有限単純群<br>The Finite Simple Groups (Graduate Texts in Mathematics) 〈Vol. 251〉

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有限単純群
The Finite Simple Groups (Graduate Texts in Mathematics) 〈Vol. 251〉

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  • 製本 Hardcover:ハードカバー版/ページ数 233 p.
  • 言語 ENG
  • 商品コード 9781848009875
  • DDC分類 512

基本説明

This textbook brings the finite simple groups to life by giving concrete constructions of most of them, sufficient to illuminate their structure and permit real calculations both in the groups themselves and in the underlying geometrical or algebraic structures.

Table of Contents

1 Introduction                                     1
1.1 A brief history of simple groups 1
1.2 The Classification Theorem 3
1.3 Applications of the Classification Theorem 4
1.4 Remarks on the proof of the 5
Classification Theorem
1.5 Prerequisites 6
1.6 Notation 9
1.7 How to read this book 10
2 The alternating groups 11
2.1 Introduction 11
2.2 Permutations 11
2.2.1 The alternating groups 12
2.2.2 Transitivity 13
2.2.3 Primitivity 13
2.2.4 Group actions 14
2.2.5 Maximal subgroups 14
2.2.6 Wreath products 15
2.3 Simplicity 16
2.3.1 Cycle types 16
2.3.2 Conjugacy classes in the alternating 16
groups
2.3.3 The alternating groups are simple 17
2.4 Outer automorphisms 18
2.4.1 Automorphisms of alternating groups 18
2.4.2 The outer automorphism of 86 19
2.5 Subgroups of Sn 19
2.5.1 Intransitive subgroups 20
2.5.2 Transitive imprimitive subgroups 20
2.5.3 Primitive wreath products 21
2.5.4 Affine subgroups 21
2.5.5 Subgroups of diagonal type 22
2.5.6 Almost simple groups 22
2.6 The O'Nan-Scott Theorem 23
2.6.1 General results 24
2.6.2 The proof of the O'Nan-Scott Theorem 26
2.7 Covering groups 27
2.7.1 The Schur multiplier 27
2.7.2 The double covers of An and Sn 28
2.7.3 The triple cover of A6 29
2.7.4 The triple cover of A7 30
2.8 Coxeter groups 31
2.8.1 A presentation of Sia 31
2.8.2 Real reflection groups 32
2.8.3 Roots, root systems, and root lattices 33
2.8.4 Weyl groups 34
Further reading 35
Exercises 35
3 The classical groups 41
3.1 Introduction 41
3.2 Finite fields 42
3.3 General linear groups 43
3.3.1 The orders of the linear groups 44
3.3.2 Simplicity of PSLn(q) 45
3.3.3 Subgroups of the linear groups 46
3.3.4 Outer automorphisms 48
3.3.5 The projective line and some 50
exceptional isomorphisms
3.3.6 Covering groups 53
3.4 Bilinear, sesquilinear and quadratic 53
forms
3.4.1 Definitions 54
3.4.2 Vectors and subspaces 55
3.4.3 Isometries and similarities 56
3.4.4 Classification of alternating 56
bilinear forms
3.4.5 Classification of sesquilinear forms 57
3.4.6 Classification of symmetric bilinear 57
forms
3.4.7 Classification of quadratic forms in 58
characteristic 2
3.4.8 Witt's Lemma 59
3.5 Symplectic groups 60
3.5.1 Symplectic transvections 61
3.5.2 Simplicity of PSp2m(q) 61
3.5.3 Subgroups of symplectic groups 62
3.5.4 Subspaces of a symplectic space 63
3.5.5 Covers and automorphisms 64
3.5.6 The generalised quadrangle 64
3.6 Unitary groups 65
3.6.1 Simplicity of unitary groups 66
3.6.2 Subgroups of unitary groups 67
3.6.3 Outer automorphisms 68
