Groups St Andrews 2001 in Oxford: Volume 1 (London Mathematical Society Lecture Note Series, Series Number 304)

Annotation. This two-volume set contains selected papers from the conference Groups St. Andrews 2001 in Oxford. Contributed by leading researchers, the articles cover a wide spectrum of modern group theory. Contributions based on lecture courses given by five main speakers are included with refereed survey and research articles Cover 1 Series-title 3 Title 5 Copyright 6 Contents of Volume 1 7 Contents of Volume 2 10 INTRODUCTION 13 PERMUTABILITY AND SUBNORMALITY IN FINITE GROUPS 15 1 Notation and terminology 15 2 Characterizations based on the normal structure 16 3 Characterizations based on the Sylow structure 16 4 Characterizations based on embedding properties 18 References 19 (PRO)-FINITE AND (TOPOLOGICALLY) LOCALLY FINITE GROUPS WITH A CC-SUBGROUP 20 1 Introduction 20 1.1 Sketching the thread 20 2 Announcement of the results 21 References 24 TABLE ALGEBRAS GENERATED BY ELEMENTS OF SMALL DEGREES 26 1 Introduction 26 1.1 Examples 27 1.2 Wreath, tensor and fibred products 28 2 Classification results about standard integral table algebras 29 2.1 The algebras... 31 2.2 The algebras... 32 3 Normalized table algebras generated by a nonreal element of degree 3 34 References 35 SUBGROUPS WHICH ARE A UNION OF A GIVEN NUMBER OF CONJUGACY CLASSES 36 1 Introduction 36 2 On n-decomposable finite groups 38 References 40 SOME RESULTS ON FINITE FACTORIZED GROUPS 41 References 44 ON NILPOTENT-LIKE FITTING FORMATIONS 45 1 Introduction 45 2 Lattice formations 46 3 Formations with the Shemetkov property 48 4 Extensions of p-nilpotent groups 50 5 Dominant Fitting classes 50 References 52 LOCALLY FINITE GROUPS WITH MIN-p FOR ALL PRIMES p 53 References 56 QUASI-PERMUTATION REPRESENTATIONS OF 2-GROUPS OF CLASS 2 WITH CYCLIC CENTRE 58 1 Introduction 58 2 Calculating p(G), c(G) and q(G) 59 3 Quasi-permutation representations of an extraspecial 2-group 61 References 63 GROUPS ACTING ON BORDERED KLEIN SURFACES WITH MAXIMAL SYMMETRY 64 1 Introduction 64 2 Preliminaries 65 3 Properties of M-groups 67 4 Families of M-groups 67 5 Relations between Hurwitz, H, and M-groups 70 References 71 BREAKING POINTS IN SUBGROUP LATTICES 73 1 Introduction 73 2 The proof of the Theorem 74 3 Final remarks 75 References 76 GROUP ACTIONS ON GRAPHS, MAPS AND SURFACES WITH MAXIMUM SYMMETRY 77 1 Introduction 77 2 Methods for dealing with finitely-presented groups 78 2.1 Computational algorithms 78 2.2 Low index subgroups 79 2.3 Low index normal subgroups 80 2.4 Schreier coset graphs 81 3 Automorphism groups of compact Riemann surfaces 84 3.1 Hurwitz’s theorem 84 3.2 Hurwitz groups 85 4 Regular maps 87 4.1 Definitions and background 87 4.2 Genus calculation 89 4.3 Group theoretic construction of regular maps 90 4.4 Non-orientable regular maps 91 4.5 Regular maps of small genus 92 4.6 Group actions on non-orientable surfaces 93 5 Symmetric graphs 96 5.1 Definitions and background 96 5.2 The trivalent case 99 5.3 Finite 7-arc-transitive graphs 100 6 Some unexpected results/surprises 102 7 Some open problems 102 Acknowledgement 103 Electronic Availability 103 References 103 ON DUAL PRONORMAL SUBGROUPS AND FITTING CLASSES 106 1 Introduction 106 2 Relevant families of subgroups being F-dual pronormal 107 3 Local normality concepts between Fitting classes 112 References 114 (p, q, r)-GENERATIONS OF THE SPORADIC GROUP O’N 115 1 Introduction 115 2 (p, q, r)-Generations for O’N 116 References 122 COMPUTATIONS WITH ALMOST-CRYSTALLOGRAPHIC GROUPS 124 1 Introduction 124 2 Investigating almost crystallographic groups 124 2.1 Representations of polycyclic almost-crystallographic groups 125 2.2 Polycyclically presented almost crystallographic groups 125 3 Constructing almost crystallographic groups 127 3.1 Constructing almost Bieberbach groups 130 3.2 An example application 131 4 Further applications 132 References 133 RANDOM WALKS ON GROUPS: CHARACTERS AND GEOMETRY 134 Introduction 134 Who cares about this stuff? 