Finite Simple Groups (eBook)

Preface 5
Contents 9
Introduction 16
A brief history of simple groups 16
The Classification Theorem 18
Applications of the Classification Theorem 19
Remarks on the proof of the Classification Theorem 20
Prerequisites 21
Notation 24
How to read this book 25
The alternating groups 26
Introduction 26
Permutations 26
The alternating groups 27
Transitivity 28
Primitivity 28
Group actions 29
Maximal subgroups 29
Wreath products 30
Simplicity 31
Cycle types 31
Conjugacy classes in the alternating groups 31
The alternating groups are simple 32
Outer automorphisms 33
Automorphisms of alternating groups 33
The outer automorphism of S6 34
Subgroups of Sn 34
Intransitive subgroups 35
Transitive imprimitive subgroups 35
Primitive wreath products 36
Affine subgroups 36
Subgroups of diagonal type 37
Almost simple groups 37
The O'Nan--Scott Theorem 38
General results 39
The proof of the O'Nan--Scott Theorem 41
Covering groups 42
The Schur multiplier 42
The double covers of An and Sn 43
The triple cover of A6 44
The triple cover of A7 45
Coxeter groups 46
A presentation of Sn 46
Real reflection groups 47
Roots, root systems, and root lattices 48
Weyl groups 49
Further reading 50
Exercises 50
The classical groups 55
Introduction 55
Finite fields 56
General linear groups 57
The orders of the linear groups 58
Simplicity of PSLn(q) 59
Subgroups of the linear groups 60
Outer automorphisms 62
The projective line and some exceptional isomorphisms 64
Covering groups 67
Bilinear, sesquilinear and quadratic forms 67
Definitions 68
Vectors and subspaces 69
Isometries and similarities 70
Classification of alternating bilinear forms 70
Classification of sesquilinear forms 71
Classification of symmetric bilinear forms 71
Classification of quadratic forms in characteristic 2 72
Witt's Lemma 73
Symplectic groups 74
Symplectic transvections 75
Simplicity of PSp2m(q) 75
Subgroups of symplectic groups 76
Subspaces of a symplectic space 77
Covers and automorphisms 78
The generalised quadrangle 78
Unitary groups 79
Simplicity of unitary groups 80
Subgroups of unitary groups 81
Outer automorphisms 82
Generalised quadrangles 82
Exceptional behaviour 83
Orthogonal groups in odd characteristic 83
Determinants and spinor norms 84
Orders of orthogonal groups 85
Simplicity of Pn(q) 86
Subgroups of orthogonal groups 88
Outer automorphisms 89
Orthogonal groups in characteristic 2 90
The quasideterminant and the structure of the groups 90
Properties of orthogonal groups in characteristic 2 91
Clifford algebras and spin groups 92
The Clifford algebra 93
The Clifford group and the spin group 93
The spin representation 94
Maximal subgroups of classical groups 95
Tensor products 96
Extraspecial groups 97
The Aschbacher--Dynkin theorem for linear groups 99
The Aschbacher--Dynkin theorem for classical groups 100
Tensor products of spaces with forms 101
Extending the field on spaces with forms 103
Restricting the field on spaces with forms 104
Maximal subgroups of symplectic groups 106
Maximal subgroups of unitary groups 107
Maximal subgroups of orthogonal groups 108
Generic isomorphisms 110
Low-dimensional orthogonal groups 110
The Klein correspondence 111
Exceptional covers and isomorphisms 113
Isomorphisms using the Klein correspondence 113
Covering groups of PSU4(3) 114
Covering groups of PSL3(4) 115
The exceptional Weyl groups 117
Further reading 119
Exercises 120
The exceptional groups 124
Introduction 124
The Suzuki groups 126
Motivation and definition 126
Generators for Sz(q) 128
Subgroups 130
Covers and automorphisms 131
Octonions and groups of type G2 131
Quaternions 131
Octonions 132
The order of G2(q) 134
Another basis for the octonions 135
