0 decides whether or not φ is true in every nonabelian free group. Tarski further conjectured that the theory of the nonabelian free groups is decidable. Tarski Conjecture 3 The elementary theory of the nonabelian free groups is decidable. Kharlampovich and Myasnikov [Kh-M ] have recently announced that all of the above Tarski conjectures are indeed true. Their proof grew out of work on equations over free groups, residually free groups and on the algebraic geometry of groups. We will outline the method of proof in section 6. A separate independent proof for Tarski conjectures 1 and 2 has been announced by Sela [S]). Theorem 3.2 If the nonabelian free group H is a free factor in the free group G then the inclusion map H → G is an elementary embedding. Hence any two nonabelian free groups are elementarily equivalent. Moreover the theory of the nonabelian free groups is decidable. In this section we review some of the inital work on the Tarski questions. The ﬁrst progress was due to Vaught who showed that the Tarski conjectures 1,2 are true if G and H are both free groups of inﬁnite rank. Theorem 3.3 If the inﬁnite rank free group H is a free factor in the free group G then the inclusion map is an elementary embedding. In particular if G and H are free groups of inﬁnite rank then T h(G) = T h(H). The basic idea in Vaught’s proof is to use the criteria for elementary embeddings given in Theorem 2.5. Recall that this says that if H0 is a subgroup of H and that to every ﬁnite subset {a1 , ..., an } of H0 and every element b ∈ H there exists an automorphism σ of H ﬁxing a1 , ..., an and mapping b into H0 , then the inclusion map from H0 into H is an elementary embedding. Applying this to free groups of inﬁnite rank suppose that F is free on an inﬁnite subset S and that G is free on an inﬁnite subset S0 of S. Then permutations of S will induce enough automorphisms to guarantee that the inclusion map of G into F is an elementary embedding. The complete details of Vaught’s proof can be found in the book by Gratzer [Gr Chapter 6]. We note that Vaught’s proof extends to inﬁnite rank free algebras in any variety of algebras of any type. The elementary chain theorem (Theorem 2.8) gives that the direct union of elementary embeddings is elementary. Therefore Vaught’s proof that the Tarski conjectures 1,2 are true for free groups of inﬁnite rank reduced the conjectures to free groups of ﬁnite rank. The next signiﬁcant progress was due to Merzljakov. Recall that a positive sentence is a ﬁrst-order sentence which is logically equivalent to a sentence constructed using (at most) the connectives ∨, ∧, ∀, ∃. The positive theory of a group G consists of all the positive sentences true in G. Merzljakov [Me] showed that the nonabelian free groups have the same positive theory.

206

FINE, GAGLIONE, MYASNIKOV, ROSENBERGER, SPELLMAN

Theorem 3.4 Two nonabelian free groups have the same positive theory. Since positive sentences are preserved by homomorphic images and since every group is a homomorphic image of a free group this theorem has the following curious corollary. If the positive sentence φ is true in some nonabelian free group and hence in all nonabelian free groups then it is true in any group whatsoever. Further work following Merzljakov centered on restricted theories of free groups. We have seen that any two nonabelian free groups satisfy the same universal theory. Sacerdote [Sa] proved that this could be extended to universal-existential sentences (see section 2). Theorem 3.5 Any two nonabelian free groups satisfy the same universal-existential theory, that is they satisfy exactly the same universal-existential sentences. In the next section we give further results about universal and existential freeness. Advances in a diﬀerent direction were given by Makanin and Razborov. Makanin [Mak1] proved that there exists an algorithm to determine, given a ﬁnite system of equations in a free group, whether the system possesses at least one solution. In other words, the Diophantine problem is decidable for free groups (compare this to undecidability of the Diophantine problems for integers). Razborov [Ra] working with the Makanin algorithm determined an algorithm to eﬀectively describe the solution sets of a ﬁnite system of equations in a free group. Kharlampovich and Myasnikov further developed the Makanin-Razborov method. Their technique allows one to transform arbitrary ﬁnite systems of equations in free groups to some ”canonical forms” and describe precisely the irreducible components of algebraic sets in free groups. These canonical forms consist of ﬁnitely many quadratic equations in a triangular form (see Deﬁnition 6.11 and Theorem 6.2). Observe that quadratic equations over free groups (Commerford and Edmund [CE]), and hyperbolic groups (see Grigorchuk and Kurchanov [GK]) are well-studied, and there are good methods for solving them. The following result is a corollary of the decidability of the Diophantine problem Theorem 3.6 (Makanin [Mak2]) 1. The existential (and hence the universal) theory of a free group is decidable. 2. The positive theory of a free group is decidable.

4

Residually free and universally free groups

In this section we consider residually and fully residually free groups and show a beautiful tie to the class of universally free groups. Recall that a group G is residually free if for each non-trivial g ∈ G there is a free group Fg and an epimorphism hg : G → Fg such that hg (g) = 1. Equivalently for each g ∈ G / Ng . A group G there is a normal subgroup Ng such that G/Ng is free and g ∈ is n-residually free for a natural number n provided to every ordered n-tuple (g1 , . . . , gn ) ∈ (G \ {1})n of non-identity elements of G there is a free group F and an epimorphism h : G → F such that h(gi ) = 1 for all i = 1, ..., n. G is fully

ELEMENTARY THEORY OF GROUPS

207

residually free or ω-residually free provided it is n-residually free for every n ∈ N. Hence G is fully residually free provided to every ﬁnite set S ⊂ G\{1} of non-trivial elements of G there is a free group FS and an epimorphism hS : G → FS such that hS (g) = 1 for all g ∈ S. Clearly fully residually free implies residually free. To continue the study and to show that these are not equivalent we need to introduce the following concept. Deﬁnition 4.1 A group G is commutative transitive , abbreviated CT, provided the relation of commutativity is transitive on the non-identity elements of G. This property holds in all free groups as well as in all Fuchsian groups and is preserved under certain amalgam operations. Further discussions of commutative transitivity can be found in [FGMRS] and [W]. The following immediate result due to Harrison [H] gives equivalent formulations of the CT property. Lemma 4.2 Let G be a group. The following three statements are pairwise equivalent. (i) G is commutative transitive. (ii) The centralizer of every nontrivial element in G is abelian. (iii) Every pair of distinct maximal abelian subgroups in G has trivial intersection. Lemma 4.3 Any fully residually free group G is CT. Commutative transitivity is given by the universal sentence ∀x, y, z((y = 1) ∧ (xy = yx) ∧ (yz = zy)) → (xz = zx)). This universal sentence holds in any free group and hence in any universally free group. Therefore universally free groups are also CT. Lemma 4.4 Any universally free group G is CT. Now using the CT property one can show that the class of fully residually free groups is a proper subclass of the class of residually free groups. Indeed, if F is a free nonabelian group then F ×F is residually free, but not commutative-transitive, hence not fully residually free. Lemma 4.5 The class of fully residually free groups is a proper subclass of the class of residually free groups. The type of argument in Lemma 4.5 using commutative transitivity is extremely powerful and used often in the study of fully residually free groups. Before giving a beautiful theorem tying together fully residually free and universally free groups we make some observations whose proofs are immediate. (1) Residually free groups are torsion free. (2) If G is residually free and g, h ∈ G then < g, h > is either free or abelian.

208

FINE, GAGLIONE, MYASNIKOV, ROSENBERGER, SPELLMAN

(3) A subdirect product and in particular a direct product of residually free groups is residually free. The situation for free products of residually free groups is more complicated. First of all the free product of residually free groups need not be residually free. Consider A =< a0 , a1 , a2 ; [a0 , a1 ] = [a0 , a2 ] = 1 > and B =< b1 , b2 ; > . The group A is the direct product of a free group of rank 1 and a free group of rank 2 and hence is residually free while B is free and so residually free. Let G = A 2 B be their free product and consider the element w = [[[b1 , a0 ], [b2 , a0 ]], [a2 , a1 ]]. If G were residually free there would be a free group Fw and a map φw : G → Fw which did not annihilate w. From φw (w) = 1 we can easily deduce that φw (a0 ) and [φw (a2 ), φw (a1 )] must be nontrivial in Fw . However since [a0 , a1 ] = [a0 , a2 ] = 1 it must follow that φw (a0 ) must commute with both of φw (a1 ) and φw (a2 ). These must then lie in the cyclic subgroup of Fw containing the nontrivial element φw (a0 ). This forces φw (a1 ) and φw (a2 ) to commute contrary to [φw (a2 ), φw (a1 )] = 1. Therefore G cannot be residually free. However if both factors are fully residually free then the free product is residually free. Suppose H and K are fully residually free groups and w = h1 k1 h2 k2 ...hn kn is a nontrivial element of G = H 2 K where hi , ki ; i = 1..., n are nontrivial elements of H and K respectively. Then there exist maps φH : H → FH and φK : K → FK with FH , FK free which do not annihilate any of the syllables in w. This can then be extended to a map φw : G → FH 2 FK which does not annihilate w, showing that G is residually free. Presently we shall show that the class of fully residually free groups is actually closed under taking free products so that G is actually fully residually free. The complete solution to the free product question was given by B. Baumslag [Ba 2] who showed that Theorem 4.6 Let H and K be non-trivial residually free groups. Then their free product H 2 K is residually free if and only if H and K are each fully residually free. In this case the free product is also fully residually free. The question of when an amalgamated free product of fully residually free groups is again residually free is still an open question (see section 8). We have observed that being fully residually free easily implies commutative transitivity. B.Baumslag [Ba 1] proved a converse - commutative transitivity together with residually free implies fully residually free. Gaglione and Spellman [GS1] and independently Remesslennikov [Re] then were able to show that these conditions in the presence of residual freeness are equivalent to universal freeness in the nonabelian case. Thus there is the following rather remarkable theorem tying together the logical condition of universal freeness to the group theoretic conditions of full residualfreeness and commutative transitivity. Theorem 4.7 [Ba1], [GS1], [Re] Let G be nonabelian and residually free. Then the following are equivalent:

ELEMENTARY THEORY OF GROUPS

209

(1) G is commutative transitive. (2) G is fully residually free. (3) G is universally free. Using the CT property Remesslennikov [Re] gives the following corollary. Corollary 4.8 A group G is fully residually free if and only if it is 2-residually free. There have been various extensions of this theorem. Gaglione and Spellman [GS1] proved if G is ﬁnitely presented then universally free implies residually free, while Remeslennikov [Re ] showed ﬁnitely generated and universally free implies residually free. Theorem 4.9 [Re] Let G be a ﬁnitely generated universally free group. Then G is fully residually free. Remeslennikov’s proof uses an embedding of the fully residually free group G into an ultrapower of a rank 2 free group and then uses the corresponding ultrapower Z of the integers Z to embed G into SL ( Z). The proof in the ﬁnitely presented 2 case is much simpler and direct. Suppose that G is a ﬁnitely presented universally free group. Let G =< x1 , ..., xn ; R1 , ..., Rm > be a ﬁnite presentation for G where Ri = Ri (x1 , ..., xn ) are words in x1 , .., xn . Since a universally free group is CT it suﬃces to show that G is residually free. Suppose W is a nontrivial element of G. Then W is given by W = W (x1 , ..., xn ) a word in the given generators. Consider now the existential sentence ∃x1 , ..., xn ((

m

Ri (x1 , .., xn ) = 1) ∧ W (x1 , ..., xn ) = 1)).

i=1

This existential sentence is clearly true in G. Since G is universally free it is also existentially free so this existential sentence must be true in all nonabelian free groups. Therefore in any nonabelian free group F there exists elements a1 , ..., an such that Ri (a1 , ..., an ) = 1 for i = 1, ..., n; and W (a1 , ..., an ) = 1. The map then from G into F given by xi → ai , i = 1, ..., n deﬁnes a homomorphism where the image of W is nontrivial. Hence G is residually free. Thus we have the following: Corollary 4.10 A ﬁnitely generated nonabelian group G is universally free if and only if it is fully residually free. Chiswell [Ch ] as a corollary then gives this characterization of universally free groups. Theorem 4.11 (Ch) A group G is universally free if and only if it is nonabelian and locally fully residually free.

210

FINE, GAGLIONE, MYASNIKOV, ROSENBERGER, SPELLMAN

Building examples of fully residually free groups and their classiﬁcation uses the following construction. Deﬁnition 4.12 Let G be a CT group, let u ∈ G \ {1} and let M = ZG (u) where ZG (u) is the centralizer of u in G. Suppose A is an abelian group. Then the group H =< G, A; rel (G), rel A, [A, z] = 1∀z ∈ M > is a centralizer extension of G by A. If A =< t > is cyclic then H = G(u, t) is the HNN extension G(u, t) =< G, t; rel (G), t−1 zt = z, for all z ∈ M > and is called the free rank one extension of the centralizer M of u in G . Theorem 4.13 ([BMR3]) Let G be a fully residually free group and A an abelian fully residually free group. Then a centralizer extension of G by A is again fully residually free. The proof of this result which is fundamental in all further considerations of fully residually free groups depends on the fact that the result can be reduced to free rank one extensions of centralizers and then on the following ”big powers” argument. It is not hard to see that in a free group F if b0 tn1 b1 ...tnk bk = 1 for inﬁnitely many values of n1 , inﬁnitely many values of n2 ... inﬁnitely many values of nk then t must commute with at least one of b0 , ..., bk . Hence the family of homomorphisms φk : F (u, t) → F from the rank one extension of the centralizer CF (u) into F , deﬁned for every positive k by φ(t) = uk and φk |F = id, is a discriminating family, as required. G.Baumslag [B1] used this type of argument to show that the orientable surface groups Sg with g ≥ 2 are all residually free. This answered a question posed by Magnus. Recall that for g ≥ 2 the group Sg which is the fundamental group of an orientable surface of genus g has the presentation Sg =< a1 , b1 , ..., ag , bg ; [a1 , b1 ]...[ag , bg ] = 1 > . Baumslag observed that each Sg embeds in S2 and residual freeness is inherited by subgroups so it suﬃces to show that S2 is residually free. He actually showed more. If F is a nonabelian free group and u ∈ F is a nontrivial element which is neither primitive nor a proper power then the group K given by K =< F 2 F ; u = u > where F is an identical copy of F and u is the correspodning element to u in F , is residually free. A one-relator group of this form is called a Baumslag double. In our terminology he proceeded by embedding K in the free rank one extension of centralizers H =< F, t; t−1 ut = u >

ELEMENTARY THEORY OF GROUPS by

211

K =< F, t−1 F t > .

The group H is then residually free and hence K is residually free. Therefore every Baumslag double is residually free. The group S2 =< a1 , b1 , a2 , b2 ; [a1 , b1 ] = [a2 , b2 ] > is a Baumslag double answering the original question. The next step in the classiﬁcation of the ﬁnitely generated fully residually free groups was the embedding of this class into a Lyndon free exponential group. Let R be a commutative ring with identity 1 = 0. An R-group is a group admitting operators from R, G × R → G, (g, α) → g α subject to the conditions (i) g α g β = g α+β , (ii) (g α )β = g αβ , (iii) g 1 = g and (iv) (h−1 gh)α = h−1 g α h. for all g, h ∈ G, α, β ∈ R . The set of R-groups forms a variety and hence the rank r free R-group, (Fr )R , exists and is unique up to an R-isomorphism. Lyndon considered the case where R = Z[t1 , .., tn ] the polynomial ring in ﬁnitely many indeterminates over the inZ[t ,...,t ] tegers Z. He explicitly constructed the free group Fr 1 n . Myasnikov and Remesslennikov [MRe] simpliﬁed Lyndon’s original construction and gave the tie to residually free and universally free groups. In particular Myasnikov and Remesslennikov [MR5] constructed Lyndon’s group FrR as the union of a countable chain Fr = G0 ⊆ G1 ... ⊆ Gn ⊆ .... of subgroups. If Gn had already been constructed then Gn+1 is obtianed from Gn as a tree extension of centralizers. The construction process preserves at each step full residual freeness and the union of the chain above remains fully residually free. We note that in general residual freeness is not preserved in direct unions. For example the additive group Q of rationals is the direct union of a chain of inﬁnite cyclic subgroups. An embedding Fr → F2 will induce an embedding FrR → F2R . The fact that all known ﬁnitely generated fully residually free groups are embeddable in the Lyndon Z[t] group F2 led Myasnikov and Remesslennikov to conjecture that there are no other ﬁnitely generated examples. This conjecture was resolved aﬃrmatively by Kharlampovich and Myasnikov [KhM1], [KhM2] using the methods of algebraic geometry over groups. They also show that the ﬁnitely generated subgroups of Z[x] are ﬁnitely presented. We state it here and discuss it further in the next F2 section. Theorem 4.14 (1) Any ﬁnitely generated fully residually free group embeds in Z[x] the free exponential group F2 .

212

FINE, GAGLIONE, MYASNIKOV, ROSENBERGER, SPELLMAN Z[x]

(2) Every ﬁnitely generated subgroup of F2

is ﬁnitely presented.

This has the following consequence. A result of Karrass and Solitar (see [LS]) asserts that if G and H have the property that every ﬁnitely generated subgroup is ﬁnitely presented and U, V are isomorphic subgroups of G and H respectively such that every subgroup of these isomorphic subgroups is ﬁnitely generated then the generalized free product < G 2 H; U = V > also has the property that every ﬁnitely generated subgroup is ﬁnitely presented. It follows from this result and the Z[t] construction of F2 as an iterated extension of centralizers that every ﬁnitely genZ[t] erated subgroup of F2 must be ﬁnitely presented. This establishes that ﬁnitely generated fully residually free groups must be ﬁnitely presented a result done independently by Sela [Se]. Theorem 4.15 [KhM3], [Se] Any ﬁnitely generated fully residually free group must be ﬁnitely presented. Using techniques involving ultraproducts combined with Nielsen reduction techniques Fine, Gaglione, Myasnikov, Rosenberger and Spellman [FGMRS] gave the complete classiﬁcation of fully residually free groups of rank 3 or less. Theorem 4.16 Let G be a fully residually free group. (1) If rank(G) = 1, then G is inﬁnite cyclic. (2) If rank(G) = 2, then either G is free of rank 2 or free abelian of rank 2. (3) If rank(G) = 3, then either G is free of rank 3, free abelian of rank 3 or a free rank one extension of centralizers of a free group of rank 2. That is G has a one-relator presentation G =< x1 , x2 , x3 ; x−1 3 vx3 = v > where v = v(x1 , x2 ) is a nontrivial element of the free group on x1 , x2 which is not a proper power. In the course of proving the classiﬁcation the following startling result is also obtained. A group G is termed n-free if every subgroup with n or fewer generators must be a free group ( of course with rank ≤ n. Theorem 4.17 Every 2-free, residually free group is 3-free. The proof of the classiﬁcation result depends upon Theorem 4.17 and the following further construction. We deﬁne the class F as the smallest class of groups containing the inﬁnite cyclic groups and closed under the following four ”operators”: (1) Isomorphism, (2) ﬁnitely generated subgroups, (3) free products of ﬁnitely many factors and (4) free rank one extensions of centralizers.

ELEMENTARY THEORY OF GROUPS

213

The class F is then classiﬁed as the class of all ﬁnitely generated groups embeddable in (Fω )Z[ν] . Hence from the theorem of Kharlampovich and Myasnikov, F contains all the ﬁnitely generated fully residually free groups. Theorem 4.18 (1) The class F is precisely the class of all ﬁnitely generated groups embeddable in (Fω )Z[ν] . (2) Every ﬁnitely generated fully residually free group lies in F. Remeslennikov, jointly with Kharlampovich and Myasnikov (see [Kh-M 1,2]) have given the following more inductive characterization of the class of ﬁnitely generated fully residually free groups. This class is properly contained in the class of groups which start with free abelian groups of ﬁnite rank and are constructed by repeated iteration of the following four operations: (1) free products, (2) amalgamated free products with abelian amalgamated subgroups at least one of which is maximal abelian, (3) free extensions of centralizers and (4) separated HNN extensions with abelian associated subgroups at least one of which is maximal abelian. An HNN extension H =< G, t; t−1 At = B > is a separated HNN extension if g −1 Ag ∩ B = {1} for all g ∈ G. This construction allows for the following inductive characterization of ﬁnitely generated fully residually free groups and allows for inductive type proofs. A fully residually free group G is of level n if it can be constructed from an inﬁnite cyclic group by n iterations of the above operations and not n − 1 such iterations. We close this section by introducing one other concept that has played a role in the study of residually free and universally free groups. Deﬁnition 4.19 A group G is CSA or conjugately separated abelian provided every maximal abelian subgroup M of G is malnormal in G. Recall that a subgroup M of a group G is malnormal if g −1 M g ∩ M = {1} implies that g ∈ M . It is clear that every abelian group is CSA. Further in CT groups, maximal abelian subgroups correspond to centralizers of nontrivial elements. Lemma 4.20 Every CSA group is CT. Lemma 4.21 Suppose G is a nonabelian CSA group. Then G contains no nontrivial normal abelian subgroups. Proof Let M1 , M2 be maximal abelian subgroups of the CSA group G. Suppose −1 m0 ∈ M1 ∩ M2 . Let m1 ∈ M1 . Then m0 = m−1 1 m0 m1 ∈ m1 M2 m1 ∩ M2 . Since G is CSA and M2 is maximal abelian we conclude that m1 ∈ M2 . Therefore M1 ⊆ M2 and by maximality M1 = M2 . Therefore G is CT. ✷

214

FINE, GAGLIONE, MYASNIKOV, ROSENBERGER, SPELLMAN

Proof Suppose M is a nontrivial normal abelian subgroup of the CSA group G. Extend M to a maximal abelian subgroup N . Since G is nonabelian we may choose g∈ / N . Then since M is normal g −1 M g = M which implies that g −1 N g ∩ N = {1}. Hence g ∈ N a contradiction. Therefore M must be trivial. ✷ Free groups and torsion-free hyperbolic groups are CSA. Further CSA can be captured by the following universal sentence ∀x, y, z(((x = 1) ∧ (y = 1) ∧ (xy = yx) ∧ (xz 1 yz = z 1 yzx)) → (xz = zx)). It follows that every universally free group is CSA. Therefore Theorem 4.2 can be extended Theorem 4.2’ Let G be nonabelian and residually free. Then the following are equivalent: (1) G is commutative transitive. (2) G is fully residually free. (3) G is universally free. (4) G is CSA. Although within the context of residually free groups CSA is equivalent to CT in general CSA is a stronger condition as the next example shows. Example 4.22 Let A =< a; a2 = 1 >, B =< b; b2 = 1 > be two cyclic groups of order 2. Then their free product D = A 2 B, the inﬁnite dihedral group, is CT being the free product of two abelian groups. However the commutator subgroup of D which is < [a, b] > is cyclic. Therefore D has a nontrivial normal abelian subgroup and hence cannot be CSA.

5

Algebraic geometry over groups and applications

Again with an eye towards studying elementary and universal properties of groups G.Baumslag, Kharlampovich, Myasnikov, and Remesslennikov [BMR2], [MR4], [KM3] have extended the ideas of classical algebraic geometry to a group theoretical setting. First let us recall some ideas from the classical setting. Let k be an algebraically closed ﬁeld and let X = (x1 , .., xn ) be an ordered n-tuple of algebraically independent commuting variables over k. An aﬃne algebraic set is a subset A ⊂ k n consisting of the common zeros of a set S of polynomials in k[X]. Classical algebraic geometry originated with the study of such solution sets. Much (but not all) of the theory remains intact if we replace the ﬁeld k with a noetherian integral domain R. From the fact that R is an integral domain one deduces that for each integer n > 0 the aﬃne algebraic sets are the closed sets of a topology on Rn called the Zarsiki topology. Since R is noetherian it follows that the Zariski topology on Rn is a noetherian topology in the sense that every descending chain of closed sets A0 ⊃ A1 ⊃ · · · ⊃ Am ⊃ · · ·

ELEMENTARY THEORY OF GROUPS

215

must stabilize after ﬁnitely many steps. Since the topology is noetherian every closed set A is the union of ﬁnitely many irreducible closed sets. A closed set is irreducible provided it cannot be expressed as the union of proper closed sets. Moreover if in the expresssion A = A1 ∪ A2 ∪ ... ∪ Am of A as a union of irreducible closed sets we never have Ai ⊂ Aj for i = j, then the expression as this union is unique apart from the ordering. A1 , .., An are then called the irreducible components of A. Dual to the notion of an aﬃne algebraic set is that of a radical ideal of R[X]. The ideal I ⊂ R[X] is a radical ideal provided that there is at least one closed subset A ⊂ Rn such that I = IR (A) = {f ∈ R[X]; f (r) = 0, ∀r ∈ A}. Corresponding to each radical ideal I in R[X] one has the aﬃne R-algebra R[X]/I. If I = IR (A) then R[X]/I is called the coordinate ring of the aﬃne algebraic set A. It turns out that A will be irreducible if and only if IR (A) is a prime ideal in R[X]. If S is any subset of R[X] then its radical is RadR (S) = IR (VR (S)) where VR (S) are the common zeros in Rn of the polynomials in S. This is the least radical ideal of R[X] containing S. There is a contravariant equivalence of aﬃne algebraic sets over R and aﬃne R-algebras. G. Baumslag, Myasnikov and Remesslennikov [BMR1] and [MR1] have created an analogous theory over groups. Their creation is not merely abstraction for the sake of abstraction as is borne out by the fact that the theory has been applied to resolve heretofore diﬃcult problems. To begin let G be a group. A G-group in analogy with that of R-algebra is a group H together with a ﬁxed embedding φ : G → H. Thus a G-group is simply a group containing a distinguished copy of G. We note that the condition that φ is monic can be relaxed (see [BPP]) but for our purposes here we assume that it is monic. The notions of G-subgroup and Ghomomorphism are the obvious ones and are in complete analogy with the classical ﬁeld situation. A G-ideal is then the kernel of a G-homormorphism. A subgroup I of a G-group H is then a G-ideal if and only if I is a normal subgroup of H which intersects the distinguished copy of G trivially. Let F be the rank n free group with free basis {x1 , ..., xn }. The elements of F will serve as the variables in the algebraic geometry over groups. We denote by G[X] the free product G 2 F . This is a G-group in the obvious way and is the analogue of the polynomial ring R[X]. We wish to build a Zariski topology on H m where H is a G-group. In order to insure that each closed set is an aﬃne algebraic set we must deﬁne the analogue of a zero divisor and hence the analogue of an integral domain. Baumslag, Myasnikov and Remeslennikov do this in the following way.

216

FINE, GAGLIONE, MYASNIKOV, ROSENBERGER, SPELLMAN

Deﬁnition 5.1 Let H be a nontrivial G-group. An element h = 1 in H is a G-zero divisor if there exists an element k = 1 in H such that [h, kg ] = 1 for all g ∈ G. Here hg = ghg −1 . Equivalently this means that [hg1 , k g2 ] = 1 for all g1 , g2 ∈ G. H is a G- domain if H contains no G-zero divisors. A group G is a domain if it is a G-domain A G-ideal I of a G-group H is a prime ideal if H/I is a G-domain. This deﬁnition of domain is a special case of anticommutativity which has been applied to associative rings and Lie algebras (see [BPP]). Suppose X = {x1 , ..., xn } and S ⊂ G[X] and H is a G-group. Then an aﬃne algebraic set in H n is given by A = VH (S) = {h ∈ H n ; w(h) = 1∀w ∈ S}. If H is a G-domain then the aﬃne algebraic sets are the closed sets in the Zariski topology on H n . We need another condition on H to guarantee that this topology be noetherian. Deﬁnition 5.2 [BMR1] Let H be a G-group. Then H is G-equationally noetherian if for every integer n > 0 and every subset S ⊂ G[x1 , ..., xn ] there is a ﬁnite subset S0 ⊂ S such that VH (S) = VH (S0 ). G itself is equationally noetherian provided it is G-equationally noetherian. It was shown in [BMR1] that nonabelian free groups and more generally linear groups over noetherian integral domains are equationally noetherian. Theorem 5.3 [BMR1] Let H be a G-group. If H is G-equationally noetherian then the Zariski topology on H n is noetherian. If follows that if H is a G-equationally noetherian G-domain then every aﬃne algebraic subset of H n is uniquely the union of ﬁnitely many irreducible components. We henceforth assume that H is a G-equationally noetherian G-domain. An ideal I in G[X] is an H-radical ideal if I = IH (A) = {w ∈ G[x]; w(h) = 1∀h ∈ A} for some aﬃne algebraic set A = VH (S) ⊂ H n . The quotient group G[X]/IH (A) is then the coordinate group of the aﬃne algebraic set A. A will be irreducible if and only if IH (A) is a prime ideal. If S is any subset of G[X] then its H-radical is given by RadH (S) = IH (VH (S)). This is the least H-radical ideal of G[X] containing S. We denote by HR(S) the coordinate group of VH (S). There is a contravariant equivalence between the categories of aﬃne algebraic sets and coordinate groups. Applicability of algebraic geometry to fully residually free groups is suggested by the following result of Kharlampovich and Myasnikov [KhM1]

ELEMENTARY THEORY OF GROUPS

217

Lemma 5.4 Let G be an equationally noetherian CSA group. Then V (S) is irreducible if and only if GR(S) is discriminated by G-homomorphisms. For the Lemma a family Λ of homomorphisms λ : H → G is discriminating if to every ﬁnite nonempty set T ⊂ H \ {1} there is a λT ∈ Λ such that λT (t) = 1 for all t ∈ T . In many applications of the algebraic geometry and particularly in the solution of the Tarski problem (see the next section) interest centers on quadratic equations. Deﬁnition 5.5 An equation S = 1 in variables from X = {x1 , .., xn } is quadratic if every variable from X occurs in S no more than twice. A quadratic equation S(X) = 1 need not contain all the variables from X, it can be empty in some variable, linear in some variables or strictly quadratic on some subset of X. Let X1 , ..., Xm be disjoint tuples of variables. A system U (X1 , ..., Xm ) = 1 of equations with coeﬃcients from a free group F of the following form S1 (X1 , ..., Xm ) = 1 S2 (X2 , ..., Xm ) = 1 ... Sm (Xm ) = 1 is said to be triangular quasi-quadratic if for every i the equation Si (Xi , .., Xm ) = 1 is quadratic in the variables from Xi . Denote by Gi the coordinate group of the subsystem Si = 1, ..., Sm = 1 of the system U = 1. The system U = 1 is said to be non-degenerate (NTQ) if for each i the equation Si (Xi , .., Xm ) = 1 has a solution in Gi+1 . Kharlampovich and Myasnikov [KhM1] use these ideas of algebraic geometry to prove that every ﬁnitely generated fully residually free group embeds into Lyndon’s Z[x] group F2 . As remarked earlier it follows from this that ﬁnitely generated fully residually free groups must be ﬁnitely presented. The proof of the embedding result follows from a sequence of theorems. We mention two of these from the tail end of the sequence (see [KhM1]). Theorem 5.6 For every ﬁnite system S(X) = 1 over a free group F one can ﬁnd eﬀectively a ﬁnite family of nondegenerate triangular quasi-quadratic systems U1 , ..., Uk and word mappings ρi : VF (Ui ) → VF (S), i = 1, ..., k such that for every b ∈ VF (S) there exists an i and c ∈ VF (Ui ) for which b = ρ(c), that is VF (S) = ρ1 (VF (U1 )) ∪ ... ∪ ρk (VF (Uk )) and all sets ρi (VF (Ui )) are irreducible. Moreover every irreducible component of VF (S) can be obtained as the closure of ρi (VF (Ui )) in the Zariski topology.