136 Acknowledgement 137 1 Random walk and representation theory 137 2 Analytic geometry 143 3 Other appearances of random transpositions 147 4 Some open problems 151 References 153 ON DISTANCES OF 2-GROUPS AND 3-GROUPS 157 1 Survey of results 157 2 Inequalities and examples 160 3 Computations 161 References 163 ZETA FUNCTIONS OF GROUPS: THE QUEST FOR ORDER VERSUS THE FLIGHT FROM ENNUI 164 1 Breaking the test ban treaty on zeta functions 164 1.1 Euler 164 1.2 Dirichlet 166 1.3 Dedekind 166 1.4 Artin, Hasse, Weil 167 1.5 Birch Swinnerton-Dyer 168 1.6 Borevich, Shafarevich, and Igusa 169 1.7 Non-commutative zeta functions 171 2 Using p-adic integrals to capture finite p-groups 172 3 Uniformity 180 4 Subgroup growth and Euler products of cone integrals 190 5 Thefuture:speculation and conjecture 193 5.1 Direct products of Heisenberg groups 193 5.2 Zeta functions of Heisenberg groups over number fields 194 5.3 Class two quotients of... 195 5.4 Grenham’s examples 196 5.5 Maximal class nilpotent lie algebras 198 5.6 Free class two three generator group 199 5.7 Functional equations 200 5.8 Class number formula 202 References 202 SOME FACTORIZATIONS INVOLVING HYPERCENTRALLY EMBEDDED SUBGROUPS IN FINITE SOLUBLE GROUPS 204 1 Introduction 204 2 Hypercentrally embedded subgroups 205 3 Factorizations of hypercentrally embedded subgroups with F-normalizers 206 4 Factorizations of hypercentrally embedded subgroups with pre-frattini subgroups 207 5 Final remarks 208 References 210 ELEMENTARY THEORY OF GROUPS 211 Contents 211 1 Introduction 211 2 First-order languages and model theory 212 3 The Tarksi problems 217 4 Residually free and universally free groups 220 5 Algebraic geometry over groups and applications 228 6 The positive solution to the Tarski problems 232 7 Discriminating, co-discriminating and squarelike groups 238 8 Open Questions 243 References 243 ANDREWS-CURTIS AND TODD-COXETER PROOF WORDS 246 1 Introduction 246 2 An AC-proof and its proof words 247 3 Some PEACE proofs and their proof words 247 4 A diffcult example 249 5 Conclusions 250 References 250 SHORT BALANCED PRESENTATIONS OF PERFECT GROUPS 252 1 Introduction 252 2 Technique 253 3 Results 254 4 Conclusions 256 Acknowledgements 257 References 257 FINITE p-EXTENSIONS OF FREE PRO-p GROUPS 258 1 Introduction 258 2 Some history of the problem 258 3 Idea of the proof and accompanying results 259 References 261 ELEMENTS AND GROUPS OF FINITE LENGTH 263 1 Introduction 263 2 Groups with small finite lengths 264 3 Elements of finite length 267 4 Groups with all elements of finite length 268 References 269 LOGGED REWRITING AND IDENTITIES AMONG RELATORS 270 1 Introduction 270 2 Definitions for logged rewriting 272 2.1 String rewriting systems 272 2.2 From group presentations to identities among relators 272 2.3 Logged rewriting and logged Knuth-Bendix completion 274 2.4 Example of logged rewriting 276 3 Resolvingcritical pairs and computing identities among relators 277 4 Computing a crossed resolution 282 5 Examples 284 6 Concluding remarks 289 References 289 ACHARACTERIZATION OF F4(q) WHERE q IS AN ODD PRIME POWER 291 1 Introduction 291 2 Preliminary results 292 3 Proof of the main theorem 294 References 297 ON ASSOCIATED GROUPS OF RINGS 298 1 Introduction 298 2 Preliminaries 298 3 Rings with periodic associated group 300 4 Associated groups with finite conjugacy classes 302 5 Rings with nilpotent associated groups 303 6 On semiperfect rings satisfying the Engel condition 305 References 306 Cover......Page 1 Series-title......Page 3 Title......Page 