Groups St Andrews 2009 in Bath: Volume 2

COVER......Page 1
HALF-TITLE......Page 3
TITLE......Page 5
COPYRIGHT......Page 6
CONTENTS......Page 7
INTRODUCTION......Page 11
1 Introduction......Page 13
2.2 Black-box groups......Page 14
2.3 Algorithm types and random elements......Page 15
2.5 The pseudo-order of a matrix......Page 16
3 The major tasks......Page 17
4 The black-box approach......Page 18
5 Geometry following Aschbacher......Page 19
5.1.2 C3 and C5......Page 20
6 Naming algorithms......Page 21
6.3 Determining the defining characteristic......Page 22
7 Constructing an involution centraliser......Page 23
8.1 Black-box classical groups......Page 24
8.2 Classical groups in their natural representation......Page 25
8.4 Alternating groups......Page 27
9.1 Classical groups......Page 28
9.2 Other algorithms......Page 29
10 Short presentations......Page 30
11 The composition tree......Page 31
12 Applications......Page 32
12.2 Constructing the automorphism group of a finite group......Page 33
References......Page 34
1 Residually finite groups......Page 40
1.2 Problems......Page 41
2 Some properties of 1-relator groups with 3 or more generators......Page 42
3 The Magnus procedure......Page 43
3.1 Ascending HNN extensions......Page 44
3.5 The Dunfield-Thurston result......Page 45
3.6 The Congruence Extension Property......Page 46
4 Probability theory and brownian motions......Page 47
4.2 Convex hull of Brownian motion and maximal Magnus indices.......Page 48
5.1 Periodic points of a word map......Page 49
5.2 Dynamics of polynomial maps over local fields I......Page 51
5.5 The main results......Page 52
5.7 Dynamics of polynomial maps over local fields II......Page 54
6.2 Some strange linear groups......Page 55
7 Two possible approaches to Problem 1.5 and some other open problems......Page 56
References......Page 57
WORDS AND GROUPS......Page 60
Notation and general remarks......Page 61
1 Fibres over finite groups......Page 62
2 Ellipticity in profinite groups......Page 67
3 Ellipticity in finite groups......Page 74
4 Algebraic groups......Page 78
5 Finite simple groups......Page 82
5.1 A model-theoretic method......Page 84
5.2 A combinatorial method......Page 85
5.3 Character theory......Page 86
References......Page 89
1 Introduction......Page 91
2 Invariants......Page 92
3.1 A reduction to nilpotent algebras......Page 93
3.2 An inductive approach......Page 94
3.3 Fingerprints and precomputing......Page 95
4.1 Computing invariants......Page 96
References......Page 97
1 Introduction......Page 100
2.1 Definitions and Notation......Page 101
2.3 Some General Lemmata......Page 102
3.1 The Janko Group J3......Page 103
3.4 The Janko Group J4......Page 104
3.5 The Fischer Groups......Page 105
4 Coxeter Groups......Page 107
5 Non-Involutory Symmetric Generation......Page 108
References......Page 109
1 Introduction......Page 111
2 Discrimination and discriminating groups......Page 112
2.1 Some ties to logic......Page 113
2.2 Some ties to the algebraic geometry of groups......Page 114
2.3 Abelian discriminating groups......Page 116
3 Positive examples: discriminating groups......Page 118
4 Negative examples: nondiscriminating groups......Page 119
5 Nontrivially discriminating groups......Page 121
6 Squarelike groups......Page 124
7 Axiomatics and the Axiomatic Closure Property......Page 126
8 Varietal discrimination......Page 128
References......Page 129
1 Introduction......Page 131
3 The case when one of the subgroups is just a pi-group......Page 133
4 Products of pi-decomposable groups of coprime orders......Page 135
5.2 The soluble case with 2…......Page 136
6 A conjecture for the general case......Page 137
References......Page 138
1 Introduction......Page 140
2 n-Recognizable Groups by Prime Graph......Page 141
3 Quasirecognizable Groups by Prime Graph......Page 142
References......Page 143
1 Introduction......Page 145
2 Finite cyclic grading with L0 = 0......Page 147
3 Finite cyclic grading with bounded L0......Page 151
4 Gradings with few non-trivial components......Page 155
5 Gradings with many commuting components......Page 157
References......Page 160
1 Introduction......Page 164
2 Contranormal subgroups, descendant subgroups and some criteria of nilpotency......Page 165
3 Generalized pronormality and transitivity of normality......Page 168
4 Pronormality and transitivity of permutability......Page 172
References......Page 174
2 Positive laws......Page 177
3 Three Questions on Engel laws......Page 178
5 Engel construction of the laws......Page 180
6 Engel construction and properties of R-laws......Page 182
7 Special kind of R-laws, called Ln......Page 185
References......Page 186
2 Results by Liebeck, Saxl and Kantor......Page 189
3 Main results......Page 190
References......Page 194
1 Introduction......Page 195
2 Classic problems......Page 196
3 Finite basis problems and related questions......Page 197
4 Small varieties......Page 199
5 Varieties and the Hopf property......Page 201
References......Page 202
1 Introduction......Page 204
2 Notation......Page 206
3 Hall subgroups of finite simple groups......Page 207
4 A criterion of Epi......Page 218
5 A criterion of Cpi......Page 220
6 A criterion of Dpi......Page 223
7 Open problems......Page 228
References......Page 231
Introduction......Page 236
1 Origin and early results......Page 237
2 Zorn’s Theorem and some generalisations......Page 238
3.1 Engel Lie rings......Page 240
3.2 Some consequences for Engel groups......Page 243
4 The structure of n-Engel groups......Page 245
5.1 The local nilpotence......Page 249
5.2 Other structure results......Page 251
6.1 The local nilpotence theorem......Page 254
6.2 Other structure results......Page 258
7.1 Generalised Burnside varieties......Page 260
7.2 Generalised Engel groups......Page 263
References......Page 264
1 Introduction......Page 267
2 The associated Lie ring of an n-Engel group......Page 269
4 The Higman algebra......Page 272
References......Page 274
1 Introduction......Page 276
3 The Algorithm......Page 278
4 Main results......Page 280
References......Page 283
1 Introduction......Page 285
2 Some defintions and basic results......Page 287
3.1 p-groups of small orders......Page 288
3.2 Groups with large cyclic subgroups......Page 289
3.4 Camina p-groups......Page 290
4 Results of Sah......Page 291
5 An upper bound for Autc(G) for a finite p-group G......Page 292
6 Some problems......Page 293
References......Page 294
1 Introduction......Page 296
2 Proof of Theorem 1.2......Page 298
3 Example......Page 302
References......Page 306