218

FINE, GAGLIONE, MYASNIKOV, ROSENBERGER, SPELLMAN

Theorem 5.7 For a system S = 1 over a free group the set V (S) is irreducible if and only if FR(S) ⊂ FR(S1 ) for a nondegenerate triangular quasi-quadratic system S1 . In the next section we will see how some of these ideas pertain to the solution of the Tarksi problem. One can relax the condition on exponential groups that the system of exponents must be a ring. Instead one can take a torsion-free additive abelian group contianing a pure inﬁnite cyclic subgroup (to be identiﬁed with Z). As another pretty application of algebraic geometry over groups we have the following result due also to Kharlampovich and Myasnikov [KhM1]. Theorem 5.8 Let S(X) = 1 be a system of equations over a free group F . Then there exists a ﬁnite set of n-tuples of parametric words k

U = (u1 , ..., un ) ∈ (F Z )n such that the set of all their specializations U is a dense subset of the variety VF (S) in the Zariski topology. It follows from this theorem that solution sets of consistent systems of equations over free groups can almost be parametrized.

6

The positive solution to the Tarski problems

As we have mentioned earlier Kharlampovich and Myasnikov [KhM1], [KhM2], [KhM3], [KhM4] and independenlty Sela [Se] have announced positive solutions to the Tarski problems (see section 3). In this section we sketch the proofs given by O.Kharlampovich and A.Myasnikov. Speciﬁcally we sketch proofs of the following two theorems. Theorem 6.1 The free group F (a1 , . . . , an ) freely generated by a1 , . . . , an is an elementary subgroup of F (a1 , . . . , an , . . . , an+p ) for every n ≥ 2 and p ≥ 0 Theorem 6.2 The elementary theory T h(F ) of a free group F even allowing constants from F in the language is decidable. Let F = F (A) be a nonabelian free group with basis A. Denote by LA the language of group theory L0 together with all elements from A as new constants. In this section following [KhM1] we collect some results (old and new) on how to eﬀectively rewrite formulas over a nonabelian freely discriminated group G into more simple or more convenient normal forms. Some of these results hold for many groups beyond the class of freely discriminated ones (we refer to [KhM1] fora more detailed discussion). To begin we ﬁx a ﬁnite set of constants A and the corresponding group theory language LA . Further let a, b be two ﬁxed elements in A.

ELEMENTARY THEORY OF GROUPS

219

Deﬁnition 6.3 A group G satisﬁes Vaught’s conjecture if the following universal sentence holds in G: (V) ∀x∀y∀z(x2 y 2 z 2 = 1 → [x, y] = 1 & [x, z] = 1 & [y, z] = 1). R. Lyndon proved that every free group satisﬁes (V) (see [LS]). Denote by T the class of all groups G such that: 1) G is torsion-free, 2) G satisﬁes Vaught’s conjecture, 3) G is CSA and 4) G has two distinguished elements a, b with [a, b] = 1. It is easy to write down axioms for the class T in the language L{a,b} . Indeed, the following universal sentences describe the conditions 1)-4) above: (TF) xn = 1 → x = 1 (n = 2, 3, . . .), (V) ∀x∀y∀z(x2 y 2 z 2 = 1 → [x, y] = 1 & [x, z] = 1 & [y, z] = 1), (CT) ∀x∀y∀z(x = 1 & y = 1 & z = 1 & [x, y] = 1 & [x, z] = 1 → [y, z] = 1), (WCSA) ∀x∀y([x, xy ] = 1 → [x, y] = 1) and (NA) [a, b] = 1. Observe that (WCSA) is a weak form of CSA but together with (CT) they provide the CSA condition. Let GROU P S denote a set of axioms of group theory. Denote by AT the union of axioms (TF), (V), (CT), (WCSA), (NA) and GROU P S. Notice that the axiom (V) is equivalent modulo GROU P S to the following quasi- identity ∀x∀y∀z(x2 y 2 z 2 = 1 → [x, y] = 1). It follows that all axioms in AT , with exception of (CT) and (NA), are quasiidentities. The next result shows that one can eﬀectively encode ﬁnite conjunctions and ﬁnite disjunctions of inequalities into a single inequality modulo AT . Theorem 6.4 For any ﬁnite set of inequalities S1 (X) = 1, . . . , Sk (X) = 1 in LA one can eﬀectively ﬁnd an inequality R(X) = 1 and an inequality T (X) = 1 in LA such that k ( Si (X) = 1) ∼AT R(X) = 1 i=1

and (

k

Si (X) = 1) ∼AT T (X) = 1.

i=1

Corollary 6.5 For every quantiﬁer-free formula Φ(X) in the language LA one can eﬀectively ﬁnd a formula Ψ(X) =

n i=1

(Si (X) = 1

∧ Ti (X) = 1)

220

FINE, GAGLIONE, MYASNIKOV, ROSENBERGER, SPELLMAN

in LA which is equivalent to Φ(X) modulo AT . In particular, if G ∈ T then every quantiﬁer-free formula Φ(X) in LG is equivalent over G to a formula Ψ(X) as above. Below we discuss canonical forms of universal formulas in the language LA modulo the theory AT . We show that every universal formula in LA is equivalent modulo AT to a universal formula in the canonical radical form. This implies that if G ∈ T is generated by A then the universal theory of G in the language LA consists of the the axioms describing the diagram of G (multiplication table for G with all the equalities and inequalities between group words in A), the set of axioms AT , and a set of axioms AR which describes the radicals of ﬁnite systems over G. We say that a universal formula in LA is in canonical radical form (or is a radical formula) if it has the following form ΦS,T (X) = ∀Y (S(X, Y ) = 1 → T (Y ) = 1) for some S ∈ G[X ∪ Y ], T ∈ G[X]. For an arbitrary ﬁnite system S(X) = 1 with coeﬃcients from A denote by ˜ S(X) = 1 an equation which is equivalent over G to the system S(X) = 1 (such ˜ S(X) exists by Corollary 6.5. Then for the radical R(S) of the system S = 1 we have R(S) = {T ∈ G[X] | G |= ΦS,T ˜ }. It follows that the set of radical sentences | G |= ΦS,T AS = {ΦS,T ˜ ˜ } describes precisely the radical R(S) of the system S = 1 over G, hence the name. Lemma 6.6 Every universal formula in LA is equivalent modulo AT to a radical formula. The next result shows how to eliminate quantiﬁers from positive universal formulas over nonabelian freely discriminated groups. Lemma 6.7 Let G be a BP-group from T . For a given word U (X, Y ) ∈ G[X ∪ Y ] one can eﬀectively ﬁnd a word W (Y ) ∈ G[Y ] such that ∀X (U (X, Y ) = 1) ∼G

W (Y ) = 1.

We now describe normal forms of general formulas and positive formulas. We show that every positive formula is equivalent modulo AT to a formula which consists of an equation and a string of quantiﬁers in front of it; and for an arbitrary formula Φ either Φ or ¬Φ is equivalent modulo AT to a formula in a general radical form (it is a radical formula with a string of quantiﬁers in front of it). (see [KhM1], [KhM2]).

ELEMENTARY THEORY OF GROUPS

221

Lemma 6.8 Every positive formula Φ(X) in LA is equivalent modulo AT to a formula of the type Q1 X1 . . . Qk Xk (S(X, X1 , . . . , Xk ) = 1), where Qi ∈ {∃, ∀} (i = 1, . . . , k). Lemma 6.9 For any formula Φ(X) in the language LA one can eﬀectively ﬁnd a formula Ψ(X) in the language LA in the following form Ψ(X) = ∃X1 ∀Y1 . . . ∃Xk ∀Yk ∀Z(S(X, X1 , Y1 , . . . , Xk , Yk , Z) = 1 → T (Z) = 1), such that Φ(X) or its negation ¬Φ(X) (and we can check eﬀectively which one of them) is equivalent to Ψ(X) modulo AT . We now consider lifting equations into generic points. Deﬁnition 6.10 Let S(X) = 1 be a system of equations over a group G which has a solution in G. We say that a system of equations T (X, Y ) = 1 is compatible with S(X) = 1 over G if for every solution U of S(X) = 1 in G the equation T (U, Y ) = 1 also has a solution in G, i.e., the algebraic set VG (S) is a projection of the algebraic set VG (S ∪ T ). Let S(X) = 1 be a system of equations over G and suppose VG (S) = ∅. The canonical embedding X → G[X] induces the canonical map µ : X → GR(S) . We are ready to formulate the main deﬁnition. Deﬁnition 6.11 Let S(X) = 1 be a system of equations over G with VG (S) = ∅ and let µ : X → GR(S) be the canonical map. Let a system T (X, Y ) = 1 be compatible with S(X) = 1 over G. We say that T (X, Y ) = 1 admits a lift to a generic point of S = 1 over G (or, shortly, S-lift over G) if T (X µ , Y ) = 1 has a solution in GR(S) (here Y are variables and X µ are constants from GR(S) ). The next result characterizes lifts in terms of the coordinate groups of the corresponding equations. Proposition 6.12 Let S(X) = 1 be an equation over G which has a solution in G. Then for an arbitrary equation T (X, Y ) = 1 over G the following conditions are equivalent: 1. T (X, Y ) = 1 is compatible with S(X) = 1 and T (X, Y ) = 1 admits S-lift over G, 2. GR(S) is a retract of GR(S,T ) , i.e., GR(S) is a subgroup of GR(S,T ) and there exists a GR(S) -homomorphism GR(S,T ) → GR(S) . In the notation of Proposition 6.1 every solution U of an equation T (X µ , Y ) = 1 in GR(S) gives rise to a retraction, i.e., GR(S) -homomorphism, φU : GR(S,T ) → GR(S) , and vice versa. This allows one to consider solutions of T (X µ , Y ) = 1 in GR(S) as retractions GR(S,T ) → GR(S) .

222

FINE, GAGLIONE, MYASNIKOV, ROSENBERGER, SPELLMAN

Deﬁnition 6.13 Let T (X, Y ) = 1 be compatible with S(X) = 1 over G. We say that T (X, Y ) = 1 admits a complete S-lift (or a separating S-lift) if the set of all solutions of T (X µ , Y ) = 1 in GR(S) (viewed as retractions) separates the group GR(S,T ) . The next result characterizes complete lifts in terms of the coordinate groups of the corresponding equations. Proposition 6.14 Let S(X) = 1 be an equation over G which has a solution in G. Then for an arbitrary equation T (X, Y ) = 1 over G the following conditions are equivalent: 1. T (X, Y ) = 1 is compatible with S(X) = 1 over G and it admits a complete S-lift, 2. GR(S,T ) !GR(S) (GR(S) )R(T (X µ ,Y )) . Let S ⊂ G[X]. Denote by var(S) the set of variables that occur in S. Deﬁnition 6.15 A set S ⊂ G[X] is called quadratic if every variable from var(S) occurs in S not more then twice. The set S is strictly quadratic if every letter from var(S) occurs in S exactly twice. A system S = 1 over G is quadratic (strictly quadratic), if the corresponding set S is quadratic (strictly quadratic). Deﬁnition 6.16 A standard quadratic equation over the group G is an equation of the one of the following forms (below d, ci are nontrivial elements from G): (st1) ni=1 [xi , yi ] = 1, n > 0;, −1 n, m ≥ 0, m + n ≥ 1, (st2) ni=1 [xi , yi ] m i=1 zi ci zi d = 1, n (st3) i=1 x2i = 1, n > 0 and −1 (st4) ni=1 x2i m n, m ≥ 0, n + m ≥ 1. i=1 zi ci zi d = 1, Lemma 6.17 Let S be a strictly quadratic word over G. Then there is a Gautomorphism f ∈ AutG (G[X]) such that f (S) is a standard quadratic word over G. The proof of this is in [CE]. Deﬁnition 6.18 Strictly quadratic words of the type [x, y], x2 , z −1 cz, where c ∈ G, are called atomic quadratic words or simply atoms. By Deﬁnition 6.6 a standard quadratic equation S = 1 over G has the form r1 r2 . . . rk d = 1, where ri are atoms, d ∈ G. This number k is called the atomic rank of this equation, we denote it by r(S). Deﬁnition 6.19 Let S = 1 be a standard quadratic equation written in the atomic form r1 r2 . . . rk d = 1 with k ≥ 2. A solution φ : GR(S) → G of S = 1 is called:

ELEMENTARY THEORY OF GROUPS

223

1. commutative if [φ(ri ), φ(ri+1 )] = 1 for all i = 1, . . . , k − 1, and noncommutative otherwise; 2. in a general position if [φ(ri ), φ(ri+1 )] = 1 for all i = 1, . . . , k − 1. Theorem 6.20 ([KhM1] ) In the following cases a standard quadratic equation S = 1 always has a solution in a general position: 1. S = 1 is of type (st1), n > 2, 2. S = 1 is of type (st2), n > 0, n + m > 1, 3. S = 1 is of type (st3), n > 3 and 4. S = 1 is of type (st4), n > 2. The following theorem describes the radical of a standard quadratic equation which has at least one solution in a freely discriminated group G. Theorem 6.21 ([KhM1]) Let G be a freely discriminated group and let S = 1 be a standard quadratic equation over G which has a solution in G. Then 1. If S = [x, y]d or S = [x1 , y1 ][x2 , y2 ] then Rad(S) = ncl(S); 2. If S = x2 d then Rad(S) = ncl(xb) where b2 = d; 3. If S = cz d then Rad(S) = ncl([zb−1 , c]) where d−1 = cb ; 4. If S = x21 x22 then Rad(S) = ncl([x1 , x2 ]); 5. If S = x21 x22 x23 then Rad(S) = ncl([x1 , x2 ], [x1 , x3 ], [x2 , x3 ]); 6. If r(S) ≥ 2 and S = 1 has a noncommutative solution then Rad(S) = ncl(S); 7. If S = 1 is of the type (st4) and all solutions of S = 1 are commutative, then Rad(S) is the normal closure of the following system: −1 {x1 . . . xn = s1 . . . sn , [xk , xl ] = 1, [a−1 i zi , xk ] = 1, [xk , C] = 1, [ai zi , C] = 1, −1 [a−1 i zi , aj zj ] = 1 (k, l = 1, . . . , n; i, j = 1, . . . , m)},

where xk → sk , zi → ai is a solution of S = 1 and the corresponding centralizer is C = CG (ca11 , . . . , camm , s1 , . . . , sn ) . The group GR(S) is an extension of the centralizer C. Deﬁnition 6.22 A standard quadratic equation S = 1 over F is called regular if either it is an equation of the type [x, y] = d (d = 1), or the equation [x1 , y1 ][x2 , y2 ] = 1, or it has a non-commutative solution and it is not an equation of the type c1 z1 cz22 = c1 c2 , x2 cz = a2 c, x21 x22 = a21 a22 . The following theorem is one of the main technical tools in the solution of the Tarski problems. Theorem 6.23 Let S(X, A) = 1 be a regular standard quadratic equation over F (A). Then every equation T (X, Y, A) = 1 compatible with S(X, A) = 1 admits a S-lift. Now we formulate the implicit function theorem over free groups in its simplest form.

224

FINE, GAGLIONE, MYASNIKOV, ROSENBERGER, SPELLMAN

Theorem 6.24 (Implicit function theorem) Let S(X) = 1 be a regular standard quadratic equation over a non- abelian free group F and let T (X, Y ) = 1 be an equation over F , |X| = m, |Y | = n. Suppose that for any solution U ∈ VF (S) there exists a tuple of elements W ∈ F n such that T (U, W ) = 1. Then there exists a tuple of words P = (p1 (X), . . . , pn (X)), with constants from F , such that T (U, P (U )) = 1 for any U ∈ VF (S).

7

Discriminating, co-discriminating and squarelike groups

In an eﬀort to further the study of universal properties in various classes of groups, G.Baumslag, Myasnikov and Remesslennikov [BMR2] introduced the concepts of discriminating and codiscriminating groups. In this section we will discuss these ideas as well as their connections to the general elemenatry theory of groups. In particular we introduce a larger class of groups called squarelike groups which properly contain the class of discriminating groups but which are in addition an axiomatic class. Deﬁnition 7.1 Let X be a nonempty class of groups and let H be a group. Then X separates H provided that for every nontivial element h ∈ H there is a Gh ∈ X and a homomrphism ϕh : H → Gh such that ϕh (h) = 1. The class X discriminates H provided that for every ﬁnite nonempty set S of nontrivial elements of H there is a group GS ∈ X and a homomorphism ϕS : H → GS such that ϕS (h) = 1 for all h ∈ S. If X = {G} a single group then we say that G separates (discriminates) H. We say that X is a separating family of groups provided that every group G separated by X lies in X . Observe that a separating family of groups is closed under isomorphism. This is so since if H ∼ = G ∈ X , then an isomorphism ϕ : H → G does not annihilate any nontrivial element of H; hence, G separates H and so H ∈ X . Recall that a prevariety is a class of groups X closed under subgroups and closed under direct products (of arbitrary indexed families (Gi )i∈I of groups from X ). Prevarieties are tied to discrimination and separation via the next easy result. Theorem 7.2 Theorem 7.1[FGMS2] Let X be a nonempty class of groups. Then X is a prevariety of groups if and only if X is a separating family of groups. Deﬁnition 7.3 The group G is termed discriminating provided it discriminates every group it separates. It should be pointed out that there is a distinct diﬀerence between this notion of a discriminating groups and the classical deﬁnition in the book of H. Neumann [N], see Deﬁnition 17.21 of [N]; i.e., a group is discriminating if and only if it discriminates the variety it generates. In particular, discrimination in the sense of Deﬁnition 7.3 is strictly stronger. If a group is discriminating in the sense of Deﬁnition 7.3, then it is not hard to show that it discriminates the variety it generates in the sense of [N]. However, there are groups (e.g., the (absolutely)

ELEMENTARY THEORY OF GROUPS

225

free group of rank 2) which are discriminating in the sense of [N] but are not discriminating in the sense of Deﬁnition 7.3. The following easy criterion given in [BMR2] is crucial in both constructing examples of discriminating groups and in showing that other groups are nondiscriminating. Lemma 7.4 A group G is discriminating if and only if its direct square G × G is discriminated by G. The following corollary follows directly. Corollary 7.5 Let G be a discriminating group and α be a cardinal. Then the Cartesian power Gα of G is also discriminating. In particular these results imply that if the direct square G × G embeds into G then G is discriminating. Using this criterion it can be shown that there are many important examples of discriminating groups. Speciﬁcally: Theorem 7.6 The following groups are discriminating: 1. Any torsion-free abelian group. 2. Higman’s universal ﬁnitely presented group G (see [LS]). 3. Thompson’s group F (see [FGMS1]). 4. The commutator subgroup of the Gupta-Sidki group Hp (see [FGMS1]). 5. Some of the Grigorchuk groups Gω ( see [FGMS1]). Notice that Thompson’s group and Higman’s group are both ﬁnitely presented. In each of the nonabelian examples in Theorem 7.6, except for the Grigorchuk groups, discrimination follows from the fact that the direct square embeds in the group. Hirshorn and Meyer [HM] have studied groups where G ∼ = G×G. These are also discriminating. However these examples are not ﬁnitely presented. P. M. Neumann [BFGS] has pointed out a collection of inﬁnitely many examples of ﬁnitely generated discriminating groups which do not embed their direct square. These examples were developed by B. H. Neumann as subgroups of Cartesian products of alternating groups (see [BFGS]). In applying the concept of discrimination to the study of universal properties it is also important to know when groups are not discriminating. The discrimination properties of various classes of groups were studied in [FGMS1]. Using the criterion of Lemma 7.4 it is easy to show that if a group G is nonabelian and commutative transitive then G cannot discriminate G × G and hence G cannot be discriminating Lemma 7.7 Any nonabelian CT group is nondiscriminating. Corollary 7.8 The following groups are nondiscriminating: 1. A nonabelian torsion-free hyperbolic group, in particular, a free group. 2. A group G in which the nontrivial elements of ﬁnite order form a ﬁnite nonempty set.

226

FINE, GAGLIONE, MYASNIKOV, ROSENBERGER, SPELLMAN

3. Nonabelian free solvable groups as well as their nonabelian subgroups. 4. The restricted wreath product of two nontrivial torsion-free abelian groups. It is known that free solvable groups are CT ([Mal], [W]). This gives part (3). Wu also shows in [W] that the wreath product of an abelian group and a torsion-free abelian group is CT. This gives part (4). Since a ﬁnitely generated nilpotent group satisﬁes the maximal condition for subgroups (2) gives us that any ﬁnitley generated nilpotent group with nontrivial torsion is nondiscriminating. Therefore we must only consider torsion-free nilpotent groups. In [FGMS1] it was further shown that all free nilpotent groups of any class are nondiscriminating. This used the following generalization of commutative transitivity which is interesting in its own right. Deﬁnition 7.9 A group G is commutative transitive of level 0 if G is commutative transitive and commutative transitive of level m with m ≥ 1, if G satisﬁes the following property if xy = yx and yz = zy and there exists w1 , ..., wm such that [y, w1 , ..., wm ] = 1 then xz = zx Here [x1 , x2 , ..., xn ] denotes the left-normed commutator [[x1 , X − 2], ....xn ]. In [FGMS1] it was shown that if a group G is commutative transitive of level m and not nilpotent of class ≤ m then G is not discriminating. Further any free nilpotent group of rank m ≥ 2 and class c ≥ 1 is commutative transitive of level c − 1. Hence: Theorem 7.10 Finitely generated nonabelian free nilpotent groups are non discriminating. Susbsequently it was shown by Baumslag, Fine, Gaglione and Spellman [BFGS] that all nonabelian ﬁnitely generated nilpotent groups are nondiscriminating. More generally A. Myasnikov and P. Shumansky [MS] have shown that all ﬁnitely genertaed nonabelian linear groups are nondiscriminating. From the criterion given in Lemma 7.4 it is clear that a universal sentence true in G must be true in G × G. Since any universal sentence true in G × G is true in any subgroup we get the following. Lemma 7.11 If G is a discriminating group then G and G × G are universally equivalent, that is share the same universal theory. In the presence of the equationally noetherian property (see Section 5) we can say more [FGMS1]. Theorem 7.12 Let G and H be ﬁnitely generated groups and G be equationally noetherian. Then G is universally equivalent to H if and only if G discriminates H and H discriminates G.

ELEMENTARY THEORY OF GROUPS

227

Corollary 7.13 Let G be a ﬁnitely generated equationally noetherian group. Then G is discriminating if and only if G and G × G are universally equivalent. Lemma 7.14 Let G and H be ﬁnitely generated groups and let G be equationally noetherian. If H is universally equivalent to G then H is also discriminating. In the case of equationally noetherian groups we tie discrimination to quasivarieties. When we consider squarelike groups the equationally Noetherian condition will no longer be necessary. Theorem 7.15 Let G be a ﬁnitely generated equationally noetherian group. Then G is discriminating if and only if qvar(G) = ucl(G). In an eﬀort further to study universal properties, attention turned to groups which satisfy the result of Corollary 7.13, [FGMS2], that is groups G which share the same universal theory as their direct squares G × G. It turns out that this is perhaps the more appropriate class in this study since the class of squarelike groups is ﬁrst-order axiomatizable and contains the class of discriminating groups as a proper subclass. Further the class of discriminating groups is not ﬁrst-order axiomatizable. Deﬁnition 7.16 A group G is termed squarelike if G and G × G have the same universal theory, that is T h∀ (G) = T h∀ (G × G). It follows from Lemma 7.11 that every discriminating group is squarelike. Further since G embeds in G×G, every universal sentence true in G×G is automatically true in G. Thus a necessary and suﬃcient condition for a group G to be squarelike is that every universal sentence true in G must also be true in G × G. In [FGMS2] the following is proved which is a summary of several results in that paper. Theorem 7.17 (1) The class of discriminating groups is a proper subclass of the class of squarelike groups. (2) The class of squarelike groups is axiomatic (3) The class of discriminating groups is non-axiomatic. To prove Theorem 7.17(1) a speciﬁc example of a non discriminating squarelike group was constructed. This construction depended on the fact that the class of squarelike groups can be shown to be closed under direct unions while the discriminating groups are not. Further this involved the following result on discriminating abelian groups given in [BMR2]. Theorem 7.18 ([BMR2]) Let A be a torsion abelian group. Suppose that for each prime p, the p-primary component of A modulo its maximal divisible subgroup contains no nontrivial element of inﬁnite p-height. Then A is discriminating if and only if the following two conditions are satisﬁed for each prime p. 1. For every positive integer k, one has that ρ(1) [p, k](A) is either 0 or ∞. 2. The rank of the maximal divisible subgroup of the p-primary component of A is either zero or inﬁnite.

228

FINE, GAGLIONE, MYASNIKOV, ROSENBERGER, SPELLMAN

Here the rank of a divisible abelian p-group is the maximal number of direct summands of the quasicyclic group Zp∞ ; moreover, the p-height of an element a of an abelian p-group A is (with respect to A) the maximal positive integer n, if it exists, such that the equation pn x = a has a solution in A. To show that the class of squarelike groups is axiomatic a result of Mal’cev is used. This says that a class of groups is axiomatic if it is closed under ultraproducts and elementary equivalence. This is shown to be true for the class of squarelike groups. Although Theorem 7.17 distinguishes the class of squarelike groups from its subclass of discriminating groups it was shown in [FGMS2] that they coincide in the presence of ﬁnite presentation. The proof of this result uses the same techniques as we exhibited after the statement of Theorem 4.9 that if G is a ﬁnitely generated universally free group then G is fully residually free. Theorem 7.19 Let G be a ﬁnitely presented group. Then G is discriminating if and only if it is squarelike. A similar argument to that proving that the class of squarelike groups is axiomatic can be used to show that the class of all groups H for which there exists a discriminating group GH elementarily equivalent to H is axiomatic. Clearly, that class is the least axiomatic class containing the discriminating groups. With this idea Theorem 7.12 can be extended to squarelike groups without assuming equationally noetherian. Theorem 7.20 Let G be a group. Then the following three conditions are equivalent. 1. G is squarelike, 2. ucl(G) = qvar(G) and 3. G is universally equivalent to a discriminating group. Several quesions were raised by the results on squarelike groups. In particular were the following two (1) Is every squarelike group the direct union of a family of discriminating groups? (2) Is every squarelike group elementarily equivalent to a discriminating group? In [FGS] question (2) is answered aﬃrmatively for all abelian groups while question (1) is answered aﬃrmatively for the class of torsion abelian groups. Speciﬁcally: Theorem 7.21 [FGS] Let A be a squarelike torsion abelian group. Then A is the union of an ω-chain of discriminating groups. Theorem 7.22 Let A be a squarelike abelian group. Then A is elementarily equivalent to a discriminating group. The ﬁnal idea is that of a codiscriminating group which was also introduced by Baumslag, Myasnikov and Remeslennikov in [BMR2].

ELEMENTARY THEORY OF GROUPS

229

Deﬁnition 7.23 A group G is codiscriminating if for any family of groups S, S separates G if and only if S discriminates G. If S is subgroup closed this means that G is fully residually S if and only if G is residually S. A domain is a group without zero divisors, that is G is a domain if given any nontrivial a, b ∈ G there exists x ∈ G such that [a, bx ] = 1. We mention two results proved in [BMR2]. Lemma 7.24 If G is a domain then G is codiscriminating. Further any nonabelian CSA group is a domain. Theorem 7.25 Every one-relator group with greater than 2 generators is a domain and hence is codiscriminating.

