Groups St Andrews 2009 in Bath: Volume 1

HALF-TITLE......Page 3
TITLE......Page 5
COPYRIGHT......Page 6
CONTENTS......Page 7
INTRODUCTION......Page 11
A SPEECH IN HONOUR OF JOHN CANNON AND DEREK HOLT......Page 13
1.1.1 The classification of the finite simple groups......Page 16
1.1.4 The orders of some finite groups of Lie type......Page 17
1.1.6 Weyl group and Dynkin diagram......Page 18
1.2 Finite reductive groups......Page 19
1.2.3 Finite reductive groups......Page 20
1.2.5 Maximal tori and the Weyl group......Page 21
1.2.7 The classification of maximal tori......Page 22
1.3.1 BN-pairs......Page 23
1.3.3 The BN-pair of GLn(k) and of SOn(k)......Page 24
1.3.6 Examples for parabolic subgroups......Page 25
2 Representations in defining characteristic......Page 26
2.2.1 A rough survey......Page 27
2.2.3 The character groups and cocharacter groups of GLn(k) and SLn(k)......Page 28
2.2.5 The roots and the coroots of GLn(k) and of SLn(k) and the roots......Page 29
2.2.6 The root datum......Page 30
2.2.9 Dominant weights and simple modules......Page 31
2.2.12 The irreducible representations of SL2(k)......Page 32
2.2.15 Characters of Weyl modules......Page 33
2.3.2 Kazhdan–Lusztig polynomials......Page 34
3.1 Harish-Chandra theory......Page 35
3.1.3 Iwahori’s example......Page 36
3.1.5 Problems in Harish-Chandra theory......Page 37
3.1.8 Goals and results......Page 38
3.2.2 Deligne–Lusztig varieties......Page 39
3.2.6 The unipotent characters of GLn(q)......Page 40
3.3.2 Jordan decomposition of conjugacy classes......Page 41
3.3.5 The irreducible characters of GLn(q)......Page 42
4.1.1 Harish-Chandra Classification: Recollection......Page 43
4.2.1 Brauer Characters......Page 44
4.2.4 Properties of Brauer characters......Page 45
4.3.3 The unipotent decomposition matrix......Page 46
4.3.5 Linear primes......Page 47
4.4.2 The q-Schur algebra......Page 48
4.4.4 Connections to symmetric group representations......Page 49
4.4.6 James’ conjecture......Page 50
4.4.9 Genericity......Page 51
References......Page 52
1 Introduction......Page 56
2.1 Definition......Page 58
2.2 Example: double self-covering of the circle......Page 59
2.3 Chebyshev polynomials......Page 60
2.4 Computation......Page 61
2.5 Examples......Page 62
2.5.2 Chebyshev polynomials......Page 63
2.5.3 The polynomial…......Page 64
2.5.4 A two-dimensional example......Page 65
3.1 Self-similar groups......Page 66
3.2 Virtual endomorphisms......Page 68
3.3 Contracting groups......Page 69
3.4 Algebraic properties of contracting groups......Page 70
3.5.3 Presentations......Page 72
3.6 Iterated monodromy group of a correspondence......Page 73
3.7.1 Dual Moore diagrams......Page 74
3.7.2 Arithmetic-geometric mean of Lagrange and Gauss......Page 75
3.7.4 Brunner-Sidki-Vieira group......Page 79
4.1 Schreier graphs......Page 80
4.2 Limit space......Page 82
4.3 Models of the limit space......Page 83
4.4 A model of the Julia set of…......Page 85
4.5 Gupta-Sidki group......Page 86
5.1 Abelian groups......Page 88
5.2 Expanding endomorphisms of manifolds......Page 90
5.3 Grigorchuk group and Ulam-von Neumann map......Page 91
5.4 Quadratic polynomials......Page 95
5.5 An example of Fornæss and Sibony......Page 101
References......Page 103
1 Introduction......Page 109
2 Interaction of Right Engel Elements with Left Engel Elements......Page 111
3 Four Engel Subsets and Corresponding Subgroups......Page 112
3.1 Left Engel Elements......Page 114
3.1.1 Left Engel Elements in Generalized Soluble and Linear Groups......Page 116
3.1.2 Left Engel Elements in Groups Satisfying Certain Min or Max......Page 117
3.1.3 Left k-Engel Elements......Page 119
3.2.1 Right Engel Elements in Generalized Soluble and Linear Groups......Page 122
3.2.2 Right Engel Elements in Groups Satisfying Certain Min or Max......Page 125
3.2.3 Right k-Engel elements......Page 126
References......Page 130
1 Introduction......Page 133
2 Proof of the main theorem......Page 136