5 Copyright......Page 6 Contents of Volume 1......Page 7 Contents of Volume 2......Page 10 INTRODUCTION......Page 13 1 Notation and terminology......Page 15 3 Characterizations based on the Sylow structure......Page 16 4 Characterizations based on embedding properties......Page 18 References......Page 19 1.1 Sketching the thread......Page 20 2 Announcement of the results......Page 21 References......Page 24 1 Introduction......Page 26 1.1 Examples......Page 27 1.2 Wreath, tensor and fibred products......Page 28 2 Classification results about standard integral table algebras......Page 29 2.1 The algebras.........Page 31 2.2 The algebras.........Page 32 3 Normalized table algebras generated by a nonreal element of degree 3......Page 34 References......Page 35 1 Introduction......Page 36 2 On n-decomposable finite groups......Page 38 References......Page 40 SOME RESULTS ON FINITE FACTORIZED GROUPS......Page 41 References......Page 44 1 Introduction......Page 45 2 Lattice formations......Page 46 3 Formations with the Shemetkov property......Page 48 5 Dominant Fitting classes......Page 50 References......Page 52 LOCALLY FINITE GROUPS WITH MIN-p FOR ALL PRIMES p......Page 53 References......Page 56 1 Introduction......Page 58 2 Calculating p(G), c(G) and q(G)......Page 59 3 Quasi-permutation representations of an extraspecial 2-group......Page 61 References......Page 63 1 Introduction......Page 64 2 Preliminaries......Page 65 4 Families of M-groups......Page 67 5 Relations between Hurwitz, H, and M-groups......Page 70 References......Page 71 1 Introduction......Page 73 2 The proof of the Theorem......Page 74 3 Final remarks......Page 75 References......Page 76 1 Introduction......Page 77 2.1 Computational algorithms......Page 78 2.2 Low index subgroups......Page 79 2.3 Low index normal subgroups......Page 80 2.4 Schreier coset graphs......Page 81 3.1 Hurwitz’s theorem......Page 84 3.2 Hurwitz groups......Page 85 4.1 Definitions and background......Page 87 4.2 Genus calculation......Page 89 4.3 Group theoretic construction of regular maps......Page 90 4.4 Non-orientable regular maps......Page 91 4.5 Regular maps of small genus......Page 92 4.6 Group actions on non-orientable surfaces......Page 93 5.1 Definitions and background......Page 96 5.2 The trivalent case......Page 99 5.3 Finite 7-arc-transitive graphs......Page 100 7 Some open problems......Page 102 References......Page 103 1 Introduction......Page 106 2 Relevant families of subgroups being F-dual pronormal......Page 107 3 Local normality concepts between Fitting classes......Page 112 References......Page 114 1 Introduction......Page 115 2 (p, q, r)-Generations for O’N......Page 116 References......Page 122 2 Investigating almost crystallographic groups......Page 124 2.2 Polycyclically presented almost crystallographic groups......Page 125 3 Constructing almost crystallographic groups......Page 127 3.1 Constructing almost Bieberbach groups......Page 130 3.2 An example application......Page 131 4 Further applications......Page 132 References......Page 133 Introduction......Page 134 Who cares about this stuff?......Page 136 1 Random walk and representation theory......Page 137 2 Analytic geometry......Page 143 3 Other appearances of random transpositions......Page 147 4 Some open problems......Page 151 References......Page 153 1 Survey of results......Page 157 2 Inequalities and examples......Page 160 3 Computations......Page 161 References......Page 163 1.1 Euler......Page 164 1.3 Dedekind......Page 166 1.4 Artin, Hasse, Weil......Page 167 1.5 Birch Swinnerton-Dyer......Page 168 1.6 Borevich, Shafarevich, and Igusa......Page 169 1.7 Non-commutative zeta functions......Page 171 2 Using p-adic integrals to capture finite p-groups......Page 172 3 Uniformity......Page 180 