8

Open Questions

(1) Is every squarelike group a direct union of discriminating groups? (2) Is the class of squarelike groups the least axiomatic class containing the discriminating groups? Equivalently must every squarelike group be elementarily equivalent to a discriminating group? (3) Is every ﬁnitely generated squarelike group discriminating? References B. Baumslag “Residually free groups” Proc. London Math. Soc. 17:3 (1967) 402–418. [Ba2] B. Baumslag “Free groups and free products” Comm. Pure and Appl. Math. 20 (1967) 635-645. [B1] G. Baumslag “On generalized free products” Math. Z. 78 (1962) 423-438. [BPP] A.Berzins, B. Plotkin and E. Plotkin “Algebraic geometry in varieties of algebras with the given algebra of constants” J. of Math. Sci. 102 (2000) 4039–4070. [BMR1] G. Baumslag, A.G. Myasnikov and V. N. Remeslennikov “Algebraic Geometry over Groups 1” J. of Algebra 219 (1999) 16–79. [BMR2] G. Baumslag, A.G. Myasnikov and V. N. Remeslennikov “Discriminating and co-discriminating groups” J. of Group Theory 3 (2000) 467-479. [BMR3] G. Baumslag, A. Myasnikov, V. Remeslennikov “Discriminating completions of hyperbolic groups” to appear. [BFGS] G. Baumslag, B. Fine, A. Gaglione and D. Spellman “Discriminating nilpotent groups”, to appear. [CK] C. C. Chang and H. J. Keisler Model Theory North-Holland, Amsterdam, 1973. [C] P. M. Cohen Universal Algebra Harper and Row, New York 1965. [CE] L. Comerford and C. Edmonds, Solutions of Equations in Free Groups, in Conf. in Group Theory Singapore 1987 Springer-Verlag 347-355, 1989. [Er] Y.L. Ershov, “Elementary group theories”, Soviet Math. Dokl. 13 (1972) 528-532. [Er1] A.Ehrenfeucht, “An application of games to the completeness problem for formalized theories”, Fund. Math. 49 129-141. [FGMRS] B.Fine,A. Gaglione,A. Myasnikov, G.Rosenberger and D. Spellman, “A Classiﬁcation of Fully Residually Free Groups of Rank Three or Less”J. of Algebra 200 (1998) 571–605. [Ba1]

230

FINE, GAGLIONE, MYASNIKOV, ROSENBERGER, SPELLMAN

[FGMS1] B. Fine, A.M. Gaglione, A.G. Myasnikov and D. Spellman, “Discriminating groups” J. of Group Theory bf 4 (2001) 463-474. [FGMS2] B. Fine, A.M. Gaglione, A.G. Myasnikov and D. Spellman, “Groups Whose Universal Theory is Axiomatizable by Quasi-Identities” to appear J. of Group Theory. [FGS] B. Fine, A.M. Gaglione, and D. Spellman, “Abelian Discriminating Groups” to appear. [GS1] A. Gaglione and D. Spellman “Generalizations of free groups: some questions” Comm. in Alg. 22:8 (1993) 3159-3169. [GS2] A. Gaglione and D. Spellman Parametric words and models of the elementary theory of non-abelian free groups in Proceedings Groups St Andrews/Galway 1993 London Math Soc.Lecture Notes Series 211 (1995) 233-248. [GS3] A. Gaglione and D. Spellman “Some Model Theory of Free Groups and Free Algebras” Houston J. Math 19 (1993) 327-356. [GS4] A. Gaglione and D. Spellman “More Model Theory of Free Groups” Houston J. Math 21 (1995) 225–245. [GS5] A. Gaglione and D. Spellman Every Universally Free Group is Tree Free in Proc. Ohio State Conference for H. Zassenhaus World Scientiﬁc 149-154 1993. [GS6] A. Gaglione and D. Spellman Even More Model Theory of Free Groups in Inﬁnite Groups and Group Rings World Scientiﬁc 37–40 1993. [G] G. Gratzer Universal Algebra Van-Nostrand, Princeton, 1968. [GK] R.I. Grigorchuk and P.F. Kurchanov: Some Questions of Group Theory Related to Geometry in Algebra VII Springer-Verlag 1990. [H] N. Harrison “Real length functions in groups” Trans. Amer. Math. Soc. 174 (1972) 77–106. [HS] G. Higman and E. Scott Existentially Closed Groups Clarendon Press, Oxford 1988. [HM] R. Hirshon and D.Meier “Groups with a quotient that contains the original group as a direct factor” Bull. Austral. Math Soc. 45 (1992) 513–520. [Kh] O. Kharlampovich, Equations over free groups and fully residually free groups, in Proc. of 12th Int. Conf. on Formal Power Series and Algebraic Combinatorics (FRSAC’00) 45-54, Moscow State Univ. 2000. [KhM1] O. Kharlampovich and A.Myasnikov “Irreducible aﬃne varieties over a free group: I. Irreducibility of quadratic equations and Nullstellensatz” J. of Algebra 200 (1998) 472-516. [KhM2] O. Kharlampovich and A.Myasnikov “Irreducible aﬃne varieties over a free group: II. Systems in triangular quasi-quadratic form and a description of residually free groups” J. of Algebra 200 (1998) 517-569. [KM3] O. Kharlampovich and A.Myasnikov Description of fully residually free groups and Irreducible aﬃne varieties over free groups Summer school in Group Theory in Banﬀ 1996, CRM Proceedings and Lecture notes 71–81 17 1999. [KhM3] O.Kharlampovich and A.Myasnikov “Hyperbolic Groups and Free Constructions” Trans. Amer. Math. Soc. 350: 2 (1998) 571-613. [KhM4] O. Kharlampovich and A. Myasnikov, “Solution of the Tarski Problem”, to appear. [KhM5] O. Kharlampovich and A. Myasnikov, Description of fully residually free groups and irreducible aﬃne varieties over a free group, in CRM Proceeding and Lecture Notes Summer School in Group Theory in Banﬀ 1996, 17 71-80 1998. [L] R.C. Lyndon, “Problems in combinatorial group theory”, Annals of Math. Studies 111 (1987), 3-33. [LS] R. C. Lyndon and P. E. Schupp, Combinatorial Group Theory Springer-Verlag

ELEMENTARY THEORY OF GROUPS

[LoS] [Mak1] [Mak2] [Mal]

[M1]

[MR1] [MR2]

[MR3]

[MR4] [MR5] [MR6] [MR7]

[MS] [N] [No] [O] [Ra] [Re] [Ro] [Sa] [Se] [S] [W]

231

1977. J. Los and R, Suszko, “On the inﬁnite sums of models”, Bull. Acad. Polon. Sci. S˙er. Sci. Math. Asctroum. Phys. 3 (1955) 201-202. G. S. Makanin, “Equations in a free group”, Izv. Akad. Nauk SSSR, Ser. Mat. 46 (1982) 1199–1273. Transl. in Math. USSR Izv. 21 (1983). G. S. Makanin, “Decidability of the universal and positive theories of a free group” Math. USSR Izvestiya 25:1 (1985) 75–88. A. I. Malcev, The elementary properties of linear groups in Certain problems in Math. and Mechanics, Izdat. Sibirsk. Otdel.Acad. Nauk SSSR,Novosibirsk 1961 110-132, English transl. in The metamathematics of algebraic systems. Collected papers: 1936-1967 North-Holland Amsterdam 1971. A. Myasnikov, “Structure of models and decidability criterion for complete theories of ﬁnite-dimensional algebras” Izvest. Akademii Nayk USSR, ser. math. 53 (1989) 379-397. A. Myasnikov, V. Remeslennikov “Elementary properties of powered nilpotent groups” Doklady Akademii Nayuk 258:5 (1981) 1056-1059, A.G. Myasnikov, V.N. Remeslennikov “Deﬁnability of the set of Mal’cev bases and elementary theories of ﬁnite dimensional algebras I” Siberian Math. Journ. 23 (1983) 711-724. A.G. Myasnikov, V.N. Remeslennikov “Deﬁnability of the set of Mal’cev bases and elementary theories of ﬁnite dimensional algebras II” Siberian Math. Journ. 24 (1983) 231-246. A. Myasnikov, V. Remeslennikov “Algebraic geometry 2: logical foundations” Journal of Algebra 234 (2000) 225-276. A.G. Myasnikov and V.N. Remeslennikov “Exponential groups 1: Foundations of the theory and tensor completion” Siberian Mat. J. 5 (1994) 1106–1118. A.G. Myasnikov and V.N. Remeslennikov, Exponential groups, preprint, New York 1994. A.G. Myasnikov and V.N. Remeslennikov, “Exponential groups 2: Extensions of centralizers and tensor completion of CSA groups”, to appear Int. J. of Algebra and Computation. A. G. Myasnikov and P. Shumansky, Linear Groups are Nondiscriminating, preprint, New York 2002. H. Neumann, Varieties of Groups Springer-Verlag, New York 1967. G.A. Noskov, “The elementary theory of a ﬁnitely generated almost solvable group”, Math. USSR-Izv. 22 (1984) 465-482. F. Oger, “Elementary equivalence for abelian-by-ﬁnite and nilpotent groups”, J. of Symbolic Logic 66 (2001) 1471-1480. A. A. Razborov, “On systems of equations in free groups” Izv. Akad. Nauk SSSR 48 (1984) 779-832. English transl: Math, USSR Izv. 25 115-162. V. N. Remeslennikov, “∃ free groups and groups with a length function,” Contemp. math. 184 (1995), 354-376. N.S.Romanovskii, “On the elementary theory of an almost polycyclic group”, Math.-USSR-Sb 39 (1981) 125-132. G.S. Sacerdote, “Elementary properties of free groups”, Trans. Amer. Math. Soc. 178 (1972) 127-138. Z. Sela, “Solution to the Tarski Problem” announcement. W. Szmielew, “Elementary properties of Abelian groups”, Fund. Math. 41 (1955) 203-271. Wu, Yu-Fen, “Groups in which commutativity is a transitive relation”, J. Algebra 207 1998 165-181.

ANDREWS-CURTIS AND TODD-COXETER PROOF WORDS GEORGE HAVAS and COLIN RAMSAY1 School of Computer Science and Electrical Engineering, The University of Queensland, Queensland 4072, Australia Email: [email protected] and [email protected]

Abstract Andrews and Curtis have conjectured that every balanced presentation of the trivial group can be transformed into a standard presentation by a ﬁnite sequence of elementary transformations. It can be diﬃcult to determine whether or not the conjecture holds for a particular presentation. We show that the utility PEACE, which produces proofs based on Todd-Coxeter coset enumeration, can produce Andrews-Curtis proofs.

1

Introduction

There are inﬁnitely many ﬁnite presentations of the trivial group, and the problem of determining whether or not a given presentation is of the trivial group is, in general, unsolvable. A presentation x1 , . . . , xn | r1 , . . . , rm is called balanced if m = n. The standard n-generator presentation of the trivial group is x1 , . . . , xn | x1 , . . . , xn , and one method of proving that a balanced presentation represents the trivial group is to reduce it to a standard presentation. Consider the following transformations on the n-tuple of relators (r1 , . . . , rn ): (AC1) replace ri by ri rj , for some j = i; (AC2) replace ri by ri−1 ; (AC3) replace ri by w−1 ri w, for some word w in the generators. Each of these AC-transformations leaves rk ﬁxed for all k = i. Andrews and Curtis [1] conjectured that any balanced presentation of the trivial group can be transformed to the standard presentation by a sequence of AC-transformations. Such a sequence of transformations is called an AC-proof. “The prevalent opinion is that the conjecture is false” [2], but no counterexample has been proved to exist. This makes the study of potential counterexamples interesting. Further, generating AC-proofs is not easy, and Miasnikov [5] describes a genetic algorithm designed to produce such proofs. Using this, Miasnikov and Myasnikov [6] have shown that the conjecture holds for all balanced presentations of the trivial group on two generators with the total length of the relators at most twelve. They also note that the presentation a, b | a3 = b4, aba = bab , of total length thirteen, is a smallest potential counterexample to the Andrews-Curtis conjecture at present. Havas and Ramsay [3] prove that it is unique up to AC-equivalence. 1

Both authors were partially supported by the Australian Research Council

ANDREWS-CURTIS AND TODD-COXETER PROOF WORDS

233

The Todd-Coxeter coset enumeration procedure [8] is a systematic method for enumerating, in a given ﬁnitely presented group, the cosets of a given ﬁnitely presented subgroup. Implicit in the workings of such an enumeration are formal proofs that particular words in the generators are in the subgroup (see, for example, Leech [4]). The utility PEACE [7] (proof extraction after coset enumeration) has been developed to automate the production of such proofs. PEACE produces proof words (see below), which can be regarded as certiﬁcates attesting to subgroup membership. For enumeration over the trivial subgroup, proof words produced by PEACE consist of products of conjugates of relators and of their formal inverses, and ACproofs can be viewed the same way. It is natural to ask whether or not PEACE proof words correspond to AC-proofs. This question seems to be diﬃcult, but we exhibit some cases where we can obtain a constructive positive solution.

2

An AC-proof and its proof words

The length thirteen presentation quoted above is one of a well-known series of balanced presentations for the trivial group, a, b | an = bn+1, aba = bab , n ≥ 0. Their triviality for n = 0, 1 is obvious, while for n ≥ 2 some work is required. To illustrate our basic technique, we take the case n = 1 as our example. So, let G = a, b | aBB, abaBAB , where we adopt the convention that A = a−1 and B = b−1. Although the triviality of G is obvious, producing an AC-proof of this requires a little work with pencil and paper. One such proof is given in Table 1; this is not the natural proof which starts by eliminating occurrences of the ﬁrst generator from the second relator, but more like the kind of proof that might be produced by machine. The r1 and r2 columns in the table illustrate the normal proof extraction process, where we freely cancel adjacent generator/inverse pairs as they occur. The r2 column illustrates an alternative where the relators are protected from cancellation by bracketing, and we only cancel outside of the relators. The word produced at the eighth transformation (or move), w = A(abaBAB)b(aBB)b(bbA)BB(bbA)a, is a proof word that b is trivial in G. To see this, simply note that w is a product of conjugates of relators (or their formal inverses) of G (so is trivial), and that it freely reduces to b. Proof words are a convenient way of presenting a proof succinctly. They can be generated by a variety of methods, and checking them is a simple mechanical process which depends only on the presentation and the group axioms. A proof word for a in G is readily extracted from Table 1, and is simply (aBB)ww. We call such proof words generated by AC-transformations AC-proofwords.

3

Some PEACE proofs and their proof words

PEACE extracts proof words from a successful coset enumeration by storing additional information in the coset table as it is constructed. This additional information is used to continually rewrite the word to be proved until a proof word is

234

HAVAS, RAMSAY

Table 1. An AC-proof for G move 0: 1: r1 → br1 B 2: r2 → r2 r1 3: r1 → r1−1 4: r1 → br1 B 5: r2 → r2 r1 6: r1 → BBr1 bb 7: r2 → r2 r1 8: r2 → Ar2 a 9: r1 → r1−1 10: r1 → r1 r2 11: r1 → r1 r2

r1 aBB baBBB baBBB bbbAB bbbbABB bbbbABB bbA bbA bbA aBB aB a

r2 abaBAB abaBAB abaBBBB abaBBBB abaBBBB aB aB abA b b b b

r2 (abaBAB) (abaBAB) (abaBAB)b(aBB)B (abaBAB)b(aBB)B (abaBAB)b(aBB)B (abaBAB)b(aBB)b(bbA)BB (abaBAB)b(aBB)b(bbA)BB (abaBAB)b(aBB)b(bbA)BB(bbA) A(abaBAB)b(aBB)b(bbA)BB(bbA)a

obtained. In general, these proof words consist of products of conjugates of relators and subgroup generators. If the subgroup is trivial then only the conjugated relators are present, as in the AC-proofwords. The particular proof word generated by a coset enumeration depends on the exact sequence of operations during the enumeration, the additional information stored, and the rewriting. So the proofs generated by PEACE are very variable. However, it readily produces a variety of proof words that b is trivial in G where the proof words contain the same mix of four relators as w. Some examples are: ba(Abb)AB(baBABa)(Abb)b(BBa)B; b(Abb)B(BabaBA)ba(Abb)B(BBa)bAB; B(aBABab)AB(bbA)a(BaB)Aba(bAb)b. Note that PEACE often uses a relator in a cyclically permuted and/or inverted form. However, any cyclings are easily removed by conjugation. (PEACE does not do this automatically, but such a feature could easily be added.) For the examples above, this yields: b(bbA)BA(abaBAB)(bbA)abA(aBB)aB; bA(bbA)aBB(abaBAB)bb(bbA)aBA(aBB)abAB; BBA(abaBAB)abAB(bbA)aB(aBB)bAbaB(bbA)bb. Although none of these match w, they are similar. The question now is whether or not there is a sequence of AC-transformations which will generate one of these. Or, can one of these proof words from PEACE be used to generate an AC-proofword, and thus an AC-proof? Note how the proof words exhibited here all contain the relator abaBAB exactly once. This observation suggests the following procedure for proving that a proof word is an AC-proofword: append copies of aBB, inverted and/or conjugated as appropriate, to abaBAB; if there are any copies of aBB preﬁxing abaBAB, then invert the current relator, repeat the previous step (appending the inverse of what is required), and then invert the result. Applying this to the ﬁrst example above readily yields the AC-transformation sequence in Table 2. Thus, we see that the proof word u = b(bbA)BA(abaBAB)(bbA)abA(aBB)aB

ANDREWS-CURTIS AND TODD-COXETER PROOF WORDS

235

Table 2. Generating a speciﬁc AC-proofword move 0: 1: r1 → r1−1 2: r2 → r2 r1 3: r1 → r1−1 4: r1 → abAr1 aBA 5: r2 → r2 r1 6: r2 → r2−1 7a: r1 → aBAr1 abA 7b: r1 → abr1 BA 8: r2 → r2 r1 9: r2 → r2−1 10: r2 → Ar2 a

r1 (aBB) (bbA) (bbA) (aBB) abA(aBB)aBA abA(aBB)aBA abA(aBB)aBA (aBB) ab(aBB)BA ab(aBB)BA ab(aBB)BA ab(aBB)BA

r2 (abaBAB) (abaBAB) (abaBAB)(bbA) (abaBAB)(bbA) (abaBAB)(bbA) (abaBAB)(bbA)abA(aBB)aBA abA(bbA)aBA(aBB)(babABA) abA(bbA)aBA(aBB)(babABA) abA(bbA)aBA(aBB)(babABA) abA(bbA)aBA(aBB)(babABA)ab(aBB)BA ab(bbA)BA(abaBAB)(bbA)abA(aBB)aBA b(bbA)BA(abaBAB)(bbA)abA(aBB)aB

produced automatically by PEACE is an AC-proofword. (In fact, Bub is also an AC-proofword, and is slightly shorter.) Since u freely reduces to b, it is a trivial matter to extend the sequence of AC-transformations given in Table 2 to an ACproof for G. Similar manipulations readily yield AC-proofs from the other two examples.

4

A diﬃcult example

In [5], Miasnikov notes that the four length twelve presentations a, b | Ab2 a = b3, a = b±1 ab±1 A are particularly diﬃcult. Let G2 = a, b | AbbaBBB, aabAB . Miasnikov gives a proof for this, in the form x, y | yxY XX, XyyxY Y Y , in nineteen moves. Two of these moves are not in (AC1)–(AC3) but are Whitehead automorphisms (see [9]), so this is not an AC-proof. However, the Whitehead automorphisms can be replaced by AC-transformations, albeit at the expense of increasing the length of the proof. Let r = AbbaBBB, s = aabAB, R = r−1 and S = s−1, and consider the following proof word for a in G2 produced by PEACE, (R)A(S)aabA(S)aB(S)a(s)bA(s)aBAA(s)a(r)AA(s)a. From this, we were able to extract the nineteen move AC-proof given in Table 3. Both this proof and the one given by Miasnikov can be shortened by noting that the ﬁrst few moves simply invert or cycle the relators, and so can be eliminated by starting with the presentations in slightly diﬀerent forms. We found our proof word by generating a number of PEACE proofs for the four presentations in the family, extracting the templates of the proof words (the one given has template RSSSsssrs) and then comparing these with a list of possible templates (easily generated by a breadth-ﬁrst search using the (AC1) and (AC2) moves). The procedure has a low success rate, seemingly due to the paucity of templates which can be generated by AC-moves, but any match gives a candidate

236

HAVAS, RAMSAY

Table 3. An AC-proof for G2 move 0: 1: r1 → r1−1 2: r2 → r2−1 3: r2 → Ar2 a 4: r1 → r1 r2 5: r2 → abr2 BA 6: r1 → r1 r2 7: r2 → aaBAr2 abAA 8: r1 → r1 r2 9: r2 → AAAr2 aaa 10: r2 → r2 r1 11: r2 → r2−1 12: r2 → ar2 A 13: r1 → r1 r2 14: r2 → r2−1 15: r2 → ar2 A 16–19: r2 → r2 r1

r1 AbbaBBB bbbABBa bbbABBa bbbABBa bbbABaBA bbbABaBA bbABA bbABA bAAA bAAA bAAA bAAA bAAA a a a a

r2 aabAB aabAB baBAA AbaBA AbaBA abAbaBABA abAbaBABA abaBAAA abaBAAA AAbaB AAbAA aaBaa aaaBa aaaBa AbAAA bAAAA b

proof word. There are usually several sequences of AC-transformations which yield a given template (these can be extracted from the breadth-ﬁrst search), and we now have to ﬁnd one where some sequence of conjugations yields the required proof word.

5

Conclusions

We have shown that it is possible to extract AC-proofs from the proofs produced by PEACE. However our technique is a triﬂe ad hoc, depending as it does on testing the pattern of the relators in a PEACE proof word against a set of possible ACproofwords and then attempting to match up the conjugation. Only in the case where there are two relators and the PEACE proofs contain one of the relators exactly once can we consistently extract an AC-proof. We have no systematic method of determining whether or not the extraction is possible in general or performing it if it is. There does not seem to be any general method for determining whether or not a given proof word can be generated by a sequence of AC-transformations and, in fact, there does not seem to be any nice characterisation of the words which can be generated by a sequence of AC-transformations. Progress in these areas might enable us to extend the work here, and use the techniques described to eliminate some of the potential counterexamples to the Andrews-Curtis conjecture. References [1] J.J. Andrews and M.L. Curtis. Free groups and handlebodies. Proceedings of the American Mathematical Society, 16:192–195, 1965. [2] G. Baumslag, A.G. Myasnikov and V. Shpilrain. Open problems in combinatorial group

ANDREWS-CURTIS AND TODD-COXETER PROOF WORDS

[3]

[4] [5] [6]

[7]

[8] [9]

237

theory. In Robert H. Gilman (ed.), Groups, languages and geometry, Contemporary Mathematics 250, American Mathematical Society, 1999, 1–27. G. Havas and C. Ramsay. Breadth-ﬁrst search and the Andrews-Curtis conjecture. Technical Report 23, Centre for Discrete Mathematics and Computing, The University of Queensland, 2001. John Leech. Computer proof of relations in groups. In Michael P.J. Curran (ed.), Topics in Group Theory and Computation, pp. 38–61. Academic Press, 1977. A.D. Miasnikov. Genetic algorithms and the Andrews-Curtis conjecture. The International Journal of Algebra and Computation, 9(6):671–686, 1999. A.D. Miasnikov and A.G. Myasnikov. Balanced presentations of the trivial group on two generators and the Andrews-Curtis conjecture. In W.M. Kantor and A. Seress (eds.), Groups and Computation III, Ohio State University Mathematical Research Institute Publications 8, Walter de Gruyter, 2001, 257–263. Colin Ramsay. PEACE 1.000: Proof Extraction after Coset Enumeration. Technical Report 22, Centre for Discrete Mathematics and Computing, The University of Queensland, 2001. J.A. Todd and H.S.M. Coxeter. A practical method for enumerating cosets of a ﬁnite abstract group. Proceedings of the Edinburgh Mathematical Society, 5:26–34, 1936. J.H.C. Whitehead. On equivalent sets of elements in a free group. Annals of Mathematics, 2nd Ser., 37(4):782–800, 1936.

SHORT BALANCED PRESENTATIONS OF PERFECT GROUPS GEORGE HAVAS and COLIN RAMSAY1 Centre for Discrete Mathematics and Computing, School of Information Technology and Electrical Engineering, The University of Queensland, Queensland 4072, Australia Email: [email protected] and [email protected]

Abstract We report some initial results from an investigation of short balanced presentations of perfect groups. We determine the minimal length 2-generator balanced presen tations for SL2 (5) and SL2 (7) and we show that M 22 , the full covering group of the sporadic simple group M22 , has deﬁciency zero. We give presentations for SL2 (7) and M 22 that are both new and of minimal length. We also determine the shortest 2-generator presentations for an inﬁnite perfect group. This is done in the context of a complete classiﬁcation of short 2-generator balanced presentations of perfect groups in terms of canonic presentations.

1

Introduction

Eﬃcient presentations for groups have been the subject of much study. A survey of such presentations for simple groups and their full covering groups appears in [4], which provides relevant background material. Balanced presentations for perfect groups are eﬃcient presentations which help us understand the answers to many associated questions. Motivated by successful enumerations of balanced presentations of the trivial group [10, 13] we realized we could in an analogous way successfully enumerate short balanced presentations of general perfect groups. Here we report some initial results from a much more substantial enumeration which is underway. We completely classify all canonic 2-generator balanced presentations of perfect groups with relator length up to 17. In particular we prove that the shortest 2-generator balanced presentations for the unique perfect group of order 120, SL2 (5), have (relator) length 12. (This result was already implicit in previous enumerations [10, 13].) We also prove that the shortest 2-generator balanced presentations for the unique perfect group of order 336, SL2 (7), have length 17. This result is new and improves greatly on the lengths of previously published balanced presentations for this group. We provide six canonic presentations of minimal length and use one to give a short eﬃcient presentation of P SL2 (7). We give a balanced presentation for the covering group of the simple group M22 . This shows that this group, M 22 , has deﬁciency zero, answering a previously unresolved question. Our canonic presentation is a unique representative of the shortest 1

Both authors were partially supported by the Australian Research Council

SHORT BALANCED PRESENTATIONS OF PERFECT GROUPS

239

possible presentations. Finally we give balanced presentations for an inﬁnite perfect group. The study of perfect groups is greatly facilitated by the classiﬁcation provided by Holt and Plesken [11]. For example, it conﬁrms that SL2 (5) and SL2 (7) are unique perfect groups with orders 120 and 336, respectively. Thus if coset enumeration reveals that a ﬁnitely presented group which is perfect has one of these orders then we know the group.

2

Technique

A presentation x1 , . . . , xn | r1 , . . . , rm is called balanced if m = n. We can enumerate all balanced presentations which are not too long and try to identify the groups. In fact we were motivated by an earlier enumeration where we studied potential counterexamples to the Andrews-Curtis conjecture. There we enumerated balanced presentations of the trivial group, the smallest perfect group. Here we are speciﬁcally not interested in the trivial group. Combinatorial explosion means that not too long is pretty short in general. By restricting ourselves to perfect groups we can easily go further, far enough to be interesting. Our general selection criteria for presentations are as follows. A) We insist that all generators are actually in the relators somewhere (ie, there is no obvious free quotient). B) The group has trivial abelian quotient (ie, is perfect). C) No relator (perhaps cycled and/or inverted) properly contains more than half of another. (Otherwise, there would exist a shorter “equivalent” presentation on the same generators.) Note that these checks (and others later) are not independent; ie, B) implies A). However, this redundant checking is often helpful, since it can speed up the running time considerably. (Removing a presentation early means we do not need to run any further checks on it. Of course, this has to be balanced against the costs of the check.) In this paper we only consider 2-generator presentations. However, similar principles apply for more generators. For two generators we add two more criteria. D) We disallow relators of length 4 or less. Examination of the list of “canonic” relators, deﬁned later, shows that these can either be reduced to one generator presentations (which are not interesting), or cannot satisfy B) above (ie, not perfect). Thus the smallest interesting presentation length is 10. E) We disallow any relator which contains a generator precisely once, since these can be reduced to one generator cases (again, not interesting). Throughout, we adopt the convention of using upper-case letters to denote inverses so that, for example, A = a−1, etc. For two generators, we start by ﬁxing on a set of generators {a, b} and an ordering a < A < b < B, and a total presentation length . We then enumerate all balanced presentations on the generator set where the sum of the relator’s lengths is , which satisfy our selection criteria. For example, to ensure that the abelian quotient is trivial we calculate the abelianised relators (ak bm, an bp ) of each presentation, and discard those with kp − mn = ±1.