3.6.4 Generalised quadrangles 68
3.6.5 Exceptional behaviour 69
3.7 Orthogonal groups in odd characteristic 69
3.7.1 Determinants and spinor norms 70
3.7.2 Orders of orthogonal groups 71
3.7.3 Simplicity of PΩn (q) 72
3.7.4 Subgroups of orthogonal groups 74
3.7.5 Outer automorphisms 75
3.8 Orthogonal groups in characteristic 2 76
3.8.1 The quasideterminant and the 76
structure of the groups
3.8.2 Properties of orthogonal groups in 77
characteristic 2
3.9 Clifford algebras and spin groups 78
3.9.1 The Clifford algebra 79
3.9.2 The Clifford group and the spin group 79
3.9.3 The spin representation 80
3.10 Maximal subgroups of classical groups 81
3.10.1 Tensor products 82
3.10.2 Extraspecial groups 83
3.10.3 The Aschbacher縫ynkin theorem for 85
linear groups
3.10.4 The Aschbacher縫ynkin theorem for 86
classical groups
3.10.5 Tensor products of spaces with forms 87
3.10.6 Extending the field on spaces with 89
forms
3.10.7 Restricting the field on spaces 90
with forms
3.10.8 Maximal subgroups of symplectic 92
groups
3.10.9 Maximal subgroups of unitary groups 93
3.10.10 Maximal subgroups of orthogonal 94
groups
3.11 Generic isomorphisms 96
3.11.1 Low-dimensional orthogonal groups 96
3.11.2 The Klein correspondence 97
3.12 Exceptional covers and isomorphisms 99
3.12.1 Isomorphisms using the Klein 99
correspondence
3.12.2 Covering groups of PSU4(3) 100
3.12.3 Covering groups of PSL3(4) 101
3.12.4 The exceptional Weyl groups 103
Further reading 105
Exercises 106
4 The exceptional groups 111
4.1 Introduction 111
4.2 The Suzuki groups 113
4.2.1 Motivation and definition 113
4.2.2 Generators for Sz(q) 115
4.2.3 Subgroups 117
4.2.4 Covers and automorphisms 118
4.3 Octonions and groups of type G2 118
4.3.1 Quaternions 118
4.3.2 Octonions 119
4.3.3 The order of G2(q) 121
4.3.4 Another basis for the octonions 122
4.3.5 The parabolic subgroups of G2(q) 123
4.3.6 Other subgroups of G2(q) 125
4.3.7 Simplicity of G2(q) 126
4.3.8 The generalised hexagon 128
4.3.9 Automorphisms and covers 128
4.4 Integral octonions 129
4.4.1 Quaternions in characteristic 2 129
4.4.2 Integral octonions 129
4.4.3 Octonions in characteristic 2 131
4.4.4 The isomorphism between G2(2) and 132
PSU3(3):2
4.5 The small Ree groups 134
4.5.1 The outer automorphism of G2(3) 134
4.5.2 The Borel subgroup of イG2(q) 135
4.5.3 Other subgroups 137
4.5.4 The isomorphism イG2(3) approximately 138
= to PΓL2(8)
4.6 Twisted groups of type 3D4 140
4.6.1 Twisted octonion algebras 140
4.6.2 The order of ウD4(q) 140
4.6.3 Simplicity 142
4.6.4 The generalised hexagon 143
4.6.5 Maximal subgroups of ウD4(q) 143
4.7 Triality 145
4.7.1 Isotopies 146
4.7.2 The triality automorphism of 147
PΩ+8 (q)
4.7.3 The Klein correspondence revisited 148
4.8 Albert algebras and groups of type F4 148
4.8.1 Jordan algebras 148
4.8.2 A cubic form 149
4.8.3 The automorphism groups of the 150
Albert algebras
4.8.4 Another basis for the Albert algebra 151
4.8.5 The normaliser of a maximal torus 153
4.8.6 Parabolic subgroups of F4(q) 155
4.8.7 Simplicity of F4(q) 157
4.8.8 Primitive idempotents 157
4.8.9 Other subgroups of F4(q) 159
4.8.10 Automorphisms and covers of F4(q) 161