The parabolic subgroups of G2(q) 136
Other subgroups of G2(q) 138
Simplicity of G2(q) 139
The generalised hexagon 141
Automorphisms and covers 141
Integral octonions 142
Quaternions in characteristic 2 142
Integral octonions 142
Octonions in characteristic 2 144
The isomorphism between G2(2) and PSU3(3):2 145
The small Ree groups 147
The outer automorphism of G2(3) 147
The Borel subgroup of 2G2(q) 148
Other subgroups 150
The isomorphism 2G2(3).5-.5.5-.5.5-.5.5-.5PL2(8) 151
Twisted groups of type 3D4 153
Twisted octonion algebras 153
The order of 3D4(q) 153
Simplicity 155
The generalised hexagon 156
Maximal subgroups of 3D4(q) 156
Triality 158
Isotopies 159
The triality automorphism of P8+(q) 160
The Klein correspondence revisited 161
Albert algebras and groups of type F4 161
Jordan algebras 161
A cubic form 162
The automorphism groups of the Albert algebras 163
Another basis for the Albert algebra 164
The normaliser of a maximal torus 166
Parabolic subgroups of F4(q) 168
Simplicity of F4(q) 170
Primitive idempotents 170
Other subgroups of F4(q) 172
Automorphisms and covers of F4(q) 174
An integral Albert algebra 175
The large Ree groups 176
The outer automorphism of F4(2) 176
Generators for the large Ree groups 177
Subgroups of the large Ree groups 178
Simplicity of the large Ree groups 179
Trilinear forms and groups of type E6 180
The determinant 180
Dickson's construction 182
The normaliser of a maximal torus 183
Parabolic subgroups of E6(q) 183
The rank 3 action 184
Covers and automorphisms 185
Twisted groups of type 2E6 185
Groups of type E7 and E8 186
Lie algebras 187
Subgroups of E8(q) 188
E7(q) 190
Further reading 190
Exercises 191
The sporadic groups 196
Introduction 196
The large Mathieu groups 197
The hexacode 197
The binary Golay code 198
The group M24 200
Uniqueness of the Steiner system S(5,8,24) 201
Simplicity of M24 203
Subgroups of M24 203
A presentation of M24 204
The group M23 205
The group M22 206
The double cover of M22 207
The small Mathieu groups 208
The group M12 208
The Steiner system S(5,6,12) 209
Uniqueness of S(5,6,12) 210
Simplicity of M12 212
The ternary Golay code 212
The outer automorphism of M12 214
Subgroups of M12 214
The group M11 215
The Leech lattice and the Conway group 216
The Leech lattice 216
The Conway group Co1 218
Simplicity of Co1 219
The small Conway groups 219
The Leech lattice modulo 2 221
Sublattice groups 223
The Higman--Sims group HS 223
The McLaughlin group McL 227
The group Co3 229
The group Co2 230
The Suzuki chain 232
The Hall--Janko group J2 233
The icosians 233
The icosian Leech lattice 234
Properties of the Hall--Janko group 235
Identification with the Leech lattice 236
J2 as a permutation group 236
Subgroups of J2 237
The exceptional double cover of G2(4) 237
The map onto G2(4) 239
The complex Leech lattice 240
The Suzuki group 242
An octonion Leech lattice 243
The Fischer groups 247
A graph on 3510 vertices 248
The group Fi22 250
Conway's description of Fi22 254
Covering groups of Fi22 255
Subgroups of Fi22 256
The group Fi23 256
Subgroups of Fi23 259
The group Fi24 259
Parker's loop 260
The triple cover of Fi24' 261
Subgroups of Fi24 263
The Monster and subgroups of the Monster 263
The Monster 264
The Griess algebra 268
6-transpositions 269
Monstralisers and other subgroups 269
The Y-group presentations 270
The Baby Monster 272
The Thompson group 273
The Harada--Norton group 275
The Held group 276
Ryba's algebra 277
Pariahs 278
The first Janko group J1 280
The third Janko group J3 281
The Rudvalis group 283
The O'Nan group 285
The Lyons group 287
The largest Janko group J4 289
Further reading 291
Exercises 292
References 295
Index 303