240

HAVAS, RAMSAY

These criteria alone allow much redundancy. To partially address this we remove some equivalent presentations by putting each presentation into a standard form. By this we mean that: the relators are freely and cyclically reduced; each relator is the lexicographic minimum of all its cyclic permutations and their inverses; the relators are in order (length plus lexicographic). For our purposes here, lists of presentations can also be pruned by retaining only a single representative of each orbit under the action of the automorphism group of the free group. For simplicitly we initially only apply the length-preserving automorphisms of the extended symmetric group of order 2n n! generated by the permutations (xi , xj )(Xi , Xj ), for all i = j, and (xi , Xi ), for all i. A canonic presentation is the least representative of such an orbit. For n = 2, this pruning decreases the list sizes by a factor of up to eight; for example, the list for = 10 reduces to one. We do this to give us lists of presentations. What next? The availability of packages for computational group theory, including GAP [8], Magma [1], Magnus [14] and testisom [12] makes it quite easy to experiment with groups. For coset enumeration we use the ACE enumerator [9] either as available in a package, or as a stand-alone program for some more diﬃcult cases. If a ﬁnitely presented group is ﬁnite then coset enumeration will reveal the fact (subject to space and time considerations and to general unsolvability results). So we simply try enumerating cosets of the trivial subgroup. There is a minor problem here: if we try to enumerate the cosets for an inﬁnite group it could go on for a long time. In our initial study we allow a maximum of ten million cosets to be deﬁned. With this limit, it takes less than one cpu minute for a coset enumeration to overﬂow on a reasonably fast machine.

3

Results

The following table indicates the results of the process. We see that most canonic balanced presentations for perfect groups in this length range actually deﬁne the trivial group. (This is another source of potential counterexamples to the AndrewsCurtis conjecture.) Presentation length 10 11 12 13 14 15 16 17

No. of canonic presentations 1 4 7 68 78 600 694 6106

Coset enumeration behaviour trivial 120 336 overﬂow 1 4 6 1 65 3 77 1 590 5 5 688 6 6057 38 6 5

Coset enumeration reveals that SL2 (5) ﬁrst appears at length 12. The unique canonic presentation is a, b | aabAb, abbbbaB. This agrees with the results of the presentation enumerations in [10, 13]. As an immediate consequence we have:

SHORT BALANCED PRESENTATIONS OF PERFECT GROUPS

241

Theorem 3.1 The shortest 2-generator balanced presentations for SL2 (5) have length 12. Much information on presentations of SL2 (5) appears in [7]; it was the ﬁrst perfect group known to have deﬁciency zero. A length 12 presentation already appears in [7, p. 69], in all editions except the ﬁrst. The situation is more complicated with SL2 (7). It ﬁrst obviously appears at length 17, six times in our table. To prove that these are indeed shortest requires further analysis of the shorter presentations which led to coset enumeration overﬂows. We have to discover enough about the groups deﬁned by these presentations to show that they are not isomorphic to SL2 (7). For our purposes this can easily be done by looking at low index subgroups. The presentations of length 14 and 15 which led to overﬂows are: P2 = a, b | ababaB, aaaaaaabb; P1 = a, b | aabAb, abbbbbbaB; P4 = a, b | abaBAB, aaaaBAbAB; P3 = a, b | abaBAB, aaaaaBAAB; P6 = a, b | aabABAb, aaaBAbAB. P5 = a, b | abaBAB, aaaBAbbAB; We can readily prove that each of these present inﬁnite groups by using programs like that in Example H20E40 [5, p. 306, vol. II]. Each presentation leads to one index 28 subgroup with inﬁnite abelianization. In fact all six groups have 83 conjugacy classes of nontrivial subgroups up to index 28, suggesting that the groups are isomorphic. Indeed testisom quickly identiﬁes isomorphisms between the groups. For example, subscripting generators according to presentation number, we have isomorphisms taking a1 → a2 b2 , b1 → a2 and a2 → b1 , b2 → B1 a1 . These groups are all isomorphic to the full covering group of the triangle group 2, 3, 7. Simply eliminating z = B 2 from a, b, z | b2 z, (ab)3 z, a7 Z, which is a presentation for 2, 3, 7, gives P2 . Theorem 3.2 The shortest 2-generator balanced presentations for SL2 (7) have length 17. Theorem 3.3 The shortest 2-generator balanced presentations for an infinite per fect group have length 14 and present 2, 3, 7. The canonic presentation of length 14 for this group is a, b | aabAb, abbbbbbaB. A previously published balanced presentation for SL2 (7) [3] has length 37. Shorter presentations with length 27 are implicit from Conder’s list [6] of onerelator quotients of the modular group combined with a result of Campbell, Havas, Hulpke and Robertson [2, Lemma 2.2]. Thus Conder’s presentation 11.34 for L2 (7): x, y | x3 , y 2 , xyxyxyxyXyXyxyXyxyXyXy leads to: x, y | x3 y 2 , xY xY xY xY XyXyxY XyxyXyXy as one presentation for SL2 (7). The six canonic presentations of length 17 revealed by our enumeration are: Q2 = a, b | aaaabbb, ababaBAbAB; Q1 = a, b | abaBAB, aaaaBBaBaBB; Q4 = a, b | aabABAb, aaaBABBBAB; Q3 = a, b | aabABAb, aaabbbAbAB; Q6 = a, b | aabaaBAB, abbbaBBBB. Q5 = a, b | aaabAbAb, abABBAbaB; It is easy to ﬁnd the orders of the generators of each of the groups (eg, by doing coset enumerations over the subgroups generated by each of the generators). Subscripting generators according to presentation number we discover that:

242

HAVAS, RAMSAY

a71 , b71 ; a82 , b62 ; a73 , b83 ; a74 , b84 ; a85 , b85 ; a76 , b14 6 are all trivial in the corresponding groups. It follows that all Qi are on diﬀerent generators except possibly Q3 and Q4 . Since a, b | aabABAb, aaabbbAbAB, aaaBABBBAB also presents SL2 (7) (readily revealed by coset enumeration), those presentations are on the same generators. Further analysis reveals that a length 17 presentation for SL2 (7) is implicit in Conder’s list. Following Conder, deﬁne u = xy and v = Xy. In terms of these generators his presentation 11.34 is: u, v | (vU )3 , (vU v)2 , uuuuvvuvuvv. Multiplying the inverse of the ﬁrst relator by the second gives the balanced presentation u, v | uV uvU v, uuuuvvuvuvv, which is equivalent to our canonic presentation Q1 . Presentation Q2 has the obvious quotient a, b | a4 , b3 , ababaBAbAB which is a short presentation for P SL2 (7), easily conﬁrmed by coset enumeration. Given that there were ﬁve coset enumeration overﬂows for length 17 canonic presentations, we need to further analyze those presentations to ensure that there are no other minimal length canonic presentations for SL2 (7). The presentations which led to overﬂow are: R2 = a, b | aabaBab, aaaaaabbAB; R1 = a, b | abaBAB, aaaaaaBBABB; R4 = a, b | aaabAbAb, aabbbbbbb; R3 = a, b | aaabaBab, aaaBAbbAB; R5 = a, b | aababAAB, abbbbaBaB. Again we can use low index subgroups to investigate the associated groups. We discover that R1 , R2 , R3 and R4 also have one index 28 subgroup with inﬁnite abelianization and have 83 conjugacy classes of nontrivial subgroups up to index 3, 7. 28. The groups deﬁned by these Ri are also all isomorphic to 2, The group presented by R5 is diﬀerent. It has an index 22 subgroup, and the corresponding permutations generate M22 , the sporadic simple group with order 443520. This is enough to conﬁrm that there are exactly six minimal length (17) canonic presentations for SL2 (7). However presentation R5 merits further analysis. We do not easily ﬁnd other low index subgroups. So we tried enumerating the cosets of a and b. Both of those enumerations complete easily enough even restricted to a maximum of 10 million cosets. Encouraged by this we try the coset enumeration over the trivial subgroup again, allowing more cosets. Using the Hard strategy of ACE we discover the group has order 5322240 (deﬁning a maximum of 20921635 cosets and total of 21611026). This is enough to tell us that R5 presents M 22 , the full covering group of M22 , and resolves in the positive the question about M 22 left open in [4]. Theorem 3.4 The presentation a, b | aababAAB, abbbbaBaB defines M 22 . It is the unique minimum length canonic presentation for the group.

4

Conclusions

We have reported some initial results from an investigation of short balanced presentations of perfect groups. We have completely classiﬁed all canonic 2-generator balanced presentations of perfect groups with relator length up to 17. Given any presentation of such length it is straightforward to determine if it is interesting and, if so, to determine its canonic equivalent.

SHORT BALANCED PRESENTATIONS OF PERFECT GROUPS

243

Our classiﬁcation has revealed that only 5 isomorphism types arise up to this length: the trivial group; SL2 (5) with order 120; SL2 (7) with order 336; M 22 with order 5322240; and one inﬁnite group, 2, 3, 7. We are continuing the enumeration of canonic presentations of various kinds and expect many new results to be forthcoming. We anticipate ﬁnding new deﬁciency zero presentations for many perfect groups, including some for groups not currently known to have deﬁciency zero. We also expect to ﬁnd other eﬃcient presentations for various groups which are associated with perfect groups, in particular for quotients including simple quotients.

Acknowledgements We are grateful to Marston Conder for helpful discussions. References [1] W. Bosma, J. Cannon, and C. Playoust. The Magma algebra system I: The user language. Journal of Symbolic Computation, 24:235–265, 1997. See also http://www.maths.usyd.edu.au:8000/u/magma/ [2] C.M. Campbell, G. Havas, A. Hulpke and E.F. Robertson. The simple group L3 (5) is eﬃcient. Communications in Algebra (to appear). [3] C.M. Campbell and E.F. Robertson. Two-generator two-relation presentations for special linear groups. In The geometric vein, Springer, New York-Berlin, 1981, 561– 567. [4] C.M. Campbell, E.F. Robertson and P.D. Williams, Eﬃcient presentations for ﬁnite simple groups and related groups. In Groups-Korea 1988, Lecture Notes in Mathematics 1398, Springer-Verlag, New York, 1989, 65–72. [5] J. Cannon and W. Bosma. Handbook of Magma Functions, School of Mathematics and Statistics, The University of Sydney, Version 2.8, Sydney, 2001. [6] M. Conder. Three-relator quotients of the modular group. Quarterly Journal of Mathematics, Oxford, Second Series, 38:427–447, 1987. [7] H.S.M. Coxeter and W.O.J. Moser, Generators and relations for discrete groups, Springer, Berlin, 1st edition, 1957; 2nd edition, 1965; 3rd edition, 1972; 4th edition, 1979. [8] The GAP Group, Aachen, St Andrews, GAP – Groups, Algorithms, and Programming, Version 4.2, 2001. See also http://www-gap.dcs.st-and.ac.uk/∼gap/ [9] G. Havas and C. Ramsay. Coset enumeration: ACE version 3.001 (2001). Available as http://www.itee.uq.edu.au/∼havas/ace3001.tar.gz [10] G. Havas and C. Ramsay. Breadth-ﬁrst search and the Andrews-Curtis conjecture. International Journal of Algebra and Computation (to appear). [11] D.F. Holt and W. Plesken. Perfect Groups, Oxford University Press, 1989. [12] D.F. Holt and S. Rees. Testing for isomorphism between ﬁnitely presented groups. In M.W. Liebeck and J. Saxl (eds.), Groups, Combinatorics and Geometry, Cambridge University Press, 1992, 459–475 [13] A.D. Miasnikov and A.G. Myasnikov. Balanced presentations of the trivial group on two generators and the Andrews-Curtis conjecture. In W.M. Kantor and A. Seress (eds.), Groups and Computation III, Ohio State University Mathematical Research Institute Publications 8, Walter de Gruyter, 2001, 257–263. [14] The New York Group Theory Cooperative. Magnus. See http://www.grouptheory.org/

FINITE p-EXTENSIONS OF FREE PRO-p GROUPS W. HERFORT1 , University of Technology, Vienna, Austria P. A. ZALESSKII2 , University of Brasilia, Brazil

Abstract Finitely generated virtually free pro-p groups are described. This generalizes Serre’s result, stating that a torsion free virtually free pro-p group is free pro-p. As a consequence of our main result certain ﬁnite subgroups and their conjugacy classes in the automorphism group of a ﬁnitely generated free pro-p group are classiﬁed.

1

Introduction

Let p be a prime number, and G a pro-p group containing an open free pro-p subgroup F . If G is torsion free, then, according to the celebrated theorem of Serre in [17], G itself is free pro-p. The main objective of the announcement is to present a description of virtually free pro-p groups without the assumption of torsion freeness. Theorem A Suppose G is a finitely generated pro-p group with an open free pro-p subgroup F . Then G is the fundamental pro-p group of a finite graph of finite p groups of order bounded by |G : F | The theorem is the pro-p analog of the description of ﬁnitely generated virtually free discrete groups proved by Karrass, Pietrovski and Solitar [12]. In fact as a consequence we obtain that a ﬁnitely generated virtually free pro-p group is the pro-p completion of a virtually free discrete group. However, the discrete result is not used (and cannot be used) in the proof. In the characterization of discrete virtually free groups Stallings theory of ends played a crucial role. One does not have such a powerful tool in the pro-p situation. Thus purely combinatorial pro-p group methods are used.

2

Some history of the problem

Let p be a prime number, and G a pro-p group containing an open free pro-p subgroup F . If G is torsion free, then, according to the celebrated theorem of Serre in [17], G itself is free pro-p. This result led Serre to conjecture that a virtually free torsion-free group is free, initiating the study of virtually free groups. Serre’s conjecture has been proved by Serre, Stalling and Swan (see [2]) and led ultimately 1 The first author would like to thank for generous support and greatful hospitality at the University of Brasilia during stays July-August in 1999 and 2000 2 The second author acknowledges the financial support of CNPq

FINITE p-EXTENSIONS OF FREE PRO-p GROUPS

245

to characterizing virtually free groups as groups acting eﬀectively with bounded vertex-stabilizers on some tree (A. Karrass ,A. Pietrovski and D. Solitar in [12] for ﬁnitely–, D.E. Cohen in [1] for countably –, and ﬁnally, G.P. Scott in [16] for arbitrary cardinality–generated G). First eﬀorts, known to the authors, to return to the pro-p situation were taken by D. Haran in [5] and A. J. Engler in [4], where certain pro-2 groups of this type were studied in the context of inﬁnite Galois theory. Studying the automorphism groups A of ﬁnite p-power order of a free (pro-p) group F is equivalent to describing the virtually free pro-p group G := F A (the holomorph). Indeed, J. L. Dyer and G.P. Scott’s result that for every automorphism α of ﬁnite order of a free group F the elements ﬁxed by α form a free factor [3] is derived from a structure theorem of G.P. Scott [16] stating ∼ (Ci × Hi ) H, (2.1) F α = i∈I

and stand for forming free product, and Hi , H are suitable free where we groups. Establishing a pro-p-analog of the J. L. Dyer -G.P. Scott result for the case of a free pro-p group F of rank 2 (the rank is deﬁned to be the minimal cardinality of a set of topological generators of F ), W. Herfort , L. Ribes and P.A. Zalesskii in [6] were led to provide a complete list of all centerless ﬁnite extensions of a free pro-p group of rank 2 in [7]. In the same paper from this list together with an application of the proﬁnite version of the Schur-Zassenhaus Theorem, as described on p.41 in [13], a classiﬁcation of all automorphisms of ﬁnite order of any free pro-p group of rank 2 is given. The pro-p-analog of Eq.(2.1) was established by the authors for ﬁnitely generated G and p = 2 in [9], by C. Scheiderer in [15] for arbitrary p using cohomological methods, and settled for arbitrary rank of F by L. Ribes together with the authors in [8]. In [10] the authors succeeded proving the Theorem A, provided G is free pro-pby-cyclic (i.e., G/F is a ﬁnite cyclic p group).

3

Idea of the proof and accompanying results

We shall give a few explanations, as far as necessary to present and discuss our results. Details and complete proofs of the results are contained in [11]. The proof of Theorem A depends on the following Proposition 3.1 Let G be a virtually free pro-p group, then G/Tor (G) is free pro-p. We do not know whether G/Tor (G) is projective whenever G is projective. More generally, we do not know whether cd(G/Tor (G)) ≤ n, whenever vcd(G) ≤ n in both the abstract and the pro-p situation.

246

HERFORT, ZALESSKII

A boolean space is the projective limit of ﬁnite sets, a profinite graph is the projective limit of ﬁnite graphs - it has a closed subset of vertices, the induced projective limit of the vertices. If Γ is a connected ﬁnite graph, the fundamental pro-p group is deﬁned to be the pro-p-completion of the discrete (usual) fundamental group of Γ. A proﬁnite, connected graph Γ with a distinguished point m0 ∈ Γ is the projective limit of ﬁnite graphs and so the canonical projection yields a distinguished point for each such ﬁnite graph. Then the projective system of pointed ﬁnite graphs gives rise (in a canonical manner) to the projective system of their fundamental groups and therefore of fundamental pro-p groups. The fundamental pro-p group π1 (Γ) is deﬁned to be the projective limit over this system of the fundamental pro-p groups of the ﬁnite graphs. If π1 (Γ) = {1}, Γ is said to be a pro-p tree. The fundamental group Π(G, Γ) of any ﬁnite graph (G, Γ) of ﬁnite p groups can be deﬁned as the pro-p-completion of it’s abstract version. Then clearly it is a virtually free pro-p group. The idea of the proof of Theorem A consists of constructing a pro-p tree T(G) on which G acts with ﬁnite vertex stabilizers such that T(G)/G is ﬁnite. After this one applies Proposition 4.4 in [18] to get the result. Let F G free pro-p group such that |G : F | = pn(G) is minimal. Deﬁne ord(G) := (n(G), rank F ) ∈ N × N, equipped with the lexicographic ordering. The construction of T(G) and the proof that it is a pro-p tree uses induction on ord(G). If G is free pro-p, then one can take a graph Cayley to serve as T(G) and Theorem A clearly holds. Suppose G is not free pro-p. Denote by TG the boolean space of all subgroups of order p in G. For each A ∈ TG , ord(CG (A)) < ord(G) can be shown, and, performing the induction step, assume that all T(CG (A)) are constructed. From the collection of all of them we need to construct a suitable candidate for the desired G-graph T(G). We construct the induced G-graph T(A) ×CG (A) G and consider the ﬁnite disjoint union of all of them for A running through a representative system of TG /G. Call the G-graph X and observe that each TA is embedded naturally. Note that X is a pro-p forest. Then by using appropriate identiﬁcations of certain pro-p subtrees of X and then inserting just enough edges in X such that X becomes connected we arrive at a proﬁnite graph Ω on which G acts with ﬁnite vertex stabilizers such that Ω/G is ﬁnite. Finally we prove that Ω is a pro-p tree and so can serve as T(G). V.A.Romankov proved in [14] that the automorphism group of a ﬁnitely generated free pro-p group of rank ≥ 2, is inﬁnitely generated. As a consequence one has that despite the fact that the automorphism group of a free group is embedded in the automorphism group of its pro-p completion, it is by no means densely embedded there! Nevertheless, Theorem A allows us to obtain a description of the conjugacy classes of ﬁnite p-groups of the automorphism group of a free pro-p group. Furthermore, we give a classiﬁcation of the conjugacy classes of automorphism groups of ﬁnite coprime order of a free pro-p group.

FINITE p-EXTENSIONS OF FREE PRO-p GROUPS

247

Theorem B Let F be a free pro-p group of rank n and Φ an abstract free group of the same rank whose pro-p-completion is F . (i) The embedding Aut(Φ) ≤ Aut(F ) induces a bijection between the conjugacy classes of finite p-subgroups of Aut(Φ) and Aut(F ). (ii) The Aut(F )-conjugacy classes of finite subgroups of Aut(F ) of order coprime to p are in one-to-one correspondence with Aut(F/Φ(F ))-conjugacy classes of finite subgroups of Aut(F/Φ(F )) ∼ = GLn (Fp ) of order coprime to π (where Φ(F ) stands for the Frattini subgroup of F ). The next theorem provides a counter example to Theorem A if one drops the assumption of ﬁnite generatedness. The counter example is based on the fact that for a pro-p group G acting on a pro-p tree T such that T /G is pro-p tree, the lifting of T /G to T does not always exist. Theorem C The pro-2 group G := D4 C2 (C2 × C2 ) contains a closed subgroup H which cannot be represented as a fundamental pro-2 group of a profinite graph of finite 2-groups. This theorem also shows that a pro-p analog of Bass-Serre’s Theorem (Theorem 6.1 in [2]) does not hold for arbitrary closed subgroups of an amalgamated free pro-p product. References [1] D. E. Cohen, Groups with free subgroups of finite index, in: Conference on Group Theory, University of Wisconsin-Parkside 1972, Lecture Notes in Mathematics, 319 Springer (1973), 26-44. [2] W. Dicks , Groups, Trees and Projective Modules, Springer 1980. [3] J. L. Dyer and G.P. Scott , Periodic automorphisms of free groups, Comm. Alg. 3(3) (1975) 195-201. [4] A. J. Engler , On the Cohomological Characterization of Real Free Pro-2-Groups, Manuscripta Math. 88, (1995), 247-259. [5] D. Haran, On the Cohomological Dimension of Artin-Schreier Structures, J. Alg. 156, 93, (19219–236) [6] W. Herfort , L. Ribes and P.A. Zalesskii . Fixed points of automorphisms of free pro-p groups of rank 2, Can. J. Math. 47, 383-404 (1995) [7] W. Herfort, L. Ribes, and P. Zalesskii, Finite Extensions of pro-p groups of rank at most 2, Israel J. Math. 107 (1998), 195-227. [8] W. Herfort , P.A. Zalesskii and L. Ribes , p - Extensions of free pro-p groups, Forum Math. 11 (1999), 49-61 [9] W. Herfort and P. Zalesskii Finitely generated pro-2 groups with a free subgroup of index 2, manuscripta math. 93, 457–464, (1997) [10] W. Herfort and P.A. Zalesskii , Cyclic Extensions of free pro-p groups, Journal of Algebra, 216, 511–547, (1999) [11] W. Herfort and P. Zalesskii Virtually free pro-p groups, (submitted for publication) (2001) [12] A. Karrass , A. Pietrovski and D. Solitar , Finite and infinite cyclic extensions of free groups, J.Australian Math.Soc. 16 (1973) 458–466.

248

HERFORT, ZALESSKII

[13] L. Ribes and P.A. Zalesskii , Profinite Groups Springer 2000. [14] L. Roman’kov , Infinite generation of automorphism groups of free pro-p groups, Siberian Mathematical Journal 34, (1993) 727-732. [15] C. Scheiderer, The Structure of Some Virtually Free Pro-p groups, Proc. Amer. Math. Soc. 127 (1999), 695-700. [16] G.P. Scott , An embedding theorem for groups with a free subgroup of finite index, Bull. London Math. Soc. 6 (1974) 304-306. [17] J.-P. Serre , Sur la dimension cohomologique des groupes profinis, Topology 3 (1965) 413-420. [18] P.A. Zalesskii and O.V. Mel’nikov, Fundamental Groups of Graphs of Profinite Groups, Algebra i Analiz, 1 (1989); translated in: Leningrad Math. J. 1 (1990), 921-940.

ELEMENTS AND GROUPS OF FINITE LENGTH MARCEL HERZOG∗1 , PATRIZIA LONGOBARDI† and MERCEDE MAJ† ∗

School of Mathematical Sciences Raymond and Beverly Sackler Faculty of Exact Sciences Tel-Aviv University, Tel-Aviv, Israel † Dipartimento di Matematica e Informatica Universit` a di Salerno via Salvator Allende, 84081 Baronissi(Salerno), Italy

1 Introduction In this survey we deﬁne group elements of ﬁnite length, or F L-elements in short, and describe properties of such elements, which were established in papers [1-4]. An F L-element may be thought of as a generalization of an F C-element, which, in turn, is a generalization of a central element of the group. In ﬁnite groups all elements are F C-elements and also F L-elements, but in the inﬁnite case these sets need not coincide. The deﬁnition of F L-elements requires certain preparations. In this survey G denotes a group, N denotes the set of positive integers and P denotes the set of primes. Let n ∈ N and let x, g1 , g2 , . . . , gn ∈ G. The sequence (xg1 , xg2 , . . . , xgn ) will be called an independent sequence for x of length n (in G) if the following conditions are satisﬁed: / xg1 , . . . , xgi−1 for i = 2, . . . n . xgi ∈ If (xg1 , . . . , xgn ), an independent sequence for x of length n, satisﬁes xg1 , . . . , xgn = xG where xG denotes the normal closure of x in G, then (xg1 , . . . , xgn ) is called an x-basis for xG of length n. We wish to emphasize, that ﬁnite x-bases for xG need not exist, and if they do exist, they may be of diﬀerent lengths. For example, if G is the dihedral group of order 16, then G = x, y | x8 = y 2 = 1, xy = x−1 and yG = y, yx2 , yx4 , yx6 = y, x2 , a dihedral group of order 8. It is easy to 2 see that the sequences (y, y x = yx2 ) and (y, y x = yx4 , y x ) are y-bases for yG of lengths 2 and 3, respectively. Similarly, if G is the inﬁnite dihedral group deﬁned by G = x, y | y 2 = 1, xy = x−1 , then yG = x2 , y is an inﬁnite dihedral group and the sequences 2n

(y, y x

= yx2

n+1

2n−1

, yx

n

= yx2 , . . . , y x = yx4 , y x = yx2 ) 2

are y-bases for yG for n = 0, 1, 2, . . . of lengths 2, 3, 4, . . . , respectively. 1 The first author is grateful to the Department of Mathematics of the University of Hawaii at Manoa for its hospitality and support, while this investigation was carried out.

250

HERZOG, LONGOBARDI, MAJ

We are ready now to deﬁne the basic notion of this survey, namely the length of x ∈ G, denoted by lG (x). If x ∈ G, lG (x) denotes the maximal ﬁnite length of an x-basis for xG , if such a maximum exists. Thus if d ∈ N, then lG (x) = d means that an x-basis for xG of length d exists and all x-bases for xG are of length m ≤ d. This implies, that any independent sequence for x is of length m ≤ d, and if m = d, then the sequence is an x-basis for xG . In the above mentioned examples, lG (y) = 3 if G is dihedral of order 16 and lG (y) does not exist in the inﬁnite dihedral case, even though ﬁnite bases for yG exist in both cases. If x ∈ G and d ∈ N, then lG (x) ≤ d means that lG (x) = k for some 1 ≤ k ≤ d. We deﬁne l(G) ≤ d to mean that lG (x) ≤ d for all x ∈ G. A group G is called of finite length (F L-group in short) if each element of G is of ﬁnite length. If lG (x) is bounded for all x ∈ G, then G is called a group of boundedly finite length (BF L-group in short). Groups satisfying l(G) ≤ n will be called groups of length n and the set of all such groups will be denoted by Jn . Finally, in similarity to the F C-center of G, denoted by F C(G), the set of all F L-elements of G will be denoted by F L(G). In Section II properties of groups belonging to J1 , J2 and J3 are described. Groups belonging to J1 are completely characterized and detailed information is rendered concerning groups in J2 . Much less is known concerning groups belonging to J3 . Section III deals with properties of F L(G), the set of F L-elements in G. It turns out that for rather extensive families of groups we have F L(G) = F C(G). In such cases clearly F L(G) is a subgroup of G. Naturally, the question arises: is that always so? We were able to show, using Ol’shanskiˇi’s geometric groups, that there exist inﬁnite simple groups G, with F L(G) not a subgroup of G. In Section IV we consider groups with all elements of ﬁnite length. Our main results are that for locally graded groups, G is an F L-group if and only if it is an F C-group and, even more surprisingly, G is a BF L-group if and only if it is a BF C-group. We also mention several interesting results concerning groups belonging to Jn for some n ∈ N. In our ﬁnal Section V we list certain open problems arising from our results.

2 Groups with small finite lengths In this section we deal with groups of length 1, 2 and 3. It follows from the deﬁnition that G ∈ J1 if and only if xG = x for each x ∈ G, or equivalently, if and only if x G for all x ∈ G. Thus G ∈ J1 if and only if G is a Dedekind group. Since Dedekind groups were completely characterized by Dedekind and Baer (see [7], Theorem 5.3.7), we have the following Theorem 1 G ∈ J1 if and only if either G is abelian or G = Q8 × E2 × A, where Q8 is the quaternion group of order 8, E2 is an elementary abelian 2-group and A is an abelian group with all its elements of odd order. In Theorem 1 E2 and A may be ﬁnite or inﬁnite.

ELEMENTS AND GROUPS OF FINITE LENGTH

251

The structure of groups in J2 is much more complicated. It follows from the deﬁnition that G ∈ J2 if and only if each x ∈ G satisﬁes either x G or x, xg G for all g ∈ G − NG (x). This family contains very complicated groups, including the inﬁnite simple groups with all proper non-trivial subgroups of a ﬁxed prime order p, called the Tarski Monsters. These strange groups were shown to exist for very large primes (p > 1075 ) by I.A.Rips and A.Yu.Ol’shanskiˇi. In contrast to the situation in the inﬁnite case, ﬁnite groups in J2 are solvable and of small derived length. Groups belonging to J2 were investigated in [1]. For ﬁnite groups, the main result was Theorem 2 Let G be a finite group in J2 . Then G is a solvable group of derived length ≤ 3. This result is the best possible, since SL(2, 3) ∈ J2 and dl(SL(2, 3)) = 3. Moreover, ﬁnite J2 -groups G with Z(G) = 1 were completely determined. Theorem 3 Let G be a finite group and suppose that Z(G) = 1. Then G ∈ J2 if and only if G is a Frobenius group with an elementary abelian kernel P of order p or p2 for some prime p and a cyclic complement D, which is of odd order dividing p + 1 if |P | = p2 . The smallest groups of each type are S3 and A4 , with |P | = 3 and |P | = 22 , respectively. Finite p-groups which belong to J2 are even more restricted. They satisfy Theorem 4 Let G be a finite p-group in J2 . Then dl(G) ≤ 2

and

cl(G) ≤ 3 .