4 Subgroup growth and Euler products of cone integrals......Page 190 5.1 Direct products of Heisenberg groups......Page 193 5.2 Zeta functions of Heisenberg groups over number fields......Page 194 5.3 Class two quotients of.........Page 195 5.4 Grenham’s examples......Page 196 5.5 Maximal class nilpotent lie algebras......Page 198 5.6 Free class two three generator group......Page 199 5.7 Functional equations......Page 200 References......Page 202 1 Introduction......Page 204 2 Hypercentrally embedded subgroups......Page 205 3 Factorizations of hypercentrally embedded subgroups with F-normalizers......Page 206 4 Factorizations of hypercentrally embedded subgroups with pre-frattini subgroups......Page 207 5 Final remarks......Page 208 References......Page 210 1 Introduction......Page 211 2 First-order languages and model theory......Page 212 3 The Tarksi problems......Page 217 4 Residually free and universally free groups......Page 220 5 Algebraic geometry over groups and applications......Page 228 6 The positive solution to the Tarski problems......Page 232 7 Discriminating, co-discriminating and squarelike groups......Page 238 References......Page 243 1 Introduction......Page 246 3 Some PEACE proofs and their proof words......Page 247 4 A diffcult example......Page 249 References......Page 250 1 Introduction......Page 252 2 Technique......Page 253 3 Results......Page 254 4 Conclusions......Page 256 References......Page 257 2 Some history of the problem......Page 258 3 Idea of the proof and accompanying results......Page 259 References......Page 261 1 Introduction......Page 263 2 Groups with small finite lengths......Page 264 3 Elements of finite length......Page 267 4 Groups with all elements of finite length......Page 268 References......Page 269 1 Introduction......Page 270 2.2 From group presentations to identities among relators......Page 272 2.3 Logged rewriting and logged Knuth-Bendix completion......Page 274 2.4 Example of logged rewriting......Page 276 3 Resolvingcritical pairs and computing identities among relators......Page 277 4 Computing a crossed resolution......Page 282 5 Examples......Page 284 References......Page 289 1 Introduction......Page 291 2 Preliminary results......Page 292 3 Proof of the main theorem......Page 294 References......Page 297 2 Preliminaries......Page 298 3 Rings with periodic associated group......Page 300 4 Associated groups with finite conjugacy classes......Page 302 5 Rings with nilpotent associated groups......Page 303 6 On semiperfect rings satisfying the Engel condition......Page 305 References......Page 306 Vol. 1. Permutabiity And Subnormality In Finite Groups/m.j. Alejandre -- (pro)-finite And (topologically) Locally Finite Groups With A Cc-subgroup/z. Arad -- Table Algebras Generated By Elements Of Small Degrees/a. Arad -- Subgroupgs Which Are A Union Of A Given Number Of Conjugacy Classes/a.r. Ashrafi -- Some Results On Finite Factorized Groups/a. Ballester-bolinches -- On Nilpotent-like Fitting Formations/a. Ballester-bolinches -- Locally Finite Groups With Min-p For All Primes P/a. Ballester-bolinches -- Quasi-permutation Representations Of 2-groups Of Class 2 With Cyclic Centre/h. Behravesh -- Groups Acting On Bordered Klein Surfaces With Maximal Symmetry/e. Bujalance -- Breaking Points In Subgroup Lattices/g. Calugareanu -- Group Actions On Graphs, Maps And Surfaces With Maximum Symmetry/m.d.e. Conder -- On Dual Pronormal Subgroups And Fitting Classes/a.d'aniello -- (p, Q, R)-generations Of The Sporadic Group O'n/m.r. Darafsheh -- Computations With Almost-crystallographic Groups/k. Dekimpe -- Random Walks On Groups: Characters And Geometry/p. Diaconis -- On Distances Of 2-groups And 3-groups/a. Drápal -- Zeta Functions Of Groups: The Quest For Order Versus The Flight