4.8.11 An integral Albert algebra 162
4.9 The large Ree groups 163
4.9.1 The outer automorphism of F4(2) 163
4.9.2 Generators for the large Ree groups 164
4.9.3 Subgroups of the large Ree groups 165
4.9.4 Simplicity of the large Ree groups 166
4.10 Trilinear forms and groups of type E6 167
4.10.1 The determinant 167
4.10.2 Dickson's construction 169
4.10.3 The normaliser of a maximal torus 170
4.10.4 Parabolic subgroups of E6(q) 170
4.10.5 The rank 3 action 171
4.10.6 Covers and automorphisms 172
4.11 Twisted groups of type イE6 172
4.12 Groups of type E7 and E8 173
4.12.1 Lie algebras 174
4.12.2 Subgroups of E8(q) 175
4.12.3 E7(q) 177
Further reading 177
Exercises 178
5 The sporadic groups 183
5.1 Introduction 183
5.2 The large Mathieu groups 184
5.2.1 The hexacode 184
5.2.2 The binary Golay code 185
5.2.3 The group M24 187
5.2.4 Uniqueness of the Steiner system 188
S(5, 8, 24)
5.2.5 Simplicity of M24 190
5.2.6 Subgroups of M24 190
5.2.7 A presentation of M24 191
5.2.8 The group M23 192
5.2.9 The group M22 193
5.2.10 The double cover of M22 194
5.3 The small Mathieu groups 195
5.3.1 The group M12 195
5.3.2 The Steiner system S(5, 6,12) 196
5.3.3 Uniqueness of 5(5,6,12) 197
5.3.4 Simplicity of M12 199
5.3.5 The ternary Golay code 199
5.3.6 The outer automorphism of M12 201
5.3.7 Subgroups of M12 201
5.3.8 The group M11 202
5.4 The Leech lattice and the Conway group 203
5.4.1 The Leech lattice 203
5.4.2 The Conway group Co1 205
5.4.3 Simplicity of Co1 206
5.4.4 The small Conway groups 206
5.4.5 The Leech lattice modulo 2 208
5.5 Sublattice groups 210
5.5.1 The Higman祐ims group HS 210
5.5.2 The McLaughlin group McL 214
5.5.3 The group Co3 216
5.5.4 The group Co2 217
5.6 The Suzuki chain 219
5.6.1 The Hall憂anko group J2 220
5.6.2 The icosians 220
5.6.3 The icosian Leech lattice 221
5.6.4 Properties of the Hall憂anko group 222
5.6.5 Identification with the Leech lattice 223
5.6.6 J2 as a permutation group 223
5.6.7 Subgroups of J2 224
5.6.8 The exceptional double cover of G2(4) 224
5.6.9 The map onto G2(4) 226
5.6.10 The complex Leech lattice 227
5.6.11 The Suzuki group 229
5.6.12 An octonion Leech lattice 230
5.7 The Fischer groups 234
5.7.1 A graph on 3510 vertices 235
5.7.2 The group Fi22 237
5.7.3 Conway's description of Fi22 241
5.7.4 Covering groups of Fi22 242
5.7.5 Subgroups of Fi22 243
5.7.6 The group Fi23 243
5.7.7 Subgroups of Fi23 246
5.7.8 The group Fi'24 246
5.7.9 Parker's loop 247
5.7.10 The triple cover of Film 248
5.7.11 Subgroups of Fi24 250
5.8 The Monster and subgroups of the Monster 250
5.8.1 The Monster 251
5.8.2 The Griess algebra 255
5.8.3 6-transpositions 256
5.8.4 Monstralisers and other subgroups 256
5.8.5 The Y-group presentations 257
5.8.6 The Baby Monster 259
5.8.7 The Thompson group 260
5.8.8 The Harada湧orton group 262
5.8.9 The Held group 263
5.8.10 Ryba's algebra 264
5.9 Pariahs 265
5.9.1 The first Janko group J1 267
5.9.2 The third Janko group J3 268
5.9.3 The Rudvalis group 270
5.9.4 The O'Nan group 272
5.9.5 The Lyons group 274
5.9.6 The largest Janko group J4 276
Further reading 278
Exercises 279
References 283
Index 291