Moreover, if p > 3, then cl(G) ≤ 2. These results are the best possible, since Q16 ∈ J2 , with dl(Q16 ) = 2 and cl(Q16 ) = 3. Finite 3-groups G in J2 with cl(G) = 3 can be almost completely determined (see Theorem 23 in [1]). A rather detailed information is available concerning the commutator subgroup of a ﬁnite p-group in J2 . It was proved in [1] that Theorem 5 Let G be a non-abelian finite p-group in J2 . Then: 1. exp(G ) = p and |G | ≤ p2 if p > 2; and 2. exp(G ) ≤ 4 and |G | ≤ 26 if p = 2. We consider now inﬁnite groups G which belong to J2 . First we exclude groups of Tarski Monster type by imposing on G the very weak condition of being locally graded. By that we mean that every non-trivial ﬁnitely generated subgroup of G has a non-trivial ﬁnite homomorphic image. We were surprised by the fact, that locally graded inﬁnite J2 -groups behave, in certain sense, better than the ﬁnite J2 -groups. In order to explain this remark, we ﬁrst prove the following simple fact:

252

HERZOG, LONGOBARDI, MAJ

Lemma 6 Let G be a group and suppose that G is of prime order. Then G ∈ J2 . Proof Let x ∈ G and suppose that x G and g ∈ G − NG (x). Since G is of prime order we have G ≤ [x, g] ≤ x, xg , whence x, xg G. Thus G ∈ J2 , as claimed. ✷ So the condition |G | = p ∈ P is suﬃcient for non-abelian groups G to belong to J2 . But is that condition also necessary? For non-trivial ﬁnite groups G in J2 with Z(G) = 1 and for non-abelian ﬁnite p-groups in J2 with p > 2 we have |G | = p or p2 , which is close, but not identical, to |G | = p. For inﬁnite locally graded groups, on the other hand, the following results hold (see [1]): Theorem 7 Let G be a finitely generated infinite non-abelian locally graded group. Then G ∈ J2 if and only if |G | is a prime. and Theorem 8 Let G ∈ J2 be a locally graded group. Then the following hold: 1. G is solvable and dl(G) ≤ 3. 2. If G is a torsion group, then it is locally ”finite of derived length at most 3”. 3. If G is not a torsion group, then |G | is either 1 or a prime. 4. If G is torsion-free, then it is abelian. Thus, for inﬁnite non-abelian locally graded groups which either contain an element of inﬁnite order or are ﬁnitely generated, the two conditions G ∈ J2 and |G | = p ∈ P are, surprisingly, equivalent. Moreover, locally graded groups in J2 are solvable of small derived lenth. In particular, ﬁnite simple groups in J2 are abelian. When we drop the ”locally graded” condition, the situation changes completely and the inﬁnite simple Tarski Monsters emerge in J2 . In [1], all non-abelian simple groups in J2 were characterized in the following sense. We deﬁne the Tarski Super Monsters (T SM in short) as the inﬁnite simple groups with all proper subgroups abelian. It follows from results of Obraztsov [5] that there exist Tarki Super Monsters, which are not Tarski Monsters. Using Ol’shanskiˇi’s methods, Obraztsov proved that given a nontrivial ﬁnite or countably inﬁnite group A without involutions and a suﬃciently large ﬁxed odd number (n > 2·1077 will do), there exists a countable 2-generator simple group G containing A, such that every proper subgroup of G is either cyclic of order dividing n or is conjugate to a subgroup of A. Thus, if we take for A an arbitrary non-cyclic ﬁnite or countably inﬁnite abelian group without involutions, then G is a Tarski Super Monster which is not a Tarski Monster. The result proved in [1] was: Theorem 9 An non-abelian simple group G is a J2 -group if and only if G is a T SM . Similarly, if we deﬁne a Perfect Tarski Super Monster (P T SM -in short) as a perfect inﬁnite group with all proper subgroups abelian, then the following theorem holds:

ELEMENTS AND GROUPS OF FINITE LENGTH

253

Theorem 10 A nontrivial perfect group G is a J2 -group if and only if G is a P T SM . Finally we mention the detailed description of non-solvable J2 groups, which was also given in [1]. We denote by Φ(N ) the Frattini subgroup of N . Theorem 11 Let G be a non-solvable J2 -group. Then G has a normal series 1 ≤ K ≤ N ≤ G such that: 1. G/N is a Dedekind group. 2. N is a 2-generated perfect group and N/K is a T SM . 3. K = Φ(N ) = Z(N ). In particular, if G is perfect, then G = N . Finite groups belonging to J3 were investigated in [2]. We proved, without using the Feit-Thompson theorem, that Theorem 12 Let G be a finite J3 -group of odd order. Then G is solvable. We needed the Thompson’s N -paper in order to establish Theorem 13 If G is a finite non-abelian simple J3 -group, then G = A5 .

3 Elements of finite length An element x ∈ G is called an F C-element (x ∈ F C(G) in short), if the size clG (x) of the conjugacy class of x in G is ﬁnite. Since lG (x) denotes the maximal number of independent conjugates of x which generate xG , it is clear that lG (x) ≤ clG (x) and hence x ∈ F C(G) implies x ∈ F L(G). Thus we always have: F C(G) ⊆ F L(G). The converse of this statement does not always hold. For example, if M is a Tarski Monster and x ∈ M − {1}, then lM (x) = 2, while clM (x) = ∞. Thus F L(M ) = M ⊃ F C(M ) = 1. In [4], we addressed the following two questions: 1. For which groups G, F L(G) = F C(G)? 2. For which groups G, F L(G) is a subgroup of G? Since F C(G) is well known to be a subgroup of G, all groups of type (1) are trivially of type (2). However, condition (1) is not necessary for (2) to hold. For example, the Tarki Monsters M are not groups of type (1), but as F L(M ) = M , they are of type (2). The main results of [4] concerning question (1) are: Theorem 14 If G is a solvable group, then F L(G) = F C(G). and Theorem 15 If G is a residually finite group, then F L(G) = F C(G). We also proved a stronger version of Theorem 14: Theorem 16 If xG is solvable for all x ∈ F L(G), then F L(G) = F C(G). The question whether locally graded groups are of type (1) remains open. Concerning question (2) we showed, using a construction of Ol’shanskiˇi in [6], that there exist inﬁnite simple groups G, with F L(G) not a subgroup of G.

254

HERZOG, LONGOBARDI, MAJ

4 Groups with all elements of finite length Groups all of whose elements are of ﬁnite length (FL-groups in short) and groups all of whose elements are of boundedly ﬁnite length (BFL-groups in short) were investigated in [3]. Here again, we tried to compare the above types of groups with the corresponding F C-groups and BF C-groups. It follows immediately from Theorems 14 and 15 that solvable or residually ﬁnite F L-groups are also F Cgroups. Our results in [3] are more speciﬁc and more general, dealing with locally graded groups. We proved: Theorem 17 Let G be a locally graded group. Then G is an F L-group if and only if G is an F C-group. and Theorem 18 Let G be a locally graded group. Then G is a BF L-group if and only if G is a BF C-group. Moreover, we obtained in [3] results concerning groups belonging to Jn for some natural n. First we mention results dealing with ﬁnite groups in Jn . Theorem 19 Let G ∈ Jn be a finite p-group. Then the following properties hold: 2 + 1; and 1. cl(G) ≤ n (n+1) 2 2. |G | is bounded by a function depending only on p and on n.

and Theorem 20 Let G ∈ Jn be a finite solvable group. Then the derived length of G is bounded by a function depending only on n. These results imply Theorem 21 Let G ∈ Jn be a locally solvable group. Then G is a solvable group and the derived length of G is bounded by a function depending only on n.

V A list of open problems concerning elements of finite length Question 1 Does F L(G) = F C(G) hold for locally graded groups? Question 2 Does there exist a function f (n) such that if x ∈ F C(G) and lG (x) = n, then clG (x) ≤ f (n)? Question 3 Does there exist a function g(n) such that if G is a BF C-group and l(G) ≤ n, then clG (x) ≤ g(n) for all x ∈ G? Question 4 Does (2) or (3) hold for solvable groups? Question 5 Does (2) or (3) hold for ﬁnite groups? Question 6 Which ﬁnite 2-groups G in J2 are of class 3? Question 7 Is it true that if G is a 2-group in J2 , then |G | ≤ 24 ?

ELEMENTS AND GROUPS OF FINITE LENGTH

255

References [1] [2] [3] [4] [5] [6] [7]

M.Herzog, P.Longobardi, M.Maj and A.Mann, “On generalized Dedekind groups and Tarski super monsters”, J. Algebra 226 2000 690-713. M.Herzog, P.Longobardi and M.Maj, On an ascending series of types of groups, in Groups-Korea ’98 181-188 Walter de Gruyter Berlin 2000. M.Herzog, P.Longobardi and M.Maj, “Groups with all elements of ﬁnite length”. To appear. M.Herzog, P.Longobardi and M.Maj “Elements of ﬁnite length in groups”. To appear. V.N.Obraztsov “An embedding for groups and its corollaries” Mat. Sb. 180 1989 529-541,560. (Russian) A.Yu.Ol’shanskiˇi The geometry of defining relations in groups Nauka, Moscow 1989. D.J.S.Robinson, A course in the theory of groups, Second Edition, Springer, New York 1996.

LOGGED REWRITING AND IDENTITIES AMONG RELATORS ANNE HEYWORTH∗1 and CHRISTOPHER D. WENSLEY† ∗

Department of Mathematics and Computer Science, University of Leicester, LE1 7RH UK Department of Informatics, Mathematics Division, University of Wales, Bangor, LL57 1UT, UK

†

Abstract We present a version of the Knuth-Bendix string rewriting procedures for group computations and apply it to the problem of computing the module of identities among relators. By lifting rewriting into the appropriate higher dimension we provide a methodology which is alternative and complementary to the popular geometric approach of pictures.

1

Introduction

Combinatorial group theory is the study of groups which are given by means of presentations; these arise naturally in a wide variety of situations including areas as diverse as knot theory [13], geometry [8] and cryptography [1]. One of the fundamental problems in computational group theory is the solution of the word problem for a given presentation. The problem is in general undecidable and consequently a number of diﬀerent approaches have been developed. Amongst the most successful is string rewriting, in particular Knuth-Bendix completion, which attempts to solve the word problem by trying to generate a conﬂuent and Noetherian rewrite system from the presentation. The advantages of this approach are twofold: i) KnuthBendix completion can be successfully applied in a large number of situations and; ii) the concrete nature of string rewriting makes these algorithms relatively easy to implement. Indeed, many computer algebra packages solve word problems in precisely this way [11, 18]. Every presentation has associated with it a CW-complex: a cellular model whose fundamental group is the group given by the presentation. The second homotopy group of the CW-complex is the module of identities among relators. This module has found applications in determining the cohomology groups, the Schur multiplier, the second Fox ideal, extensions of the group and making judgements about the eﬃciency of the given presentation [2]. One of the central problems in this area has been the search for presentations of the module of identities amongst relations. Again, this problem is in general undecidable, and research has typically been concentrated in the following areas. • Geometric: Elements of the module are represented by spherical pictures. Obtaining a generating set is then a geometric problem, solved by selecting 1 Supported by EPSRC grant GR/R29604/01 Kan: A Categorical Approach to Computer Algebra

LOGGED REWRITING AND IDENTITIES AMONG RELATORS

257

a set of canonical pictures and arguing that these generate all other pictures [3, 22]. These methods have been especially successful for group constructions where known collections of pictures can be combined. • Algebraic and Topological: The module of identities among relators is the kernel of the free crossed module on the inclusion of the relators to the free group. A covering crossed resolution can be constructed based upon the Cayley graph and its projection gives information about the identities among relators [6]. This is not an entirely algorithmic method as various technical choices need to be made – one beneﬁt of logged rewriting is that it provides a mechanism for making these choices. • Matrix: The kernel of the crossed module is isomorphic to the kernel of the Whitehead-Reidemeister-Fox derivative which is deﬁned on free modules over the integral group ring. A combination of Todd-Coxeter and Kannen-Bachen algorithms have been used to ﬁnd a basis for the module of identities among relators for some presentations of groups up to order 30 [10]. Given the successful application of string rewriting to the computation of a group from a group presentation the natural question to ask is whether string rewriting can also be applied to the problem of computing the module of identities among relations. Our crucial insight is that an identity among relators is a record of a rewrite resulting from the resolution of critical pairs. Hence we develop a reﬁnement of string rewriting, called logged rewriting which analyses not just when a string rewrites to another string, but also the structure of the rewrite. By suitably reﬁning the Knuth-Bendix algorithm we obtain a procedure for generating a complete logged rewriting system. We show that this procedure terminates whenever the standard Knuth-Bendix completion on the underlying rewriting system terminates (Section 2). Thus, we obtain from the critical pairs of the complete logged rewriting system a complete set of generators for module of identities among the relators of the presentation. As with the geometric methods this approach does not require the group itself to be ﬁnite (Section 3). This approach is appealing from both the theoretical and practical perspectives. On the theoretical level, just as the module of identities amongst relators is a higher dimensional version of the group, so logged rewriting is the analogous higher dimensional version of string rewriting. That is, the shift of levels in the rewriting is directly related to the shift of levels in the underlying computational domain. When compared to the topological work of Brown and Razak Salleh we can see that rewriting determines the ﬁrst level of the contracting homotopy for their covering crossed resolution and logged rewriting determines the second level (Section 4). From the practical perspective, the use of logged rewriting means that these procedures are easily implemented; we chose to use GAP, the system for computational discrete algebra popular for computational group theory [12]. They will be made available as a GAP4 share package IdRel. In the meantime we are happy to allow interested researchers to test our current prototype. We would like to thank Ronnie Brown, Neil Ghani and Mike Johnson for many helpful discussions.

258

2 2.1

HEYWORTH, WENSLEY

Definitions for logged rewriting String rewriting systems

For the basic notions of string rewriting systems we refer the reader to [4] or [9], including here only suﬃcient detail to ﬁx our notation. For A a set we denote by F (A) the free group on A, and by A∗ the free monoid on A, both having the empty word λA as identity element. We denote the unreduced length of a word w in F (A) or A∗ by |w|. A rewrite system on A∗ is a subset R ⊆ A∗ × A∗ and generates a reduction relation →R = {(ulv, urv) : (l, r) ∈ R, u, v ∈ A∗ } on A∗ . If (w1 , w2 ) ∈ →R then we write w1 →R w2 , and say that w1 reduces ∗ to w2 . The reﬂexive, transitive closure of →R is written →R , and the reﬂexive, ∗ symmetric, transitive closure ↔R coincides with the congruence =R . The monoid with presentation monA, R is the factor monoid A∗ / =R . We say that R is a rewrite system for a group G on A if G ∼ = A∗ /=R . A well-ordering > on A∗ is an ordering such that no inﬁnite sequence w1 > w2 > · · · exists. An ordering > on A∗ is admissible if y > z implies uyv > uzv for all u, v ∈ A∗ . A rewrite system R is compatible with an admissible well-ordering if l > r for all (l, r) ∈ R. Two rules (l1 , r1 ), (l2 , r2 ) of a rewrite system R overlap if for some u1 , u2 , v1 , v2 ∈ A∗ we can write w = u1 l1 v1 = u2 l2 v2 where either u1 = v1 = λA or u2 = v2 = λA (a type 1 overlap) or, u1 = v2 = λA or u2 = v1 = λA (a type 2 overlap). Applying the two rules, we obtain the critical pair (u1 r1 v1 , u2 r2 v2 ) resulting from the overlap. ∗ This critical pair can be resolved by R if there exists z such that u1 r1 v1 →R z ∗ and u2 r2 v2 →R z. If a rewrite system R is compatible with an admissible well-ordering >, then →R is Noetherian. A rewrite system generating a Noetherian reduction relation is complete if and only if all of its critical pairs can be resolved. The Knuth-Bendix completion procedure attempts to transform a given ﬁnite, Noetherian rewrite system into an equivalent complete one. It ﬁnds all the overlaps and attempts to resolve the resulting critical pairs by reducing both sides with respect to →R . If a pair fails to resolve, then the reduced critical pair is added to R and the search for overlaps begins again. If the procedure terminates then the rewrite system is complete. 2.2

From group presentations to identities among relators

This section deﬁnes the module of identities among relators which is the next structure associated to a group presentation after the group itself. Let P = grpX, ω : R → F (X) be a presentation of a group G, where the set R provides labels for the relators, and the function ω identiﬁes the relators as words in the free group. The rewrite system associated to P is deﬁned in the usual way.

LOGGED REWRITING AND IDENTITIES AMONG RELATORS

259

¯ = {x+ : x ∈ X} {x− : x ∈ X}. Then deﬁne Speciﬁcally, let X ¯ , R± = {(x x− , λX¯ ) : x ∈ X} ω (ρ), λX¯ ) : ρ ∈ R} , Rinit = {(¯ Rinit∗ = R± ∪ Rinit . ¯ ∗ → F (X) and the canonical section Here we use the monoid morphism ν : X ∗ ¯ ∗ which are ¯ µ : F (X) → X which maps reduced words of F (X) to words in X ¯ = µ ◦ ω which reduced with respect to R± , Then we can deﬁne the composite ω ¯ ∗ . We note that the free group identiﬁes the relators as words in the free monoid X ¯ ∗ / =R , giving ¯ ∗ / =R and G is isomorphic to X F (X) is isomorphic to to X ± init∗ ∗ ¯ that Rinit∗ is a rewrite system for G on X . We seek to express reductions as consequences of the original group relators. The key idea is that ‘consequences of relators’ are products of conjugates of relators, or elements of the free crossed module on ω : R → F (X). Recall that in general, a crossed module consists of two groups C and F with a morphism δ : C → F and an action of F on C such that CM1) CM2)

δ(cv ) = v −1 δ(c)v , dδc = c−1 dc .

We now deﬁne Y := R × F (X) and Y¯ = {y + : y ∈ Y } {y − : y ∈ Y }. It will be convenient to write (ρ, u) ∈ Y¯ as (ρ )u . A consequence of relators is an element of the free monoid Y¯ ∗ and has the form (ρ11 )u1 · · · (ρnn )un

where

ρi ∈ R, ui ∈ F (X), i ∈ {+, −}.

The identity of Y¯ ∗ is denoted λY¯ and corresponds to the consequence of applying no relators. It is convenient to deﬁne the ‘inverse’ of a consequence of relators n un · · · (ρ−1 )u1 . The free group F (X) c = c+ = (ρ11 )u1 · · · (ρnn )un to be c− := (ρ− n ) 1 ¯ acts on Y , by (ρ, u)v = (ρ, uv)

and

((ρ )u )v = (ρ )uv ,

v ∈ F (X).

¯ ∗ to be that induced by We deﬁne the monoid morphism δ¯ : Y¯ ∗ → X ¯ + )u ) = µ(u)− (¯ ω ρ)µ(u) , δ((ρ

¯ − )u ) = µ(u)− (¯ δ((ρ ω ρ)− µ(u) .

Some consequences of relators one would naturally consider to be equivalent, for example applying a relator and its inverse should amount to the same as doing nothing. This is formalised by factoring Y¯ ∗ by the congruence generated by the inverse rules and the second crossed module law (the Peiﬀer rules): ¯ P = {(y y − , λY¯ ) : y ∈ Y¯ } ∪ {(y − z η y , (z η )δy ) : y , z η ∈ Y¯ } .

If c =P d, then we will say that the consequences of relators c and d are Peiﬀer equivalent. It is usual to identify consequences of the relators as elements of the free crossed module (δ2 : C(R) → F (X)) on the function ω : R → F (X) as in [5]. Here it is given by:

260

HEYWORTH, WENSLEY

• C(R) = (Y¯ ∗ /=P ), where [c] denotes the class of c ∈ Y¯ ∗ under =P , • the action of F (X) on C(R) is given by [c]v = [cv ] , ¯ . • δ2 [c] = ν δc A consequence of relators which maps to the identity in the free group is known as an identity among relators. The equivalence classes of identities among relators for a presentation P of a group G form the ZG-module of identities among relators Π2 (P). Our description of this crossed module, in terms of monoids and congruences is appropriate for rewriting techniques but is equivalent to the traditional description using groups and Y-sequences, detailed expositions of which may be found in [5, 15]. 2.3

Logged rewriting and logged Knuth-Bendix completion

We will now deﬁne logged rewriting systems and give the rewriting algorithms. A logged rewriting system for a presentation records each rewrite as a consequence of relators. Definition 2.1 (Logged rewriting system) A logged rewriting system for the group presentation P of G is a set of triples L = {(l1 , c1 , r1 ), . . . , (ln , cn , rn )} , where c1 , . . . , cn ∈ Y¯ ∗ and RL = {(l1 , r1 ), . . . , (ln , rn )} is a rewrite system for G on ¯ such that X ¯ i )ri for 1 i n . li =R (δc ±

The system L is complete if its underlying string rewriting system RL is complete. The initial logged rewriting system Linit∗ of P is given by ¯ , L± (P) = { (x x− , λY¯ , λX¯ ) : x ∈ X} ω ρ, ρ+ , λX¯ ) : ρ ∈ R } , Linit (P) = { (¯ Linit∗ (P) = L± (P) ∪ Linit (P) . Note that the rules in L± (P) deﬁne F (X); this is necessary as we are computing ¯ ∗ . The logged parts of the rules in L± (P) are a quotient of F (X) from inside X ¯ the identity in Y : they are not relevant to the group presentation so application of these rules will eﬀectively not be recorded. ¯ ∗ , there Algorithm 2.2 Given a logged rewriting system L and a word w ∈ X ∗ ¯ , together with is an algorithm that will ﬁnd an irreducible word irr(w) ∈ X ∗ a consequence of relators log(w) ∈ C(R), such that w →RL irr(w) and w =R± ¯ δ(log(w))irr(w).

LOGGED REWRITING AND IDENTITIES AMONG RELATORS

261

Logged rewriting ¯ 0 )w0 . R1: (Initialise) Set i = 0, w0 = w, d0 = λY¯ . Clearly w = (δd ¯ ∗ such that R2: (Search) Determine whether there exist (li , ci , ri ) ∈ L and ui , vi ∈ X wi = ui li vi (i.e. wi is reducible with respect to RL ). R3: (Rewrite and Record) If the search was successful then set wi+1 = ui ri vi and di+1 = u−

di ci i ; then increment i. R4: (Loop) Repeat Search (R2), Rewrite and Record (R3) until Search fails. R5: (Terminate) Output irr(w) = wi and log(w) = di .

¯ i )wi : It is easily veriﬁed at each stage of the algorithm that w = (δd −

¯ i )(δc ¯ ui )ui ri vi = (δd ¯ i )ui (δ(c ¯ i ))ri vi = (δd ¯ i )ui li vi = (δd ¯ i )wi . ¯ i+1 )wi+1 = (δd (δd i ∗

We denote this logged reduction by: “w →L irr(w) by log(w)”. Note that in general neither irr(w) nor log(w) need be unique, but when L is complete then irr(w) is the unique irreducible word in the equivalence class of w under the congruence generated by L. Also, if we choose a strategy for reduction (such as leftmost, with the set of rules being an ordered list) then we can guarantee that the log resulting from the reduction of a given word will be the same each time. We now extend the standard process of completing a rewrite system, which adds unresolved critical pairs resulting from overlaps, by recording the logged part of each rule used. This logged Knuth-Bendix procedure results in a rewrite system which stores information on how the rules were constructed. The information is not unique because a rule may be derived in many diﬀerent ways as we shall see in Section 3. Algorithm 2.3 (Logged Knuth-Bendix completion) Given a logged rewriting system L for a group presentation P with an ordering > on the strings, this algorithm attempts to complete the rewrite system. When using rewrite system Lj with input w in Algorithm 2.2, we denote the output by irrj (w), logj (w). In order to ﬁx a search order, we consider Lj as a list rather than as a set. The following diagram illustrates the main step in the algorithm: u−

c1 1

u1 r1 v1

u1 l1 v1 = l2 u2 c2

r2 v2

b1

b2

z1

z2 −

− u In the case z2 > z1 , we would add the rule (z2 , b− 2 c2 c1 b1 , z1 ).

262

HEYWORTH, WENSLEY Logged Knuth-Bendix completion

K1: (Initialise) Set j = 1 and L0 = L(P). K2: (Find critical pairs) Set C = ∅. Take each pair (l1 , c1 , r1 ), (l2 , c2 , r2 ) from Lj in turn, and consider the |l1 |+|l2 |−1 juxtapositions of l2 against l1 . We thus ﬁnd all possible overlap words such that u1 l1 v1 = l2 v2 where (type 1) v2 = λX¯ , or (type 2) v1 = λX¯ . For each pair, put z1 = irrj−1 (u1 r1 v1 ), b1 = logj−1 (u1 r1 v1 ), z2 = irrj−1 (r2 v2 ) and − b2 = logj−1 (r2 v2 ). If z1 = z2 , add (z2 , b2 − c2 − (c1 + )u b1 + , z1 ) to C. K3: (Add new rules) Set Lj = Lj−1 . For each (z2 , d, z1 ) in C, either add the logged rule (z2 , d+ , z1 ) to Lj if z1 < z2 , or add the logged rule (z1 , d− , z2 ) to Lj if z1 > z2 . K4: (Inter-reduction) For each (l, c, r) ∈ Lj , test whether l and r reduce to the same word with respect to Lj \ (l, c, r). If they do, then remove (l, c, r) from Lj . K5: (Loop) Repeat: increment j, Find (K2) unresolvable critical pairs, Add (K3) logged rules, and Remove (K4) redundant rules until all critical pairs are resolved. K6: (Terminate) Output the ﬁnal logged rewriting system Lcomp (P) = Lj .

Note that for each j, the triples (l, c, r) which are added to Lj satisfy the require¯ It can be seen immediately from the description of the algorithm ment l =R± (δc)r. that, if the middle term of each triple is omitted, the standard completion Rcomp of the rewrite system is obtained, which gives the following result. Lemma 2.4 The algorithm for completing a logged rewriting system L terminates if and only if the Knuth-Bendix algorithm for the underlying rewrite system RL terminates. 2.4

Example of logged rewriting

We illustrate the procedures with the following example. The quaternion group Q8 is presented by Q = grpX, ω : R → F (X) where X = {a, b}, R = {ρ1 , ρ2 , ρ3 , ρ4 }, ω : ρ1 → a4 , ρ2 → b4 , ρ3 → abab−1 , ρ4 → a2 b2 . The initial logged rewriting system is Linit∗ (Q) = { (a+ a− , λY¯ , λX¯ ), (a− a+ , λY¯ , λX¯ ), (b+ b− , λY¯ , λX¯ ), (b− b+ , λY¯ , λX¯ ), +4 + + + + − + +2 +2 + (a+4 , ρ+ ¯ ), (b , ρ2 , λX ¯ ), (a b a b , ρ3 , λX ¯ ), (a b , ρ4 , λX ¯ ) }. 1 , λX

Starting the logged completion procedure, we ﬁnd the ﬁrst overlap in the word a− a+4 , which contains the two left hand sides a− a+ , a+4 , giving the rule (a+3 , a+ − (ρ+ 1 ) , a ). As we shall see in Remark 3.10, the logged part may be simpliﬁed to + (ρ1 ), since we could have used overlaps in a+4 a− . The next three reductions, a+ − on words a− a+ b+ a+ b− , a− a+2 b+2 , and b− b+4 , give rules (b+ a+ b− , (ρ+ 3 ) , a ), + a+ − + b+ − + +2 +3 (a b , (ρ4 ) , a ), and (b , (ρ2 ) , b ) respectively. The next overlap is the ﬁrst with two non-inverse rules and the new logged rewrite rule as deﬁned in

LOGGED REWRITING AND IDENTITIES AMONG RELATORS −2

263

−2

+ a + a , a+2 ) so c = (ρ− . This may be veriﬁed by: step K3 is (b+2 , (ρ− 1 )(ρ4 ) 1 ) (ρ4 ) −2 ∗ − + a +2 −4 +2 +2 +2 −2 +2 ¯ δ((ρ1 )(ρ4 ) ) a = (a ) (a (a b )a ) a →L± b+2 . Continuing with the logged completion for Q, a total of 44 rules are formed, of which 22 become redundant. A complete logged rewriting system Lcomp (Q) is obtained, containing the list of 16 rules ψi = (li , ci , ri ) shown in Table 1.