From Ennui/m.p.f. Du Sautoy -- Some Factorizations Involving Hypercentrally Embedded Subgroups In Finite Soluble Groups/l.m. Ezquerro -- Elementary Theory Of Groups/b. Fine -- Andrews-curtis And Todd-coxeter Proof Words/g. Havas -- Short Balanced Presentations Of Perfect Groups/g. Havas -- Finte P-extensions Of Free Pro-p Groups/w. Herfort -- Elements And Groups Of Finite Length/m. Herzog -- Logged Rewriting And Identities Among Relators/a. Heyworth -- A Characterization Of F4(q) Where Q Is An Odd Prime Power/a. Iranmanesh -- On Associated Groups Of Rings/y.b. Ishchuk Vol. 2. Gracefulness, Group Sequencings And Graph Factorizations/g. Kaplan -- Orbits In Finite Group Actions/t.m. Keller -- Groups With Finitely Generated Integral Homologies/d.h. Kochloukova -- Invariants Of Discrete Groups, Lie Algebras And Pro-p Groups/d.h. Kochloukova -- Groups With All Non-subnormal Subgroups Of Finite Rank/l.a. Kurdachenko -- On Some Infinite Dimensional Linear Groups/l.a. Kurdachenko -- Groups And Semisymmetric Graphs/s. Lipschutz -- On The Covers Of The Finite Groups/m.s. Lucido -- Groupland/o. Macedonska -- On Maximal Nilpotent [pi]-subgroups/j. Medina -- Characters Of P-groups And Sylow P-subgroups/a. Moretó -- On The Relation Between Group Theory And Loop Theory/m. Niemenmaa -- Groups And Lattiices/p.p. Pálfy -- Finite Generalized Tetrahedron Groups With A Cubic Relator/g. Rosenberger -- Character Degrees Of The Sylow P-subgroups Of Classical Groups/j. Sangroniz -- Character Correspondences And Perfect Isometries/l. Sanus -- The Characters Of Finite Projective Symplectic Group Psp(4, Q) -- M.a. Shahabi -- Exponent Of Finite Groups With Automorphisms/p. Shumyatsky -- Classifying Irreducible Representations In Characteristic Zero/a. Turull -- Lie Methods In Group Theory/m.r. Vaughan-lee -- Chevalley Grops Of Type G2 As Automorphism Groups Of Loops/p. Vojtechovsky Edited By C.m. Campbell, E.f. Robertson, G.c. Smith. Includes Bibliographical References. This first volume of the two-volume book contains selected papers from the international conference'Groups St Andrews 2001 in Oxford'which was held at the University of Oxford in August 2001. Five main lecture courses were given at the conference, and articles based on their lectures form a substantial part of the Proceedings. This volume contains the contributions from Marston Conder (Auckland), Persi Diaconis (Stanford) and Marcus Du Sautoy (Cambridge). The series of Proceedings of Groups St Andrews conferences have provided snapshots of the state of research in group theory throughout the past twenty years. As with earlier volumes, these refereed volumes also contain accessible surveys of contemporary research fronts, as well as a diverse collection of short research articles. They form a valuable reference for researchers, especially graduate students, working in group theory. This first volume of the two-volume book contains selected papers from the international conference 'Groups St Andrews 2001 in Oxford' which was held at the University of Oxford in August 2001. Five main lecture courses were given at the conference, and articles based on their lectures form a substantial part of the Proceedings. This volume contains the contributions from Marston Conder (Auckland), Persi Diaconis (Stanford) and Marcus Du Sautoy (Cambridge). The series of Proceedings of Groups St Andrews conferences have provided snapshots of the state of research in group theory throughout the This first volume contains selected papers from the conference 'Groups St Andrews 2001 in Oxford'. The articles are contributed by leading researchers and cover a wide spectrum of modern group theory. There are articles based on lecture courses given by five main speakers together with refereed survey and research articles