References......Page 138
1 Introduction......Page 139
3 Groups with three class sizes......Page 140
4 Groups with four class sizes......Page 143
References......Page 145
1 Introduction......Page 148
2 Cryptography Basics......Page 149
3.1 Conjugacy and exponentiation......Page 151
3.2 Computing a common commutator......Page 152
3.4 Logarithmic signatures......Page 153
3.5 Symmetric schemes......Page 154
4.1 Analysis of braid based schemes......Page 155
4.2 Stickel’s scheme......Page 157
5 Next Steps......Page 158
References......Page 160
2 Groups versus Lattice-ordered Groups......Page 165
3 Right-Orderable Groups......Page 168
4 Amalgamation of Right-Orderable and Right-Ordered Groups......Page 169
5 Right Orders versus Lattice Orders......Page 172
References......Page 174
1 Introduction......Page 176
2 Some preliminaries......Page 178
3.1 The minimum genus problem......Page 179
3.2 The maximum order problem......Page 181
4 Minimum real genus......Page 182
4.1.1 Abelian, cyclic by dihedral, dihedral by dihedral and dicyclic groups......Page 183
4.1.2 Simple groups......Page 185
4.1.3 Other families of groups......Page 186
4.2 Small real genus......Page 187
4.4 The groups in each real genus......Page 188
5 Large groups of automorphisms in prescribed families......Page 189
5.1 The class of all finite groups......Page 190
5.2 Supersolvable groups......Page 191
5.3 Nilpotent groups......Page 192
5.4 p-groups......Page 193
5.5 Solvable groups......Page 194
References......Page 195
1 Introduction......Page 198
2 Conder’s approach revisited......Page 199
3 Our methods and the easy cases......Page 200
4 Commentary on the easy cases......Page 201
5 Harder presentations......Page 203
6 A large finite group......Page 206
7.2 Knuth–Bendix applications......Page 207
8 Concluding remarks......Page 209
References......Page 210
1 Introduction......Page 213
2 Preparatory results......Page 214
3 A generalization of Huppert’s fundamental theorem......Page 215
4 Balance in finite groups......Page 218
5 Further results......Page 223
References......Page 226
1 Direct products......Page 228
2 Semidirect products......Page 233
References......Page 237
1 Introduction......Page 239
2 Main results......Page 240
References......Page 242
2 Notation and Terminology......Page 243
3 Autocommutators......Page 244
4 Applications of the Cauchy–Frobenius Lemma......Page 246
5 On a Theorem of Burnside......Page 247
6 On a Theorem of Thompson......Page 248
7 Commuting Automorphisms......Page 249
8 Orders of Automorphism Groups......Page 251
9 Orbits of Automorphism Groups......Page 253
10 Miscellanea......Page 255
References......Page 256
1 Introduction......Page 259
2 n-Abelian, n-Soluble and n-Nilpotent Groups......Page 260
3 n-Levi and n-Bell Groups......Page 262
4 Locally Graded Bell Groups......Page 264
5 Some combinatorial characterizations......Page 266
6 An arithmetic approach......Page 267
References......Page 269
1 Introduction......Page 271
2.2 The Tits alternative......Page 273
4 The congruence homomorphism......Page 274
5.1 Finiteness testing over number fields......Page 275
5.3 Computing the order......Page 276
6.1 Nilpotency testing......Page 277
7 Solvable groups......Page 278
7.2 Other algorithms......Page 279
8.1 Mal’cev correspondence......Page 280
9 Testing virtual properties; a computational analogue of the Tits alternative......Page 281
10 Irreducibility testing and related problems......Page 282
References......Page 283
1 Introduction......Page 286
2 Linear groups with finite G–orbits......Page 287
3 Linear groups with finite dimensional G–orbits......Page 289
4 Subgroups of infinite central dimension......Page 291
5 Generalizations of finitary linear groups......Page 294
References......Page 296
1 Introduction.......Page 298
2 Orderable Engel groups.......Page 299
3 Engel conditions in combinatorial problems.......Page 305
References......Page 309