ψi ψ1 ψ2 ψ3 ψ4 ψ5 ψ6 ψ7 ψ8 ψ9 ψ10 ψ11 ψ12 ψ13 ψ14 ψ15 ψ16

li a+ a− a− a+ b + b− b − b+ b+ a+ b+2 b+ a− a− b+ a−2 a− b− b− a+ b− a− b−2 a+3 a+2 b+ a+2 b−

ci λY¯ λY¯ λY¯ λY¯ a+ (ρ− ) (ρ+ )a− (ρ+ 3) 1 4 + a−2 (ρ− 1 ) (ρ4 ) (ρ− 3) + a− (ρ− 1 ) (ρ4 ) (ρ− 1) − a−2 + a+ b− (ρ− (ρ+ (ρ− 1 ) (ρ2 ) 4 ) (ρ3 ) 3) − − + a (ρ4 ) (ρ3 ) a+ b+ (ρ− 3) − (ρ4 ) (ρ+ 1) (ρ+ 4) a−2 (ρ+ ) (ρ− 4) 1

ri λX¯ λX¯ λX¯ λX¯ a+ b− a+2 a+ b+ a+ b− a+2 a+ b+ a+ b+ a+ b− a+2 a− b− b+

Table 1. Complete logged rewriting system Lcomp (Q) So, for example, a+ b+ b+ a+ reduces to λX¯ as follows: ∗

−

∗

−

a+ b+ b+ a+ →Lcomp (Q) a+4 (by c6 a ) →Lcomp (Q) a+ a− (by c14 a ) ∗

→Lcomp (Q) λX¯ (by λY¯ ) −

−3

−

+ a a a = (ρ+ )a . with the logged information: log(a+ b+ b+ a+ ) = (ρ− (ρ+ P 1 ) (ρ4 ) 1) 4

3

+

Resolving critical pairs and computing identities among relators

The identities among the relators of a group presentation P are the consequences of relators (elements of C(R)) which are in the kernel of δ2 . The ZG-module of identities among relators is denoted Π2 (P). If a word can be reduced in two diﬀerent ways then the two reductions are referred to as a critical pair. Critical pairs (whether or not their divergence can be resolved) characterise the property of conﬂuence. They also yield information on how the deﬁning relators of the group can be combined in a way which creates a ‘loop’ of rewriting, or identity.

264

HEYWORTH, WENSLEY

Lemma 3.1 (Digraphs and identities from pairs of reduction sequences) Given two words w and z, any pair of logged reductions from w to z yields an identity among the relators. Proof Let c1 and c2 be the logs of the two diﬀerent reductions of w to ¯ 1 )z =R another word z. By the deﬁnition of logged rewriting w =R± (δc ± + − ¯ ¯ (δc2 )z, so δ(c1 c2 ) =R± λX¯ . Thus we have a digraph (shown right) and − an associated identity c+ 1 c2 ∈ Π(P).

w c1

c2

z 2

We now consider more carefully what happens when the two reduction sequences result from a critical pair and its resolution. Lemma 3.2 (Overlaps and Identities) If (l1 , c1 , r1 ) and (l2 , c2 , r2 ) are rules of a complete logged rewriting system L, which may both be applied to a word w then i) if the rules do not overlap then resolution of the critical pair yields the trivial identity, ii) if the rules do overlap then there is a resolution of the critical pair which yields a conjugate of the identity given by a resolution of the overlap word. Proof In the ﬁrst case the rules do not overlap − c1 x ¯ ∗ such on w. Then there exist x, y, z ∈ X that w = xl1 yl2 z and the logged reductions are − w →L xr1 yl2 z by c1 x , xr1 yl2 z →L xr1 yr2 z xr1 yl2 z − − by c2 (xr1 y) , w →L xl1 yr2 z by c2 (xl1 y) and − x xl1 yr2 z →L xr1 yr2 z by c1 . The digraph − c2 xr1 y is shown on the right; it yields the identity − − x− − (xl1 y)− which is Peiﬀer c1 x c2 (xr1 y) (c− 1 ) (c2 ) equivalent to λY¯ . In the second case the rules do overlap on − c1 (u1 ) ¯ ∗ such w. Then there exist u1 , v1 , v2 , x, y, z ∈ X that w = xyz and either y = u1 l1 v1 = l2 or y = u1 l1 = l2 v2 . In either case we can write xu1 r1 v1 y = u1 l1 v1 = l2 v2 and the logged reductions of − b1 y are y →L u1 r1 v1 by c1 u1 and y →L r2 v2 by c2 . By completeness, there are logged reduc∗ ∗ ˆ by b1 and r2 v2 →L w ˆ by tions u1 r1 v1 →L w − − − u b2 such that ι = c1 1 b1 (b2 )(c2 ) is an identity − c1 (u1 x) – the digraph is shown the right. The critical pair of reductions on w are w →L xu1 r1 v1 z by − − c1 (u1 x) and w →L xr2 v2 z by c2 x . They can be xu1 r1 v1 z − ∗ ˆ by b1 x and by resolved by xu1 r1 v1 z →L xwz ∗ x− ˆ by b2 . The digraph is shown xr2 v2 z →L xwz − b1 x − on the right and the associated identity is ιx .

w

c2 xl1 y

λY¯ xr1 yr2 z

w

xl1 yr2 z −

c1 x

c2

r2 v2

ι w ˆ xwz −

ιx

xwz ˆ

−

b2

−

c2 x

xr2 v2 z −

b2 x

2

LOGGED REWRITING AND IDENTITIES AMONG RELATORS

265

Computing identities among relators using logged rewriting is possible because the overlaps yield the only interesting identities - all others are either trivial (the ﬁrst digraph above) or can be obtained by from all the identities resulting from resolving overlaps (second digraph above) by combining overlap identities, their inverses and their conjugates (third digraph above). Theorem 3.3 (Identities among relators) Let Lcomp (P) be a ﬁnite complete logged rewriting system for a group presentation P. Then the critical pairs of L = Linit∗ (P) ∪ Lcomp (P) determine a complete set of ZG-module generators for Π2 (P). Before proving the theorem we present the algorithm for computing the identities from L and some preliminary results. Algorithm 3.4 Given the initial and complete logged rewriting systems for a group presentation, this algorithm ﬁnds all critical pairs and computes a set of identities. Identities from critical pairs I1: (Initialise) Let C(P) be the set of all critical pairs obtained by comparing the logged rules of Linit∗ (P) with Lcomp (P), together with all the critical pairs of Lcomp (P). Set I(P) = ∅. I2: (Resolve and compute) Remove the ﬁrst critical pair from C(P) and resolve it to − ˆ Add b+ obtain two diﬀerent logs b1 , b2 of reductions of the overlap word w to w. 1 b2 to I(P). I3: (Loop) Repeat Resolve (I2) and Compute (I3) until no critical pairs remain in C(P). I4: (Terminate) Output I(P).

Lemma 3.5 (Digraph of reduction sequences) For any pair of reduction sequences, there exists a ﬁnite collection of digraphs D1 , . . . , Dt resulting from overlaps (as in Lemma 3.2) such that the digraph of the original pair of reduction sequences can be recovered. Proof Given two reduction sequences w → w1 → · · · → wm → λX and w → wm+1 → · · · → wn → λX , we deﬁne a digraph D. The vertices are the distinct words occurring in these sequences, and there is an edge labelled ci from wi to wj , if wi → wj is a reduction step with the log ci in one of the two given reduction sequences. The pair of reduction sequences yield the identity ι in the way described in Lemma 3.2. We now deﬁne a ﬁnite collection of digraphs Di (pictorially, this can look like subdivison of D into small conﬂuence diagrams, the proof was originally ¯ ∗. phrased in ‘diamonds’). Note that the vertices are ordered with respect to > in X

266

HEYWORTH, WENSLEY

We note ﬁrst that Lcomp (P) is ﬁnite, so there are only ﬁnitely many rules which can be applied; secondly, any ﬁnite word can only be reduced in a ﬁnite number of ways; ﬁnally, the system is Noetherian, so there are no inﬁnite reduction sequences. This means that the procedure will terminate, giving a collection of 2 digraphs D1 , . . . , Dt which all result from overlaps as in Lemma 3.2. Constructing a collection of covering digraphs D1: (Initialise) Given D be as deﬁned above, set i = 1. D2: (Find a critical pair) Consider the maximum vertex vi of D. If vi is the source of two distinct arrows of D then we consider the corresponding two reductions vi → vi,1 by bi,1 and vi → vi,2 by bi,2 . The critical pair can be resolved since Lcomp is a complete rewrite system, and we use a strategy so that we will always reduce a word in the ∗ ∗ same way. Then we have vi,1 →L zi by di,1 and vi,2 →L zi by di,2 . D3: (Deﬁne Di ) Deﬁne Di to be the digraph consisting of the four vertices and four arrows considered in this step (these may not always be distinct: for example if vi,1 = zi ). Note that at least one of the edges bi,1 or bi,2 will be from the original sequences. / {cm+1 , . . . , cn } and bi,2 , di,2 ∈ / {c1 , . . . cm }, We make the convention that bi,1 , di1 ∈ in order to orientate the identities which will later be associated with these digraphs. D4: (Update D) Remove the vertex vi from D and remove the two edges labelled bi,1 and bi,2 . If zi is not already a vertex of D then add it to the digraph. If the edges di,1 : vi,1 → z1 or di,2 : vi,2 → zi are not in D then add them. D5: (Loop) Increment i and repeat step K2. D6: (Terminate) The algorithm terminates when there are no vertices which are the source of more than one arrow. D7: (Output) A ﬁnite set of digraphs D1 , . . . , Dt is output.

The key feature of the digraphs Di is that their edges can be matched, using the convention we chose. If we construct the digraph created by all the arrows bi,1 , di1 and the inverses of the arrows bi,2 and di,2 , and then where we have an arrow and an inverse arrow cancel them – we obtain the digraph which is the ﬁrst sequence of reductions and the inverse of the second – an identity loop. The next Lemma shows what this means in terms of the identities among relators, as we ﬁll in the digraphs. First we recall the Primary Identity Property which helps us to identify when identities are Peiﬀer equivalent (see [5]). Definition 3.6 (Primary identity property) An identity (ρ11 )u1 (ρ22 )u2 · · · (ρrr )ur satisﬁes the Primary Identity Property if the set {1, . . . , r} can be partitioned into pairs (i, j) such that for each pair ρi = ρj , i = −j , and ui =R uj in F (X). Any identity which satisﬁes this property is Peiﬀer equivalent to λY¯ .

LOGGED REWRITING AND IDENTITIES AMONG RELATORS

267

Lemma 3.7 Let D and D1 , . . . , Dt be as in Lemma 3.5. For each digraph Di , deﬁne − the identity ιi = bi,1 ci,1 c− i,2 bi,2 . The product of the identities is Peiﬀer equivalent to the identity ι associated with the original digraph D. Proof Consider the product ι1 · · · ιt . All of the logs which label edges which were not in the original digraph D will occur exactly twice and all of the logs which label the original edges of the digraph occur exactly once. This follows from the algorithm. Moreover, the logs which occur twice will be raised to a positive power in one occurrance and a negative power in the other. By pairing the duplicated logs and applying the Primary Identity Property we ﬁnd that Thus ι1 · · · ιt is Peiﬀer equivalent to the identity ι associated with the original digraph. 2 Proof of Theorem 3.3 Let I(P) be the set of identities among relators that results from the critical pairs of L (see Algorithm 3.4). Let [ι] ∈ Π2 (P). We will prove that ι is Peiﬀer equivalent to a product of conjugates of the identities and inverses of identities of I(P). ¯ We have two logged reduction sequences Set w = δι. ∗

w →L λX¯ by ι

∗

and w →L± λX¯ by λY¯ .

In other words w is reducible to λX¯ in at least two ways (assuming R = ∅) – by repeatedly cancelling x+ x− and x− x+ pairs or by rewriting each element (ρ )u in the ¯ ρ → λX¯ by ρ+ and (¯ ω ρ)− → λX¯ Y-sequence to λX¯ in turn, using the logged rules ω − + − − + by ρ , followed by the inverse rules to cancel x x and x x pairs. From the two reduction sequences, using the algorithm of Lemma 3.5 we can construct a collection of digraphs D1 , . . . Dt whose associated identities ι1 , . . . , ιt are elements of I(P). By Lemma 3.7 the product ι1 · · · ιt is Peiﬀer equivalent to ι as required. 2 Remark 3.8 (Filling in the Diamond Lemma) The proof is like a higher dimensional version, or a tiling, of the proof of the Diamond Lemma, thinking of the identities among relators as ﬁllers for the digraphs which are ‘diamond shaped’ conﬂuence diagrams. Any identity among relators ι can be expressed as a conﬂu¯ to λ ¯ ) which can then be ﬁlled with the ence diagram (two ways of reducing δ(ι) X identities from I(P) because at each point where the word may be reduced in more than one way there is an overlap identity. This any identity is a product (in many diﬀerent ways) of conjugates of overlap identities. Remark 3.9 (Sesquicategory models) This result is closely associated with the sesquicategory model [23] of the rewriting system. The fact that we work with a group presentation means that the 2-cells are identities among relators. This idea is considered in more detail in [16], where identities among the relations for monoid presentations are considered. ¯ ∗ is the smallest initial subword v of w Recall that the root of a word w ∈ X ¯ ρ there are overlaps in the such that w = v m for some m ∈ Z+ . When w = ω words v −1 v m and v m v −1 , giving two logged critical pairs (v m−1 , (ρ+ )v , v −1 ) and

268

HEYWORTH, WENSLEY

(v m−1 , (ρ+ ), v −1 ), so we obtain the root module identity ιρ = (ρ+ )v (ρ− ). Note that, if w = v, then [ιρ ] = [λY¯ ]. Remark 3.10 (Simplification of identities) The element log(w) is constructed as an element of Y¯ ∗ , the free pre-crossed module on ω. Since we are working in the crossed module C(R) we may choose a simpler representative from [log(w)], by cancelling inverse pairs or by using the Peiﬀer rules or the root module identities. So if (l, c, r) ∈ L(P) and c contains c1 as a subsequence, then applying one of the simpliﬁcation rules S1: (ρ )v

+

=P ρ and (ρ )v

−

=P ρ , where v is the root of ωρ, −

−

S2: (ρ11 )u1 (ρ22 )u2 =P (ρ22 )u2 (ρ11 )u1 u2 (¯ωρ2 ) 2 u2 or (ρ22 )u2 u1 (¯ωρ1 )

−1 u 1

(ρ11 )u1 ,

to c1 gives an alternative logged rewriting system for P. We note that the identity list I(P) may be simpliﬁed by deleting duplicates and conjugates; by applying the two simpliﬁcation rules; and by searching to see if one identity occurs as a subsequence of another.

4

Computing a crossed resolution

Brown and Razak Salleh have given in Theorem 1.1 of [6] a generating set I for Π2 (P). This construction requires a morphism k1 from the free groupoid on the Cayley graph of P to the free crossed module on P. We show how such a k1 may be obtained from the logged information in L(P), and Algorithms 2.2 and 2.3 which implement these constructions. Identities among the relators in P are ∗ found by taking suitable words w such that there is a logged reduction w →L(P) λX¯ by log(w) where [log(w)] is one of the required identities. The point here is that, for ﬁnite groups, the work of Brown and Razak Salleh gives an alternative way to obtain a complete set of generators for Π2 (P). Logged rewriting, when applied here make the above theory algorithmic, also capturing the information necessary for their higher constructions. Logged rewriting systems can be used alone, as described in the previous section, to obtain a ﬁnite set of generators for Π2 (P) when the rewrite system can be completed, or if the group is ﬁnite, they can be used to implement the procedure described in [6] which not only computes a set of generators for Π2 (P), but also provides information about the construction of a small crossed resolution for the group. Consider the short exact sequence C(R)

δ2

F (X)

φ

G

1

where δ2 : C(R) → F (X) is the free crossed module as before and φ : F (X) → G is the factor morphism. By exactness, any element of F (X) that represents the identity in G is expressible as a (non-unique) product of conjugates of relators. logged rewriting systems provide a method for doing this. The reader is referred to [6] for

LOGGED REWRITING AND IDENTITIES AMONG RELATORS

269

details of contracting homotopies and crossed resolutions. Here we merely use the formulae and results of that paper. The main result we use is that a complete set of generators for the module of identities among relators may be obtained from the set of relator cycles of the Cayley graph, by deﬁning particular functions and applying them to the relator cycles. The theorem quoted below holds in general, but for inﬁnite groups will generate an inﬁnite number of identities among relations, which is in general very diﬃcult to handle. Therefore, we restrict our attention to ﬁnite groups and so we may assume that the rewrite system completes, with respect to a short-lex ordering. Note that, assuming elements of the group G are given by a representative word in F (X), then a complete rewrite system for G deﬁnes not only a normal form function N on F (X), but also a section σ of θ : F (X) → G. of the presentation P = grpX, ω : R → F (X) has the The Cayley graph X are written as pairs [g, x ], where g is elements of G as its vertices. Edges of X the source vertex, x is a monoid generator identiﬁed with the edge label, g = gx is the target vertex, and the direction is chosen so that σ(g) < σ(g ) with respect are taken to be the [g, x+ ] with no to short-lex. Note that in [6] the edges of X requirement that g < g , since the group elements, rather than normal forms are of all paths on X the inverse of [g, x ] is [g , x− ] used. In the free groupoid F (X) and paths are written [g, u] : g → gφ(u) for g ∈ G, u ∈ F (X). We now quote Theorem 1.1 of [6] which deﬁnes a generating set of identities among relators. Theorem 4.1 The module Π2 (P) is generated as a ZG-module by elements −

sep[g, ρ+ ] = (ρ− )(σg) (k1 [g, ω ¯ ρ]) for all g ∈ G, ρ ∈ R, where i) σ : G → F (X) is a section of the quotient morphism φ : F (X) → G, to the group C(R), ii) k1 is a groupoid morphism from F (X) iii) δ2 k1 [g, x ] = (σg) x (σ(g(φx )))− for all [g, x ] ∈ X. The identities sep[g, ρ+ ] may be seen as separation elements in the geometry of the Cayley graph with relators. Our aim is to show that a logged rewriting system for a presentation provides constructions of the functions σ and k1 satisfying the conditions given in the theorem, thus making possible the computation of a generating set for Π2 (P). Theorem 4.2 (Separation morphism) If the group presentation P has a complete logged rewriting system L(P) then the logged information determines a mor → C(R) satisfying conditions ii) and iii) of Theorem 4.1 so that phism k1 : F (X) the elements sep[g, ρ+ ] for all g ∈ G and ρ ∈ R form a complete set of generators for Π2 (P) as a ZG-module.

270

HEYWORTH, WENSLEY

Proof First, let σ be the section of φ given by the complete rewrite system and let and with source N be the normal form function. Now consider the edge [g, x ] ∈ X vertex g and target vertex g = gφ(x ). Observe that σ(g)x ≥ N (σ(g)x ) according to our deﬁnition of the graph, above. So we have a logged rule (gx , c(g, x ), g ) in L(P), and we therefore set k1 [g, x ] = c(g, x ). These choices induce a groupoid → C(R), and δ2 k1 [g, x ] = (σg)x (σg )− by the deﬁnitions of morphism k1 : F (X) logged rewriting. Therefore, by Theorem 4.1, the elements sep[g, ρ+ ] for g ∈ G, ρ ∈ 2 R are a complete set of generators for Π2 (P). We now present an algorithm for computing a set of generating identities among relators for a group presentation. Algorithm 4.3 (Identities among relators) Given a presentation P= grpX, ω : R → F (X) of a ﬁnite group G, a set of consequences of relators is determined whose Peiﬀer equivalence classes generate Π2 (P) as a ZG-module. Identities from Cayley graph cycles C1: (Logged rewriting system) Apply Algorithm 2.3 to obtain the completion Lcomp (P) from the logged rewriting system Linit (P). Let L and N be the log and normal form functions determined by Algorithm 2.2 using Lcomp (P). C2: (Cayley graph) The Cayley graph is represented by a list of edges, which are pairs ¯ (with respect to Lcomp (P)), x ∈ X, ¯ and [g, x ] where g is an irreducible word in X g < g(φx). C3: (Deﬁnition of k1 ) The map k1 is deﬁned on the edges by k1 [g, x ] = [λY¯ ] if gx is irreducible with respect to Lcomp (P), and by the logged rule (gx , k1 [g, x ], g ) ∈ L(P) otherwise. C4: (Determination of identities) All pairs [g, ρ+ ] where g is a vertex and ρ is a relator are considered. The path ρ+ is written as a product of edges, and we obtain a log − ¯ ρ]) representing the identity determined by [g, ρ+ ]. ι[g,ρ+ ] = (ρ− )(σg) (k1 [g, ω C5: (Simpliﬁcation) The identities are sorted by length and then lexicographically on the relators. An identity is discarded if it is the empty list; if it is equal to an earlier identity; or if it is the inverse of an earlier identity. Furthermore, a proper subsequence is deleted if it is a conjugate of another identity, and the list is resorted. C6: (Output) The resulting list YP of logs which is a complete set of generators for Π2 (P) as a ZG-module, is output.

We note that this complete set of generators is usually not minimal but we are not concerned with that here: this is a problem resolved in the sequel to this paper [17], using methods of Gr¨ obner bases and preﬁx saturation.

5

Examples

In this section we consider three examples. First we return to the presentation Q of Q8 and the complete logged rewriting system obtained in Example 2.4. We show the

LOGGED REWRITING AND IDENTITIES AMONG RELATORS

271

results obtained using the GAP implementation, obtaining 32 identities, which can be reduced to 18. Secondly we consider the free abelian group on two generators and verify that all the identities are trivial. Finally we consider the inﬁnite, 2-generator, 1-relator trefoil group and obtain a logged rewriting system which is complete with respect to a wreath product order. of the quaternion presentation Q with: Example 5.1 The Cayley graph X ¯ : ρ1 → a+4 , ρ2 → b+4 , X = {a, b}, R = {ρ1 , ρ2 , ρ3 , ρ4 }, ω ρ3 → a+ b+ a+ b− , ρ4 → a+2 b+2 . is shown below, where the elements of Q8 are taken to be the irreducible words obtained in Example 2.4: a−

a− b−

b− a+

b− b−

a+2 b+

a+ b−

a− a−

a+ a−

a+ b+ b+

λX¯

b+ b−

a+

b+ b+

a+

a+

The edges [g, x ] for which N ((σg)x ) = (σg)x are marked with double lines in the graph. The image of these edges under k1 is the identity. The images of the other edges, calculated using the logged complete rewrite system Lcomp (Q), are shown in the Table 2. [g, x ] [a− , a− ] [a− , b+ ] [a− , b− ] [b+ , a+ ] [b+ , a− ] [b+ , b+ ] [b− , a+ ] [b− , a− ] [b− , b− ]

g a+2 a+ b− a+ b+ a+ b− a+ b+ a+2 a+ b+ a+ b− a+2

(σg)x (σg )− a− a− a− a− a− b+ b+ a− a− b− b− a− b+ a+ b+ a− b+ a− b− a− b+ b+ a− a− b− a+ b− a− b− a− b+ a− b− b− a− a−

k1 [g, x ] (ρ− 1) − a− (ρ1 ) (ρ+ 4) −3 − a a− (ρ− (ρ+ 1 ) (ρ2 ) 4) + a (ρ− ) (ρ+ )a− (ρ+ 3) 1 4 (ρ− 3) + a−2 (ρ− 4 ) (ρ2 ) + a− (ρ− 4 ) (ρ3 ) −2 − a a− (ρ− (ρ− 4 ) (ρ3 ) 4) (ρ− 4)

logged rule ψ9 = ψ14 ψ8 ψ10 ψ5 ψ7 ψ6 = ψ16 ψ11 ψ12 ψ13 = ψ15

Table 2. Chosen values for k1 [g, x ] Where Each of the 32 relator cycles [g, ρ+ ] is split into its component edges in X. the direction of the cycle is contrary to the direction of the generator on an edge,

272

HEYWORTH, WENSLEY

the inverse edge is used. We thus obtain the identity ¯ ρ])g . ι[g,ρ+ ] = (ρ− ) (k1 [g, ω For example: + + + + + +2 + − − + − − + + − − − − − [a+ b+ , ρ+ 3 ] → [a b , a ][b , b ][a , a ][a , b ] = [b , a ] [b , b ][a , a ] [a , b ],

and the image of this product under k1 is read oﬀ from the chosen values on the four edges, giving ι14 in Table 3. The resulting list of 32 identities may be shortened by omitting trivial identities and duplicates. The identity ι[a− , ρ+ ] 2

=

−

− a a (ρ− (ρ+ 2 ) (ρ1 ) 2)

−4

a (ρ+ 1)

−

satisﬁes the primary identity property (see Remark 3.6) and so may be omitted. The remaining 18 identities, ordered by length, are given in Table 3. The list represents a complete set of generators for the ZG-module of identities among relators for the presentation Q of Q8 . In fact this set can be reduced to 6 generators, but the reduction requires methods dealt with in [17]. cycle [λX¯ , ρ+ 3] [a+2 , ρ+ 1] [λX¯ , ρ+ 2] + − [b , ρ2 ] [b+ , ρ+ 2] [a+ b− , ρ+ 3] [a+2 , ρ+ ] 4 [a− , ρ+ 4] [b+ , ρ+ 4] [a+ b+ , ρ+ 4] [b+ , ρ+ ] 1 [a+ b+ , ρ+ 1] + + [b , ρ3 ] [a+ b+ , ρ+ 3] [b− , ρ+ ] 3 [a+ b− , ρ+ 4] [b− , ρ+ ] 1 [a+ b− , ρ+ 1]

identity + a+ ι1 = (ρ− 1 ) (ρ1 ) − a+2 ι2 = (ρ1 ) (ρ+ 1) − − a−2 (ρ+ ) ι3 = (ρ2 ) (ρ4 ) (ρ+ 2) 4 − − b a−2 b− (ρ+ )b− ι4 = (ρ− (ρ+ 2 ) (ρ4 ) 2) 4 − b+ a−2 b+ (ρ+ )b+ ι5 = (ρ− (ρ+ 2 ) (ρ4 ) 2) 4 − b− a− b− (ρ+ )a+ b− ι6 = (ρ− (ρ+ 3 ) (ρ4 ) 3) 4 +2 + a − a+2 + ι7 = (ρ− ) (ρ ) (ρ ) (ρ 4 1 4 2) − a−2 + a−4 + a− ι8 = (ρ− ) (ρ ) (ρ ) (ρ 4 4 2 1) − + a+ b+ − b+ a−2 b+ (ρ+ )b+ ι9 = (ρ4 ) (ρ3 ) (ρ1 ) (ρ+ 3) 4 − + a+ b+ + a+2 b+ a+ b+ (ρ+ )b+ ι10 = (ρ4 ) (ρ3 ) (ρ3 ) (ρ− 1) 4 + a+ b+ b+ (ρ+ )a−2 b+ (ρ+ )a− b+ (ρ+ )b+ ι11 = (ρ− (ρ− 1 ) (ρ3 ) 1) 3 3 3 + a+ b+ a+2 b+ (ρ− )a+ b+ (ρ+ )a− b+ (ρ+ )b+ ι12 = (ρ− (ρ+ 1 ) (ρ3 ) 3) 1 3 3 + a+ b+ b+ (ρ+ )a− b+ (ρ− )a−2 b+ (ρ+ )b+ ι13 = (ρ− (ρ− 3 ) (ρ3 ) 1) 4 2 4 + a+ b+ a+ b+ (ρ+ )a− b+ (ρ− )a−2 b+ (ρ+ )b+ ι14 = (ρ− (ρ− 3 ) (ρ3 ) 4) 2 2 4 − − − b + a b− − a− b− + a−3 b− + b− ι15 = (ρ− ) (ρ ) (ρ ) (ρ ) (ρ ) (ρ ) 3 4 3 4 2 1 − b− a− b− (ρ+ )b− (ρ− )b− (ρ+ )a−2 b− (ρ+ )b− ι16 = (ρ− (ρ+ 4 ) (ρ4 ) 3) 3 4 2 4 − b− a− b− (ρ+ )b− (ρ+ )a+ b− (ρ− )b− (ρ+ )a−2 b− (ρ+ )b− ι17 = (ρ− (ρ+ 1 ) (ρ4 ) 3) 3 3 1 3 4 − b− a− b− (ρ+ )b− (ρ+ )a+ b− (ρ+ )a+2 b− (ρ− )a+ b− (ρ+ )b− ι18 = (ρ− (ρ+ 1 ) (ρ4 ) 3) 3 3 3 1 4 Table 3. Non-trivial identities ι[g,ρ+ ]

LOGGED REWRITING AND IDENTITIES AMONG RELATORS

273

Example 5.2 Our second example is the inﬁnite abelian group with presentation A = grpX, ω : R → F (X) where X = {x, y} and R = {ρ} with ωρ = xyx−1 y −1 . The initial logged rewriting system is: Linit (A) = { (x+ x− , λY¯ , λX¯ ), (x− x+ , λY¯ , λX¯ ), (y + y − , λY¯ , λX¯ ), (y − y + , λY¯ , λX¯ ), (x+ y + x− y − , ρ+ , λX¯ ) }.

Logged Knuth-Bendix completion terminates, yielding Lcomp (A) = { (x+ x− , λY¯ , λX¯ ), (x− x+ , λY¯ , λX¯ ), (y + y − , λY¯ , λX¯ ), (y − y + , λY¯ , λX¯ ), (y + x+ , ρ− , x+ y + ), (y + x− , (ρ+ )x , x− y + ), +

+ y + x−

(y − x+ , (ρ+ )x

, x+ y − ), (y − x− , (ρ− )x

+ y+

, x− y − ) }.

¯ which are irreducible with respect It can be deduced that the set of elements of X to the complete rewrite system is {xn y m : n, m ∈ Z}. This enables us to say that has edges of the form [x+n y +m , x+ ] and [x+n y +m , y + ]. the (inﬁnite) Cayley graph X We must now deﬁne k1 on all such edges. First note that there are two cases when k1 maps an edge to λY¯ , ﬁrstly when the edge is of the form [x+n y +m , y + ], since x+n y +(m+1) is irreducible, and secondly when the edge is of the form [x+n , x+ ]. It remains to determine k1 [x+n y +m , x+ ] when m > 0 and when m < 0. Using the formula for k1 in the proof of Theorem 4.2, k1 [x+n y +m , x+ ] = Y (x+n y +m x+ N (x+n y +m x+ )− )) = Y (x+n y +m x+ y −m x−(n+1) ). When m > 0, logged rewriting of x+n y +m x+ y −m x−(n+1) gives k1 [x+n y +m , x+ ] = (ρ− )y

−(m−1) x−n

(ρ− )y

−(m−2) x−n

· · · (ρ− )y

− x−n

−n

(ρ− )x

by repeated application of the logged rule y + x+ → x+ y + by ρ− . For m < 0, + + − repeated application of y − x+ → x+ y − by (ρ+ )x y x gives k1 [x+n y +m , x+ ] + y + x− y −(m+1) x−n

= (ρ+ )x

+ y + x− y −m x−n

+ y + x− y −2 x−n

(ρ+ )x

· · · (ρ+ )x

+ y + x− y − x−n

(ρ+ )x

.

There are inﬁnitely many pairs [g, ρ+ ] for this group, but each has the form [x+n y +m , ρ+ ]. The boundary of the relator cycle is [x+n y +m , x+ y + x− y − ] = [x+n y +m , x+ ] [x+(n+1) y +m , y + ] [x+n , y +(m+1) , x+ ]− [x+n y +m , y + ]− . −m −n

+ + − −(m+1) −n

x The image under k1 of this cycle is (ρ+ )y x when m > 0, and (ρ+ )x y x y when m < 0. The identities formula in Theorem 4.1 for sep[g, ρ+ ] thus gives us

(ρ− )y

−m x−n

+n y +m )−

(ρ+ )(x

=P λY¯ ,

+ y + x− y −(m+1) x−n

(ρ− )x

+n y +m )−

(ρ+ )(x

=P λY¯ .

So we have veriﬁed algebraically that all the identities among relators for this one relator group are trivial.

274

HEYWORTH, WENSLEY

Example 5.3 The application of logged rewriting systems to the work of [6] allows the direct computation of a ﬁnite set of generators for the module of identities among relators for presentation of a ﬁnite group. In the case of inﬁnite groups the computation is more diﬃcult as it relies upon the logged reduction of general expressions such as x+n y +m x+ y −m x−(n+1) . An inﬁnite group, where the problem of proving asphericity algebraically using the Brown-Razak set of generators for the module of identities among relators is not straightforward, is the trefoil knot group with presentation T = grpX = {x, y}, ω : R = {ρ} → F (X), ρ → x3 y −2 . The complete rewrite system for T with respect to this syllable ordering has been obtained using the COSY completion system at Kaiserslautern [21] with a wreath product or syllable ordering (see [20]). The logged information was calculated by hand (since we have not yet implemented this ordering in our programs), resulting in the complete logged system: Lcomp (T ) = { (y + y − , λY¯ , λX¯ ), (y − y + , λY¯ , λX¯ ), (x+3 , (ρ+ ), y +2 ), −

− y+

(y +2 x+ , (ρ− )(ρ+ )x , x+ y +2 ), (y − x+ , (ρ− )x

(ρ+ )y , y + x+ y −2 ), +

(x− , (ρ− )x , x+2 y −2 )}. +

Although the normal forms of the group elements are too irregular to attempt to describe the Brown-Razak identities, we can consider the identities that arise from the logged reduction of the critical pairs of Linit∗ (T ). There are eighteen words on which the rules overlap; these are: y + y − y + , y − y + y − , y − y +2 x+ , y + y − x+ , y − x+3 , y +2 x+3 , x+3 x− , x− x+3 , y +2 x+ x− , y − x+ x− , x− x+ , x+3 y −2 y + , x+3 y −2 x+ 4y −2 , x+ 5y −2 , x− x+3 y −2 x+3 y −2 x+ , y − x+3 y −2 . We examine the identities given by each overlap (as in Lemma 3.2). The ﬁrst two overlaps immediately give the trivial identity λY¯ . The third overlap word is + − + − + y − y +2 x+ , which can be reduced to y + x+ by λY¯ or by ((ρ− )y (ρ+ )x y (ρ− )x y + (ρ+ )y , giving the identity − y+

(ρ− )y (ρ+ )x +

− y+

(ρ− )x

(ρ+ )y , +

which is equivalent under =P to λY¯ . Similarly, y + y − x+ can be reduced to x+ by − − λY¯ or by (ρ− )x (ρ+ )(ρ− )(ρ+ )x , giving the identity −

−

(ρ− )x (ρ+ )(ρ− )(ρ+ )x . The ﬁfth overlap word is y − x+3 , and can be reduced to y + in two ways giving the identity − y+

(ρ− )x

− y +2 x− y −

(ρ+ )y (ρ− )x +

(ρ+ )y

+2 x− y −

−

− y +2 x+ y −3

(ρ+ )y

−2

−2

(ρ+ )y (ρ− )x

The sixth overlap word y +2 x+3 gives the identity −

−

−2

−2

(ρ− )(ρ+ )x (ρ− )x (ρ+ )x (ρ− )x (ρ+ )(ρ+ )y (ρ− )y .

+2 x+ y −3

(ρ− )y . +

LOGGED REWRITING AND IDENTITIES AMONG RELATORS

275

Using the Primary Identity Property it is straightforward to check that all these identities are Peiﬀer equivalent to λY¯ . We can continue in this way, checking that all 18 overlap identities are trivial and thus oﬀering a direct algebraic proof that the trefoil group is aspherical.

6

Concluding remarks

A collection of functions implementing some of the algorithms described is included in the ﬁrst author’s thesis [15], written using the computational group theory program GAP3. These functions have been rewritten for GAP4 [12], added to, and submitted as a share package IdRel. Once the set of identities I has been obtained, the next stage is to determine a minimal generating set for I and to express each identity in terms of these generators. Again a logged reduction can be performed, and the resulting information used to obtain identities among the identities. The details of this computation will be given in [17]. There are connections between the work presented here and that of Squier, Lafont, Street, Groves, Cremanns and Otto on ﬁnite derivation types (see [19, 24, 14, 9]). In brief, we know from their work that when a group can be presented by a ﬁnite complete rewriting system it is of ﬁnite derivation type and has the property F P3 . This suggests that our algorithms, which construct the ﬁnite sets of generators which deﬁne these properties, can be exploited further in the future. The paper [16] will pursue this research further, by examining the case of monoid presentations and relating the ideas to work on sesquicategories, derivation schemes and ﬁniteness conditions. References [1] M. Anshel, Constructing Public Key Cryptosystems via Combinatorial Group Theory, Department of Computer Sciences CCNY-CUNY New York 1999 available from: http://cryptome.unicast.org/cryptome022401/pkc-cgt.htm [2] Y.K. Baik and S.J. Pride, Generators of the second homotopy module of presentations arising from group constructions, University of Glasgow Preprint 92/49, 1992 [3] W.A. Bogley and S.J. Pride, Calculating generators of Π2 , in: Low-Dimensional Homotopy Theory and Combinatorial Group Theory, (C. Hog-Angeloni, W. Metzler, A. Sieradski (eds.), Cambridge University Press, 1993) 157-88 [4] R. Book and F. Otto, String-Rewriting Systems Springer-Verlag, New York, 1993 [5] R. Brown and J. Huebschuman, Identities Among Relations, in: Low-Dimensional Topology, London Math. Soc. Lecture Notes 46 (ed. R. Brown and T.L. Thickstun, Cambridge University Press, Cambridge, 1982) 153-202 [6] R. Brown and A. Razak Salleh, On the Computation of Identities Among Relations and of Free Crossed Resolutions of Groups, LMS J. Comput. Math. 2 (1999) 28-61 [7] R. Brown and C.D. Wensley, On Finite Induced Crossed Modules and the Homotopy 2-Type of Mapping Cones, Theory Appl. Categ. 1 (1995) 54-71 [8] D.J. Collins, Combinatorial Group Theory and Applications to Geometry in: Encyclopaedia of Mathematical Sciences, (ed. A.I. Kostrikin and I.R. Shafarevich) Springer [9] R. Cremanns and F. Otto, For groups the property of having ﬁnite derivation type is equivalent to the homological ﬁniteness condition F P3 , J. Symb. Comput. 22 (1996) 155-177

276

HEYWORTH, WENSLEY

[10] G. Ellis and I. Kholodna, Three-dimensional presentations for the groups of order at most 30, LMS J. Computation and Math. 1999 [11] D.B.A. Epstein, J.W.Cannon, D.F. Holt, S.V.F. Levy, M.S. Paterson and W.P. Thurston, Word processing in groups, A.K. Peters, Natick, Mass, 1992 [12] The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4, Aachen, St Andrews, 1998, http://www.gap.dcs.st-and.ac.uk/~gap. [13] N.D. Gilbert and T. Porter Knots and Surfaces Oxford University Press, 1994 [14] J.R.J. Groves, An algorithm for computing homology groups, J. Algebra 194 (1997) 331-361. [15] A. Heyworth, Applications of rewriting systems and Gr¨ obner bases to computing Kan extensions and identities among relations, Ph.D. thesis, University of Wales, Bangor, 1998 [16] A. Heyworth and M. Johnson, Logged Rewriting for Monoids, (in preparation) [17] A. Heyworth and B. Reinert, Reduction of generating sets of ZG-modules and applications to identities among relations, (in preparation) [18] D. Holt, The Warwick automatic groups software, in: Geometrical and Computational Perspectives on Inﬁnite Groups, (ed. Gilbert Baumslaug et al, DIMACS Series in Discrete Mathematics and Theoretical Computer Science 25, 1995) 69-82 [19] Y. Lafont, A ﬁniteness condition for monoids presented by complete rewriting systems (after C.C. Squier), J. Pure and Applied Algebra, 98 (1995) p229-244 [20] P. Le Chenadec, Canonical forms in ﬁnitely presented algebras Pitman, London, 1986 [21] K. Madlener and B. Reinert, String rewriting and Gr¨ obner bases - a general approach to monoid and group rings, Proc. Workshop on Symbolic Rewriting Techniques, Monte Verita 1995, (Birkh¨ auser, 1998) p127-180 [22] S.J. Pride, Identities among relations, in: Proc. Workshop on Group Theory from a Geometrical Viewpoint, International Centre of Theoretical Physics, Trieste, 1990, (ed. E. Ghys, A. Haeﬂiger, A. Verjodsky World Scientiﬁc, 1991) 687-716 [23] J.G. Stell, Modelling Term Rewriting Systems by Sesquicategories, Technical Report TR94-02, University of Keele, 1994 [24] R. Street, Categorical structures, Handbook of Algebra, Vol.1 Elsevier, 1996 529-577

A CHARACTERIZATION OF F4 (q) WHERE q IS AN ODD PRIME POWER A. IRANMANESH∗ and B. KHOSRAVI∗∗ ∗

Department of Mathematics, Tarbiat Modarres University, P.O. Box: 14115-137, Tehran, Iran E-mail: [email protected] ∗∗ Center for Theoretical Physics and Mathematics P.O. Box 11365-8486, AEOI, Tehran, Iran

Abstract The order components of a ﬁnite group were introduced in [5]. We prove that F4 (q) is uniquely determined by its order components where q is an odd prime power. A main consequence of our result is the validity of Thompson’s conjecture for the groups under consideration. AMS Subject Classification: 20D05,20D60 Keywords : Finite group, simple group , prime graph, order component.

1 Introduction If n is an integer, then π(n) is the set of prime divisors of n and if G is a ﬁnite group π(G) is deﬁned to be π(|G|). The prime graph Γ(G) of a group G is a graph whose vertex set is π(G), and two distinct primes p and qare linked by an edge if and only if G contains an element of order pq. Let πi , i = 1, 2, . . . , t(G) be the connected components of Γ(G). For |G| even, π1 will be the connected component containing 2. Then |G| can be expressed as a product of some positive integers mi , i = 1, 2, . . . , t(G) with π(mi ) = πi . The integers mi ’s are called the order components of G. The set of order components of G will be denoted by OC(G). If the order of G is even, we will assume that m1 is the even order component and m2 , . . . , mt(G) will be the odd order components of G. The order components of non-abelian simple groups having at least three prime graph components are obtained by G. Y. Chen [9, Tables 1,2,3] , and we presented the order components of non-abelian simple groups with two order components in Tables 1-3 in [12] by [14,19]. The following groups are uniquely determined by their order components: Suzuki-Ree groups [7], Sporadic simple groups [4], PSL2 (q) [9], E8 (q) [8], F4 (q) [q = 2n ] [11], 2 G2 (q) [2], Ap where p and p − 2 are primes [13] and PSL3 (q) where q is an odd prime power [12]. In this paper, we prove that F4 (q) are also uniquely determined by their order components, where q is an odd prime power, that is we have: The Main Theorem. Let G be a ﬁnite group, M = F4 (q)where q is an odd prime power and OC(G) = OC(M ). Then G ∼ = M.

278

IRANMANESH, KHOSRAVI

2 Preliminary results In this section we bring some preliminary lemmas to be used in the proof of the main theorem. Definition 2.1 [10] A ﬁnite group G is called a 2-Frobenius group if it has a normal series G > K > H > 1, where K and G/H are Frobenius groups with kernels H and K/H, respectively. Lemma 2.2 [19; Theorem A] If G is a ﬁnite group and its prime graph has more than one component, then G is one of the following groups: (a) a simple group; (b) a Frobenius or 2-Frobenius group; (c) an extension of a π1 −group by a simple group; (d) an extension of a simple group by a π1 −solvable group; (e) an extension of a π1 −group by a simple group by a π1 −group. Lemma 2.3 [19; Corollary] If G is a solvable group with at least two prime graph components, then G is either a Frobenius group or a 2-Frobenius group and G has exactly two prime graph components one of which consists of the primes dividing the lower Frobenius complement. Lemma 2.4 [1; Theorem 1] Let G be a Frobenius group of even order, H and K be Frobenius complement and Frobenius kernel of G respectively. Then t(G) = 2, and prime graph components of G are π(H), π(K) and G has one of the following structures: (a) 2 ∈ π(K), all Sylow subgroups of H are cyclic. (b) 2 ∈ π(H), K is an abelian group, H is a solvable group, the Sylow subgroups of odd order of H are cyclic and the 2−Sylow subgroups of H are cyclic or generalized quaternion groups. (c) 2 ∈ π(H), K is an abelian group and there exists H0 ≤ H such that |H : H0 | ≤ 2, H0 = Z × SL(2, 5), (|Z|, 2 · 3 · 5) = 1 and the Sylow subgroups of Z are cyclic. Lemma 2.5 [1; Theorem 2] Let G be a 2-Frobenius group of even order. Then: (a) t(G) = 2; (b) There exists normal series 1 H K G such that π1 = π(G/K) ∪ π(H), π(K/H) = π2 , G/K and K/H are cyclic, |G/K| | |Aut(K/H)|, (|G/K|, |K/H|) = 1 and |G/K| < |K/H|. Moreover H is a nilpotent group. Lemma 2.6 [6; Lemma 8] Let G be a ﬁnite group with t(G) ≥ 2 and N a normal subgroup of G. If N is aπi −group for some prime graph component πi of G and m1 , m2 , . . . , mr are some order components of G but not a πi -number, then m1 m2 · · · mr is a divisor of |N | − 1. Lemma 2.7 Let M = F4 (q) and D(q) = q 4 − q 2 + 1 where q is an odd prime

A CHARACTERIZATION OF F4 (q) WHERE q IS AN ODD PRIME POWER 279 power. Then: (a) If p ∈ π(M ), then |Sp | ≤ q 24 where Sp ∈ Sylp (M ); (b) If p ∈ π1 (M ) and pα | |M |, then pα + 1 ≡ 0 (modD(q)) if and only if pα = q 6 or q 18 ; (c) If p ∈ π1 (M ) and pα | |M |, then pα − 1 ≡ 0 (mod D(q)) if and only if pα = q 12 or q 24 . Proof. (a) Observe that |M | = q 24 (q + 1)4 (q − 1)4 (q 2 + 1)2 (q 4 + 1)(q 2 + q + 1)2 (q 2 − q + 1)2 (q 4 − q 2 + 1) and if pα | |M |, then pα is a divisor of q 24 , 9 × 27 (q + 1)4 , 9 × 27 (q − 1)4 , 29 (q 2 + 1)2 , 210 (q 4 + 1), 34 (q 2 + q + 1)2 ,34 (q 2 − q + 1)2 or q 4 − q 2 + 1. Therefore, (a) follows. For proof (b), we assume that there exists p ∈ π1 (M ), pα | |M | and pα + 1 ≡ 0 (mod D(q)). Obviously pα > D(q). Numerical calculations show that if q ≤ 13, then there is any pα such that pα + 1 ≡ 0 (mod D(q)). Thus, suppose that q > 13 : (1) If pα |32 27 (q + 1)4 then pα = 32 27 (q + 1)4 /t where t = 27 or 32 × 2s where 4 32 (0 ≤ s ≤ 7), since 16 (q + 1)4 < (q+1) < q 4 − q 2 + 1. Hence 32 27 (q + 1)4 /t < 2 pα + 1 = hD(q), where ht ≤ 32 28 . Easy calculations show that this is impossible. (2) If pα |32 27 (q − 1)4 , then we get a contradiction similar to (1). (3) If pα |29 (q 2 + 1)2 , then pα |211 or pα |(q 2 + 1)2 /4, but (q 2 + 1)2 /4 < D(q), and this is impossible. (4) If pα |210 (q 4 + 1), then we get a contradiction similar to (3). (5) If pα |34 (q 2 − q + 1)2 , then pα = 34 (q 2 − q + 1)2 /t where t = 1, 3, 9, 25, 27, 49, 81, since (q 2 − q + 1)2 < D(q). Similar to case (1), easy calculation shows that this is impossible. (6) If pα |q 24 , then pα = q 6 or q 18 , by a similar method. (c) Just as in part (b) we conclude that pα must be equal to q 12 or q 24 . Lemma 2.8 Let G be a ﬁnite group, M = F4 (q) where q is an odd prime power and OC(G) = OC(M ). Then G is neither a Frobenius group nor a 2−Frobenius group. Proof . Suppose that G is a Frobenius group. By Lemma 2.4, OC(G) = {|H|, |K|} where H and K are the Frobenius complement and the Frobenius kernel of G, respectively. 2 | |H| since |H| < |K| . If 2||K|, then |H| = q 4 − q 2 + 1 and |K| = q 24 (q 8 − 1)(q 6 − 1)2 (q 4 − 1). If P is a p−Sylow subgroup of H, then |P | < |K|, but |K| | (|P | − 1), which is a contradiction. Therefore G is not a Frobenius group. Let G be a 2-Frobenius group. By Lemma2.5, there is a normal series 1 H K G such that |K/H| = q 4 − q 2 + 1 < 81(q 2 + q + 1)2 and |G/K| < |K/H|. Thus there exists a prime number p such that p | 81(q 2 + q + 1)2 and p||H|. If P is a p−Sylow subgroup of H, since H is nilpotent, P must be a normal subgroup of K with P ⊆ H and |K| = (q 4 −q 2 +1)|H|. Therefore, (q 4 −q 2 +1) | (|P |−1)by Lemma 2.6 and hence |P | = q 12 or q 24 , which is impossible since |P | ≤ 81(q 2 + q + 1)2 . Therefore G is not a 2-Frobenius group. Lemma 2.9 Let G be a ﬁnite group. If the order components of G are the same as those of M = F4 (q) where q is an odd prime power, then G has a normal series

280

IRANMANESH, KHOSRAVI

1 H K G such that satisfying the following two conditions: (a) H and G/K are π1 −groups, K/H is a non-abelian simple group and H is a nilpotent group. (b) The odd order component of M is equal to some of those of K/H, especially, t(K/H) ≥ 2. Proof . (a) This part of lemma follows from Lemmas 2.2,2.3, 2.4 and 2.8 because the prime graph of M has two prime graph components. (b) If p and q are prime numbers, then if K/H has an element of order pq, then G has an element of order pq and hence by the deﬁnition of prime graph components, an odd order component of G must be an odd order component of K/H. If q is odd, then the number of order components of M is equal to two, therefore t(K/H) ≥ 2.

In the next section we prove the Main Theorem.

3 Proof of the main theorem By Lemma 2.9, G has a normal series 1 H K G such that H and G/K are π1 -groups, K/H is a non-abelian simple group where t(K/H) ≥ 2, and the odd order component of M is some of the odd order components of K/H. We now proceed the proof in the following steps : Step 1. If K/H ∼ = An where n = p, p + 1, p + 2 and p ≥ 5 is a prime number, then p or p − 2 equals to q 4 − q 2 + 1. If p = q 4 − q 2 + 1, then p − 1 = q 2 (q 2 − 1) and p − 2 = q 4 − q 2 − 1. Easy calculation shows that (p − 2, |G|) |24 5 and hence p − 2|24 5 which implies that p − 2 = 5. In this case, we have q 4 − q 2 − 6 = 0 which is impossible since has no solution in Z. If p − 2 = q 4 − q 2 + 1, then p − 4 = q 4 − q 2 − 1 and q 4 − q 2 − 1 must divide 24 5 which is impossible. Step 2. K/H ∼ Ar (q ) and 2 Ar (q ). For example if K/H ∼ = Ar (q ), then we = distinguish the following 6 cases: p 2.1. K/H ∼ = Ap −1 (q ) where (p , q ) = (3, 2), (3, 4) . Then q − 1 ≡ 0 (mod p 12 24 D(q)) which implies that q = q or q , by Lemma 2.7(c). If p > 5, then p (p −1)

q 2 Hence

> q 24 , which is impossible by Lemma 2.7(a). If p = 5, then q 5 = q 12 .

(q 2

(q 5 − 1) (q 12 − 1) = , 6 + 1)(q − 1) (q − 1)(5, q − 1)

q 5 = q 12 ,

which is impossible. If p = 3, then q = q 4 and (q − 1)(3, q − 1) = (q 6 − 1)(q 2 + 1), or q = q 8 and (q − 1)(3, q − 1) = (q 6 − 1)(q 2 + 1)(q 12 + 1), which both of them are impossible. p 2.2. K/H ∼ = Ap (q ) where (q − 1)|(p − 1). Then q = q 12 or q 24 . If p > 3, then p +1 > 2 and we get a contradiction by Lemma 2.7. If p = 3, then q = q 4 and 2 q − 1 = (q 6 − 1)(q 2 + 1) which is impossible.

A CHARACTERIZATION OF F4 (q) WHERE q IS AN ODD PRIME POWER 281 2.3. K/H ∼ = A1 (q ), where 4|(q + 1). If D(q) = q 2−1 , then q = q 12 or q 24 , which implies that 2 = (q 6 − 1)(q 2 + 1) or 2 = (q 6 − 1)(q 2 + 1)(q 12 + 1), and this is impossible. If D(q) = q = q 4 − q 2 + 1, then q + 1 = q 4 − q 2 + 2 and 4 |q 4 − q 2 + 2 which is a contradiction. 2.4. K/H ∼ = A1 (q ) where 4|(q − 1). If D(q) = q 2+1 , then q = q 6 or q 18 and hence 2 2 12 6 2 = q + 1 or 2 = (q + 1)(q − q + 1) which is impossible. If D(q) = q , then q = q 4 − q 2 + 1 and thus q − 1 = q 4 − q 2 and q + 1 = q 4 − q 2 + 2. But easy calculations show that q 4 − q 2 + 2 must divide 74 2 which is impossible. 2.5. K/H ∼ = A1 (q ) where 4|q . If D(q) = q − 1, then q = q 12 or q 24 . For example, 12 if q = q , then 1 = (q 6 − 1)(q 2 + 1) which is impossible. If D(q) = q + 1, then q = q 6 or q 18 . For example, if q = q 6 , then 1 = q 2 + 1, which is impossible. 2.6. K/H ∼ = A2 (4). Then D(q) must be equal to 3, 5, 7, 9 which = A2 (2) or K/H ∼ is impossible. Step 3. K/H ∼ Br (q ) and Cr (q ) andDr (q ). For example if K/H ∼ = Br (q ), then = we consider two cases: 3.1 K/H ∼ = Bm (q ) where m = 2k ≥ 4 and q is an odd number. Then q m = q 6 or 18 q . For example, if q m = q 6 , then q 2 + 1 = 2 which is impossible. 3.2 K/H ∼ = Bp (3). Then 3p = q 12 or q 24 , which is impossible since p is an odd prime. Step 4. If K/H ∼ = 2 B2 (q ) where q = 22t+1 > 2, then we consider 3 cases: 4.1 D(q) = q − 1. Then q = q 12 or q 24 . But |G| < |K/H| and this is a contradiction. √ 4.2. D(q) = q − 2q + 1 . Then q 2 + 1 ≡ 0 (mod D(q)). Therefore, q = q 3 or q 9 . For example if q = q 3 , then (q 2 + 1) = q + 2q + 1, q = q3

have no common √ solution in Z. 4.3. D(q) = q + 2q + 1. We proceed similar to 4.2. Step 5. K/H ∼ 2 Dr (q ) and G2 (q ) and 3 D4 (q ). For example if K/H ∼ = 2 Dr (q ), = then we consider following cases: r 5.1 K/H ∼ = 2 Dr (q ) where r = 2t > 2. Then q = q 6 or q 18 . But q r = q 18 , by Lemma 2.7(a). If q r = q 6 , then (q 6 + q 3 + 1, |G|)|3, which is a contradiction. 5.2 K/H ∼ = 2 Dr (2) where r = 2t + 1 ≥ 5. Then 2r−1 = q 6 or q 18 , which is a contradiction since q is odd number. 5.3 K/H ∼ = 2 Dp (3) where 5 ≤ p = 2r + 1.Then 3p = q 6 or q 18 which is impossible. 5.4 K/H ∼ = 2 Dr (3) where r = 2t + 1 = p, t ≥ 2. Then 3r−1 = q 6 or q 18 , and we get a contradiction by Lemma 2.7(a). 5.5 Similarly, we can show that K/H ∼ = is not equal to2 Dp (3) where p = 2t +1, t ≥ 2 2 r or Dp+1 (2) where p = 2 − 1, r ≥ 2. Step 6. If K/H ∼ = E7 (2),E7 (3), 2 E6 (2) or 2 F4 (2) , then D(q) must be equal to 13, 17, 19, 73, 127, 757, 1093 which is impossible unless D(q) = 73. In this case q = 3. But 19||E7 (2)| and 19 ||3 D4 (3)|, which is a contradiction. Step 7. If K/H ∼ = F4 (q ), then q = q . Thus |G| = |F4 (q)| = |K/H| = |K|/|H|

282

IRANMANESH, KHOSRAVI

which implies that |H| = 1 and|K| = |G| = |F4 (q)|. Therefore, K = F4 (q) and hence G =F4 (q). 2 Step 8. If K/H ∼ = 2 F4 (q ) where q = 22r+1 > 2, then D(q) = q ± 2q 3 + q ± √ 2q + 1. Therefore q 6 + 1 ≡ 0 (mod D(q)) and hence q = q or q = q 3 which is impossible since q is odd. √ Step 9. If K/H ∼ = 2 G2 (q ) where q = 32r+1 , then D(q) = q ± 3q + 1. Thus q = q 2 or q 6 which is impossible since the power is even. 6 3 Step 10. K/H ∼ = E6 (q ), then D(q) = (q + q + = E6 (q ) or 2 E6 (q ). If K/H ∼ 9 36 12 12 > q which is impossible. Similar 1)/(3, q − 1) and hence q = q . But q treatment as the ﬁrst case leads to a contradiction. Step 11. If K/H is a sporadic simple group, then D(q) must be equal to 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47,59, 67, 71, which has no solution greater than 2. Step 12. If K/H ∼ = E8 (q ), then since all odd order components are less than or equal to q 9 we have q < q 9 or q 12 < q 90 which is a contradiction by Lemma 2.6(a). The proof of the main theorem is now completed. Remark 3.1 It is a well known the conjecture of J.G.Thompson that if G is a ﬁnite group with Z(G) = 1 and M is a non-abelian simple group satisfying N (G) = N (M ) where N (G)={n| G has a conjugacy class of size n }, then G ∼ = M. We can give positive answer to this conjecture by this characterization, for the groups under discussion. Corollary 3.2. Let G be a ﬁnite group with Z(G) = 1,M = F4 (q) where q is an odd prime power and N (G) = N (M ), then G ∼ = M. Proof. By [5; Lemma 1.5] if G and M are two ﬁnite groups satisfying the conditions of Corollary 3.2, then OC(G) = OC(M ). So the main theorem implies this corollary. Wujie Shi and Bi Jianxing in [17] put forward the following conjecture: ∼ M if Conjecture. Let G be a group and M a ﬁnite simple group, then G = and only if (i) |G| = |M | (ii)πe (G) = πe (M ), where πe (G) denotes the the set of orders of elements in G. This conjecture is correct for all groups of alternating type [18], Sporadic simple groups [15], and some simple groups of Lie types [16]. As a consequence of the main theorem, we prove the validity of this conjecture for the groups under discussion. Corollary 3.3 Let G be a ﬁnite group and M = F4 (q) where q is an odd prime power. If |G| = |M | andπe (G) = πe (M ), then G ∼ = M. Proof. By assumption we must have OC(G) = OC(M ), then this corollary also proved by the main theorem.

A CHARACTERIZATION OF F4 (q) WHERE q IS AN ODD PRIME POWER 283 References [1] [2]

[3] [4] [5] [6] [7] [8] [9] [10]

[11] [12] [13] [14] [15]

[16] [17] [18] [19]

G.Y. Chen, On Frobenius and 2-Frobenius group, J. Southwest China Normal Univ. 20 (5) (1995) 485-487. G.Y. Chen, A new characterization of simple group of Lie type 2 G2 (q) ,Proc. of 2nd Ann. Meeting of Youth of Chin. Sci. and Tech. Assoc. Press of Xinan Jiaotong Univ. (1995) 221-224. G.Y. Chen, A new characterization of G2 (q), [q ≡ 0(mod 3)], J. Southwest China Normal Univ.(1996) 47-51. G.Y. Chen, A new characterization of sporadic simple groups, Algebra Colloq. 3:1 (1996) 49-58. G.Y. Chen, On Thompson’s conjecture, J.Algebra, 15 (1996) 184-193. G. Y. Chen : Further reﬂections on Thompson’s conjecture, J.Algebra 218, 276-285 (1999). G.Y. Chen, A new characterization of Suzuki-Ree groups, Sci. in China (ser A), 27:5 (1997) 430-433. G.Y. Chen, A new characterization of E8 (q), J. Southwest China Normal Univ. 21 (3),(1996)215-217.(Chinese) G.Y. Chen, A new characterization of PSL2 (q), Southeast Asian Bulletin of Math, 22 (1998) 257-263. K.W. Gruenberg and K.W. Roggenkamp, Decomposition of the augmentation ideal and of the relation modules of a ﬁnite group, Proc. London Math. Soc. 31 (1975) 146-166. A. Iranmanesh and B. Khosravi, A characterization of F4 (q) where q is even, Far East J. Math. Sci.(FJMS) 2:6 (2000) 853-859. A. Iranmanesh, S.H. Alavi and B. Khosravi, A characterization of PSL3 (q) where q is an odd prime power, to appear journal of pure and applied algebra. A. Iranmanesh and S.H. Alavi, A new characterization of Ap where p and p − 2 are primes, Korean J.Comput. & Appl. Math. 8:3 (2001) 665-673. A.S. Kondtratev, Prime graph components of ﬁnite groups, Math. USSR-sb. 67:1 (1990) 235-247. W. Shi , A new characterization of the sporadic simple groups ,in Group Theory, Proceeding of the Singapore Group Theory Conference held at the National University of Singapore , 1987 W. Shi and Bi Jianxing , A new characterization of some simple groups of Lie type, Contemporary Math. 82 (1989) 171-180. W. Shi and Bi Jianxing , A characteristic property for each ﬁnite projective special linear group, Lecture notes in Mathematics 1456 (1990) 171-180 . W. Shi and Bi Jianxing , A new characterization of the alternating groups , Southeast Asian Bull. Math.16:1 (1992) 81-90 . J.S. Williams, Prime graph components of ﬁnite groups, J.Algebra 69 (1981) 487-513.

ON ASSOCIATED GROUPS OF RINGS YURII ISHCHUK Department of Mechanics and Mathematics, Ivan Franko National University of Lviv, Universytetska, 1, Lviv 79000, Ukraine E-mail: [email protected]

Abstract We consider the construction of associated group of a ring with identity element. The characterization of rings with a periodic, FC-group, or nilpotent associated group are given. It is shown that if the adjoint group R◦ of a semiperfect ring R with some ﬁniteness conditions is an Engel group then it is nilpotent and R is a Lie nilpotent ring.

1

Introduction

Let R be an associative ring with an identity element. The set of all elements of R forms a semigroup with the identity element 0 ∈ R under the operation a ◦ b = a + b + ab for all a and b of R. The group of all invertible elements of this semigroup is called the adjoint group of R and is denoted by R◦ . Clearly, if R has the identity 1, then 1 + R◦ coincides with the group of units U (R) of the ring R and the map a → 1 + a with a ∈ R is an isomorphism from R◦ onto U (R). Many authors have studied rings with prescribed adjoint groups (or equivalently, groups of units) (see, for example, [1-16]). This paper is concerned with the question of how properties of associated group inﬂuence some characteristics of ring structure. The idea of associated group was introduced in [1] for radical rings. We extend this construction to arbitary associative rings with identity element. In Sections 3,4,5 we obtain some results on rings determined by their associated groups which are periodic, an FC-group, or a nilpotent group. In Section 6 we examine relations between a commutator law of a ring R and a group-commutator law of its adjoint group R◦ . It is proved that if the adjoint group R◦ of a semiperfect ring R with some ﬁniteness conditions is an Engel group then it is nilpotent and R is a Lie nilpotent ring.

2

Preliminaries

Let R be an associative ring (not necessarily with identity element) and R◦ its adjoint group. In the same way as in [1] we consider the set of pairs G(R) = {(x, y) | x ∈ R, y ∈ R◦ } and deﬁne an operation by the rule (x, y)(u, v) = (y · u + u + x, y ◦ v).

(2.1)

ON ASSOCIATED GROUPS OF RINGS

285

Definition 2.1 Let R be an associative ring. Then G(R) = A B is a group with the neutral element (0, 0) with respect to the operation (2.1), where A = {(x, 0) | x ∈ R} ∼ = R◦ . = R+ , B = {(0, y) | y ∈ R◦ } ∼ Following [1], the group G(R) will be called the associated group of the ring R. Lemma 2.2 Let R be an associative ring with associated group G(R). If S is a subring of R with associated group G(S) = X Y then following statements are true: (i) G(S) ≤ G(R), X ≤ A, Y ≤ B; (ii) if S is a left ideal of the ring R, then X G(R); (iii) if X G(R), then rS ≤ S for all r ∈ R; (iv) if S is a right ideal of the ring R, then G(S) A Y ; (v) if G(S) A Y , then S ◦ R ≤ S; (vi) if S a two-sided ideal of the ring R, then G(S) G(R), X G(R); (vii) CA (B) = {(a, 0) | a ∈ Annr (R◦ )}, CB (A) = {(0, b) | b ∈ R◦ and b ∈ Annl (R)}; in particular, if R is a ring with identity, then CB (A) = (0, 0) and if R is a domain, then CB (A) = CA (B) = (0, 0). Proof. (i) is immediate from Deﬁnition 2.1. (ii) Let S be a left ideal of the ring R and rs ∈ S for all elements r ∈ R and for all elements s ∈ S. Then for an arbitrary element (a, b) ∈ G(R) and arbitrary element (x, 0) ∈ X we have (a, b)−1 (x, 0)(a.b) = (b(−1) x + x, 0) ∈ X,

(2.2)

hence X is a normal subgroup in G(R). (iii) If X G(R), then from (2.2) it follows that b(−1) x ∈ S for all b ∈ R◦ and all x ∈ S. (iv) Let S be a right ideal of the ring R and sr ∈ S for all s ∈ S, r ∈ R. Then for all elements (x, y) ∈ X Y and all (a, c) ∈ A Y we have (a, c)−1 (x, y)(a, c) = (−c(−1) a − a, c(−1) )(x, y)(a, c) = (ya + c(−1) ya + c(−1) x + x, y + c(−1) y + yc + c(−1) yc) ∈ X Y,

(2.3)

because c, y ∈ S. Therefore G(R) A Y . (v) If c = 0, then from (2.3) it follows that S ◦ R ≤ S. (vi) Since S is a two-sided ideal of the ring R, then for arbitrary elements (x, y) ∈ X Y and (u, v) ∈ G(R) we have (u, v)−1 (x, y)(u, v) = (yu + v (−1) yu + v (−1) x + x, y + v (−1) y + yv + v (−1) yv) ∈ G(S).

(2.4)

In particular, if y = 0 then (u, v)−1 (x, 0)(u, v) = (v (−1) x + x, 0) ∈ X, hence X G(R).

286

ISHCHUK

(vii) Let (a, 0) ∈ CA (B), then for arbitrary elements (0, b) ∈ B we have (0, b) = (a, 0)−1 (0, b)(a, 0) = (ba, b)

(2.5)

and consequently ba = 0 for all b ∈ R◦ . Therefore a ∈ Annr (R◦ ). The converse statement is also true. Let (0, b) ∈ CB (A), then for all elements (a, 0) ∈ A we have (a, 0) = (0, b)−1 (a, 0)(0, b) = (b(−1) a + a, 0)

(2.6)

and hence b(−1) a = 0 for all a ∈ R. It follows that 0 = 0 · a = (b + b(−1) + bb(−1) )a = ba, hence b ∈ Annl (R).

(2.7) 2

Lemma 2.3 Let R be a ring and let I be an ideal of R such that I ≤ J(R). Then G(R/I) ∼ = G(R)/G(I).

(2.8)

Proof Let G(R) = A B (respectively G(I) = X Y, G(R/I) = C D) be the associated group of the ring R (respectively of the ideal I, of the quotient-ring R/I). Then G(R) G(I) = AB G(I) ∼ = AG(I) G(I) · BG(I) G(I) = (2.9) (AXY XY ) (BXY XY ) = (AY XY ) (XB XY ). Moreover,

∼ R◦ /I ◦ = ∼ B/Y = ∼ XB/XY, D∼ = (R/I)◦ = + ∼ + + ∼ ∼ ∼ C = (R/I) = R /I = A/X = AY /XY.

(2.8) is immediate from the above equations.

(2.10) 2

The next corollary follows from Lemma 4.2 [3]. Corollary 2.4 Let S be unital subring of ring R such that |R+ : S + | < ∞. Then |G(R) : G(S)| < ∞.

3

Rings with periodic associated group

By analogy to Lemma 1.1 [3] the following lemma can be proved. Lemma 3.1 Let R be a ring and J = J(R) its Jacobson radical. Then G(R) is a periodic group if and only if J is a nil ideal with periodic additive group J + and the group G(R/J) is periodic. Remark 3.2 It is clear that for any ring R with identity the following statements are equivalent: (1) the group G(R) is periodic if and only if so is the group of units U (R);

ON ASSOCIATED GROUPS OF RINGS

287

(2) charR is ﬁnite. Let us recall that a ﬁeld T is absolute if T is a ﬁeld of prime characteristic p and T is an algebraic extension of its simple subﬁelds. Hence the multiplicative group T ∗ of an absolute ﬁeld T is the periodic p -group. Lemma 3.3 Let R be a commutative ring with identity. Suppose that R has no zero divisors and Q(R) its field of quotients. Then G(Q(R)) is a periodic group if and only if R is an absolute field. Proof (⇐) Suﬃciency of the lemma is clear. (⇒) Suppose that G(Q(R)) is a periodic group, then for all elements r ∈ R there exist n = n(r) ∈ N such that rn = 1. Therefore the element r is invertible in R. The lemma is proved. 2 Theorem 3.4 Let R be a ring with identity and suppose R has no zero divisors. Then G(R) is periodic group if and only if the following statements are equivalent: (1) P [x] is a field, where P is simple subfield of R; (2) the element x ∈ R is algebraic over P ; (3) x ∈ U (R). Proof Necessity. Suppose that the group G(R) is periodic. Then charR = p, where p is prime. (1) ⇒ (2). If P [x] is a ﬁeld, then the element x is invertible. It follows that xn = 1 for some n ∈ N, hence x is algebraic over P . (2) ⇒ (1). If x is algebraic over P , then the domain P [x] is ﬁnite and therefore it is a ﬁeld. Implications (3) ⇒ (2) and (1) ⇒ (3) are obvious. Suﬃciency. Let statements (1), (2) and (3) be equivalent in the ring R. Assume to the contrary that a is an element of inﬁnite order in the adjoint group R◦ . Then 1 + a ∈ U (R), hence P [1 + a] is a ﬁeld and condition (2) implies that the element a is algebraic over P . This is a contradiction, so the theorem is proved. 2 Corollary 3.5 Let R be a ring with identity, P is its prime subring. If R has no zero divisors, then R◦ = {0} if and only if the following statements are true: (1) P ∼ = GF (2); (2) every arbitrary element x ∈ R − P is transcendental over P ; (3) P [x] is not a field for arbitrary element x ∈ R − P . Proof Suppose R◦ = {0}, then 2 = −2 and therefore charR = 2. Assume that there exists an element a ∈ R − P algebraic over P . Then P [a] is a ﬁnite ring without zero divisors. This means that P [a] is a ﬁeld and a ∈ U (R), giving a contradiction. So condition (2) is true. Condition (3) is obvious. The converse is trivial. 2 The rings R with torsion free additive group R+ and periodic group of units U (R) were studied in paper [5].

288

ISHCHUK

Remark 3.6 If K[G] is the group ring of a non-trivial group G over a skew ﬁeld K of zero characteristic, then the group of units U (K[G]) is not periodic. Indeed, if charK = 0, then the prime subﬁeld P of skew ﬁeld K is isomorphic to Q, but Q∗ is not a periodic group. Corollary 3.7 Let K[H] be a group algebra of a group H over a skew field K. Then the following statements are equivalent: (1) G(K[H]) is a periodic group; (2) U (K[H]) is a periodic group; (3) K is an absolute field, H is a locally finite group. Proof (1) ⇔ (2) is obvious. (2) ⇒ (3). Since the groups H and K ∗ can be embedded in U (K[H]), it follows from Lemma 2.1 [15] that K is an absolute ﬁeld and H is a periodic group. ∞ Let y1 , . . . , yn be arbitrary elements of the group H. Since K = Ki , where i=1

Ki is ﬁnite ﬁeld and Ki [y1 , . . . , yn ] is ﬁnite domain (hence ﬁeld), the subgroup y1 , . . . , yn ≤ H is ﬁnite. (3) ⇒ (2). Clearly, for all elements x ∈ K[H] there exists a ﬁnite subﬁeld F of the ﬁeld K such that x ∈ F [C] for a certain ﬁnite subgroup C of the group H. Since the subring F [C] is a ﬁnite, the group U (K[H]) is periodic. 2

4

Associated groups with finite conjugacy classes

A group G is called an F C-group if every conjugacy class is ﬁnite, i.e., if |G : CG (x)| < ∞ for all element x ∈ G. Lemma 4.1 Let R be a ring with identity. Then G(R) is an F C-group if and only if G(R) is a locally normal group. ∼ R◦ . If the group R◦ is not periProof Let G(R) = A B, where A ∼ = R+ and B = odic, then by Corollary 3.10 [20] CB (A) = 1. But this contradicts Lemma 2.2 (vii). Therefore, the subgroup R◦ is periodic. Let (a, 0) be an arbitrary element of A. Since (a, 0)n ∈ Z(G(R)) for some n = n(a) ∈ N, we obtain (na, b) = (na, 0)(0, b) = (0, b)(na, 0) = (bna + na, b).

(4.1)

bna = 0

(4.2)

Hence for arbitrary non-zero elements a ∈ R. If charR = 0, then (−2)e ∈ R◦ , where e is the identity element of the ring R. From (4.2), if we put a = e it follows that nb = 0 for arbitrary b ∈ R◦ . This contradicts the fact that the order | − 2e|+ is inﬁnite. Therefore charR = n is ﬁnite. Thus G(R) is a locally normal group. The converse is trivial. The lemma is proved. 2

ON ASSOCIATED GROUPS OF RINGS

289

Corollary 4.2 Let R be a ring with identity. Then G(R) is a fibrewise finite group if and only if R is a finite ring. Corollary 4.3 Let R be a ring with identity. Suppose R has no zero divisors, then G = G(R) is an F C-group if and only if R◦ = {0} or R is a finite field. Indeed, if the adjoint group R◦ is not trivial, then it follows from Lemma 4.1 and fact, that quotient-group G/CG (xG ) (where xG = g −1 xg | g ∈ G) of the F C-group G is ﬁnite for all x ∈ G. Theorem 4.4 Let R be a ring with identity. If G = G(R) is an F C-group, then G = AB is a locally normal group with finite commutator, moreover, the subgroup B is finite, |G : Z(G)| < ∞ and B ∩ Z(G) = 1. Proof Let G = G(R) = A B be an F C-group. Then for all element g ∈ G the quotient-group G/CG (g G ) is ﬁnite. Lemma 4.1 implies that subgroup B is ﬁnite. By Lemma 3.10 [20] |G : Z(G)| < ∞ and by the theorem of Baer the commutator 2 G is ﬁnite. Corollary 4.5 Let K[H] be a group algebra of a group H over a field K. Then G(K[H]) is an F C-group if and only if the algebra the K[H] is finite. Proof Since the groups H and K ∗ can be embedded in the adjoint group (K[H])◦ , we see that H and K ∗ are ﬁnite by Theorem 4.4. Therefore, the algebra K[H] is ﬁnite as well. The converse is trivial. 2

5

Rings with nilpotent associated groups

Lemma 5.1 Let T be a skew field. Then G(T ) is nilpotent group if and only if T ∼ = GF (2). Proof (⇐) is obvious. (⇒). If the associated group G(T ) is nilpotent, then T is a ﬁeld of characteristic p for some prime p. Since the ﬁeld GF (p) is embedded in T then by Lemma 2.2 we have |GF (p)| = p − 1 = 1, so p = 2. Let p ∼ = GF (2) be a prime subﬁeld of T , then it follows from exercise 9 [19], that T ⊇ P is a ﬁnite algebraic extension and T = P. 2 Remark 5.2

  1, ∼ U (Z2n ) = Z2 ,   Z2 × Z2n−2 ,

n = 1; n = 2; n ≥ 3.

(5.1)

The equation above implies that G(Z2n ) is a nilpotent 2-group. Remark 5.3 If p is an odd prime and n ∈ N, then U (Zpn ) ∼ = Zpn−1 (p−1) . From Lemma 2.2 (vii) it follows that the group G(Zpn ) is not nilpotent.

(5.2)

290

ISHCHUK

Lemma 5.4 Let R be a ring with identity e and suppose that R has no zero divisors. Then G(R) is a nilpotent group if and only if charR = 2 and R◦ = {0}. Proof Let G(R) = A B be a nilpotent group. Then CA (B) = 1 by Proposition 1.6 [20]. From Lemma 2.2 (vii) it follows that B is the identity group and consequently R◦ = {0}. Moreover, charR = 2. Conversely, if R◦ = {0} then 2 G(R) ∼ = R◦ is an abelian group. The lemma is proved. Below N (R) will denote the set of all nilpotent elements of a ring R. Theorem 5.5 Let R be a ring with identity e. If the associated group G(R) is nilpotent, then charR = 2m (m ∈ N). If, in addition, ring R is a commutative, then R◦ = N (R). Proof Let additive order |e|+ = m for some m ∈ N ∪ {0}, then group G(Zm ) is embedded in G(R) (where Z0 = Z). Follow Lemma 5.4 m = 0. If m = 2a pa11 . . . pal l is a canonical decomposition of m, then by Theorem 3 [19] l U (Zm ) ∼ = U (Z2a ) × U (Zap11 ) × . . . × U (Zapl ),

(5.3)

∼ where = Zpai −1 (p −1) and U (Z2a ) is described in Remark 5.2. Remark 5.3 i i implies a1 = . . . = al = 0 and m = 2a . ¯◦) ¯ = R/2R. If a torsion part T (R ¯ ◦ ) = {0} then by Lemma 2.2 (vii) T (R Let R ◦ n ¯ Conversely, let x ¯ then x ¯ ) ⊂ N (R). ¯ ∈ N (R), ¯ = ¯0 is a 2-group and therefore T (R s for some n ∈ N. It follows, that the adjoint power x ¯(2 ) = ¯0, where s ∈ N such that ¯ ◦ ) = N (R). ¯ n ≤ 2s . Hence T (R ¯ is ideal of R. Let Suppose R is a commutative ring. Then, clearly, N (R) ¯ ¯ ¯ is torsion G(D) = A B is a group associated with a ring D = R/N (R), then B ¯ ¯ ¯ ¯ free and CB¯ (A) = 1. It means, B is embedded in the group Aut(A) of the subgroup ¯ A. ¯ is not identity subgroup, then [A, ¯ B] ¯ = A. ¯ It contradict to the nilpotency of If B ¯ is an identity subgroup and R◦ = N (R). The theorem the group G(D). Hence B is proved. 2 U (Zapii )

Remark 5.6 Let R = Q[a], where a2 = 0. Then R is a local Artinian ring. From the results in [21] we have R = B + J(R), where the ﬁeld B = Q. It follows that R◦ = B ◦ × J(R)◦ is a mixed abelian group. Assume (a, 0) is nonzero element of G(R), then (a, 0)−1 (0, −2)(a, 0) = (−a, 0)(0, −2)(a, 0) = (−2a, −2) ∈ / T (G(R)).

(5.4)

Since (0, −2) ∈ T (G(R)), then G(R) is a nilpotent group. Remark 5.7 Let F = GF (pn ), n ≥ 2 and suppose σ is the Frobenius automorphism of the ﬁeld F . Suppose F [x, σ] is a skew polynomial algebra such that xa = σ(a)x for all a ∈ F . Then R = F [x, σ]/(x2 ) is a local Artinian ring. Since R = J(R) + B, where ﬁeld B ∼ = F , then U (R) ∼ = (1 + J(R)) B ∗ , where 1 + J(R) ∗ n is a p-group, |B | = p − 1. As a corollary of [11] we have that the group U (R) is not nilpotent.

ON ASSOCIATED GROUPS OF RINGS

6

291

On semiperfect rings satisfying the Engel condition

Let R be an associative ring and R◦ its adjoint group. Let [x, y] = x ◦ y ◦ x ◦ y be a commutator of x and y in R◦ , where x ◦ x = x ◦ x = 0 = y ◦ y = y ◦ y. For any positive integer n we can deﬁne the n-th commutator [x, n y] by the rule [x, n+1 y] = [[x, n y], y]. A group G is called an Engel group if every element x of G is an Engel one (i.e. for each g in G there is an integer n = n(x.g) ≥ 0 such that [x, n g] = 1). Let us recall that a ring R is semilocal if the quotient R/J(R) is right Artinian, and semilocal ring R is semiperfect if all idempotents of R/J(R) can be lifted modulo J(R) to idempotents of R. Theorem 6.1 Let R be a semiperfect ring such that its Jacobson radical J(R) = J is nilpotent and the quotient R/J = K1 ⊕ . . . ⊕ Kl is the direct sum of fields Ki (i = 1, . . . l) and the adjoint group R◦ is an Engel group. If all fields Ki are algebraic over their simple subfields then (i) [R◦ , R◦ ] ≤ J ◦ ; (ii) [(J m )◦ , R◦ ] ≤ (J m+1 )◦ ; (iii) [J ◦ , m R◦ ] ≤ (J m+1 )◦ . ∼ K ∗ × · · · × K ∗ is an abelian group then (i) holds. Proof Since (R/J(R))∗ = 1 l m For (ii), let J(R) = 0 and J(R)m+1 = 0 for some positive integer m. Suppose that ai − ia = 0 (6.1) for some a ∈ U (R) and i ∈ J(R)m . Let in is an element of J(R) deﬁned by the rule [1 + in , a] = [1 + i, n a] for every positive integer n. Then there is a positive integer s such that [1 + is , a] = 1 + is − ais a−1 = is+1 = 1, but [1 + is+1 , a] = 1 + is+1 − ais+1 a−1 = 1. Hence, ais+1 = is+1 a, ais − is a = −is+1 a = 0. Let Pw be the simple subﬁeld of Kw , α = a + J(R) be an arbitrary element of R/J(R). Then α = a1 a2 . . . au +J(R), where αw = aw +J(R) ∈ Kw (w = 1, . . . , u). From (6.1) it follows that aw i − iaw = 0 (6.2) for some w (1 ≤ w ≤ u). Then as before [1 + iz , aw ] = 1 + iz − aw iz a−1 w = iz+1 = 1,

(6.3)

and [1 + iz+1 , aw ] = 1 + iz+1 − aw iz+1 a−1 w = 1. Hence, aw iz+1 = iz+1 aw ,

(6.4)

aw iz − iz aw = −iz+1 aw = 0

(6.5)

and X k +b1 X k−1 +· · ·+bk

be the minimum polynomial of αw = aw +J(R) Let fw (X) = over Pw . A map D : Pw (αw ) → J(R)m , given by the rule D(g) = giz − iz g, g ∈

292

ISHCHUK

Pw (αw ), deﬁnes a homomorphism of Pw -modules Pw (αw ) and J(R)m . Combining (6.4) and (6.5), we obtain 2 = D(αw )αw αw D(αw ) = −αw iz+1 αw = −iz+1 αw

and consequently k−1 + · · · + bk−1 )D(αw ). 0 = D(fw (αw )) = (kαw

(6.6)

k−1 + · · · + b Since an element kαw k−1 of Pw (αw ) is nontrivial, from (6.6) follows that = 0, a contradiction with (6.3). Thus aw i−iaw = 0 D(αw ) = 0, and so iz −aw iz a−1 w and hence (ii) holds. Also (iii) follows from (ii). 2

For any ring R, R(2) is the ideal generated by all (r, s) = rs − sr with r, s in R, and inductively, R(n) is the ideal generated by all (r, s) with r ∈ R(n−1) , s ∈ R. The ring R is said to be strongly Lie nilpotent if R(m) = 0 for some m. In view of several known results, namely, the result of Gupta and Levin who have proved that the unit group of a Lie nilpotent ring is nilpotent of at most the same class, and the results that some Lie identities in a ring R imply the corresponding group-commutator laws in the adjoint group R◦ , it is natural to ask whether an Engel condition of the adjoint group R◦ inﬂuence Lie nilpotency of a ring R. From Theorem 6.1 follows Theorem 6.2 Let R be a semiperfect ring satisfying the conditions of Theorem 6.1, then R◦ is a nilpotent group and R is a strongly Lie nilpotent ring. In closing, we mention several problems about adjoint groups of associative rings. • Is the Artinian ring R with the Engel adjoint group R◦ Lie nilpotent? • Is the perfect ring R with the Engel adjoint group R◦ Lie nilpotent? References [1] Y.P.Sysak, Products of infinite groups, Preprint N 82.53 of the Institute Math. NASU, 1982. [2] J.Krempa, Finitely generated groups of units in group rings, Preprint of the Institute Math. Warsaw Univ., Warsaw, 1985. [3] J.Krempa, Unit groups and commutative ring extensions, Preprint of the Institute Math. Warsaw Univ., Warsaw, 1985. [4] J.Krempa, On finite generation of unit groups for group rings, London Math. Soc. Lecture Note 212 – Cambridge Univ. Press 1995, 352–367. [5] J.Krempa, Rings with periodic unit groups, Abelian groups and modules. (A. Facchini, C. Menini, eds), Kluwer: Dordrecht e.a., 1995, 313–321. [6] K.R.Pearson and J.R.Schneider Rings with a cyclic group of units, J. Algebra. 16(1) (1970), 243–251. [7] I.Fisher and K.E.Eldridge Artinian rings with cyclic quasi-regular groups, Duke Math. J., 36(1), (1969), 43–47. [8] S.A.Jennings, Radical rings with nilpotent associated groups, Trans. Royal Soc. Canada, 24(3), (1955), 31–38.

ON ASSOCIATED GROUPS OF RINGS

293

[9] J.F.Watters, On the adjoint group of a radical ring, J. London Math. Soc., 43, (1968), 725–729. [10] H.Lane, On the associated Lie ring and the adjoint group of a radical ring, Can. Math. Bull., 27(2), (1984), 215–222. [11] G.Groza Artinian rings having a nilpotent groups of units, J. Algebra, 121(2), (1989), 253–262. [12] X.Du The centres of a radical ring, Can. Math. Bull., 35(2), (1992), 174–179. [13] F.Catino On the centres of a radical ring, Arch. Math.,60, (1993), 330–333. [14] B.Amberg and O.Dickenshied, On the adjoint group of a radical ring, Canad. Math. Bull, 38(3), 1995, 262–270. [15] O.D.Artemovych and Yu.B.Ishchuk, On semiperfect rings determined by adjoint groups, Matematychni Studii, 8(2), (1997), 162–170. [16] Yu.B.Ishchuk, Semiperfect rings with periodic locally nilpotent group of units, Matematychni Studii, 7(2), (1997), 125–128. [17] D.J.S.Robinson, Finiteness conditions and generalized soluble groups, P1. New York e.a.: Springer, 1972. [18] D.J.S.Robinson, A course in the theory of groups, New York e.a.: Springer, 1982. [19] L.Fuchs, Inﬁnite abelian groups, M.:Mir, 1977. ˇ [20] S.N.Cernikov, Groups with prescribed properties of subgroups systems, M.:Nauka, 1980. [21] I.S.Cohen On the structure and ideal theory of complete local rings, Trans. Amer. Math. Soc., 59(1), 54–106. [22] N.D.Gupta and F.Levin, On the Lie ideals of a ring, J.Algebra. 81 (1983), 225–231. [23] David M. Riley and Vladimir Tasic, The transfer of a commutator law from a nil-ring to its adjoint group, Canad. Math. Bull., 40(1) (1997), 103–107. [24] R.K.Sharma and J.B.Shrivastava, Lie solvable rings, Proc. Amer. Math. Soc. 4(1), (1985), 1–8.

Groups St Andrews 2001 in Oxford