Wilson The Finite Simple Groups

1. Auflage 2009
ISBN: 978-1-84800-988-2
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 251, 310 Seiten

Reihe: Graduate Texts in Mathematics

The finite simple groups are the building blocks from which all the finite groups are made and as such they are objects of fundamental importance throughout mathematics. The classification of the finite simple groups was one of the great mathematical achievements of the twentieth century, yet these groups remain difficult to study which hinders applications of the classification.

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. This is the first time that all the finite simple groups have been treated together in this way and the book points out their connections, for example between exceptional behaviour of generic groups and the existence of sporadic groups, and discusses a number of new approaches to some of the groups. Many exercises of varying difficulty are provided.

Autoren/Hrsg.

Wilson, Robert

Inhaltsverzeichnis

1;Preface;5
2;Contents;9
3;Introduction;16
3.1;A brief history of simple groups;16
3.2;The Classification Theorem;18
3.3;Applications of the Classification Theorem;19
3.4;Remarks on the proof of the Classification Theorem;20
3.5;Prerequisites;21
3.6;Notation;24
3.7;How to read this book;25
4;The alternating groups;26
4.1;Introduction;26
4.2;Permutations;26
4.2.1;The alternating groups;27
4.2.2;Transitivity;28
4.2.3;Primitivity;28
4.2.4;Group actions;29
4.2.5;Maximal subgroups;29
4.2.6;Wreath products;30
4.3;Simplicity;31
4.3.1;Cycle types;31
4.3.2;Conjugacy classes in the alternating groups;31
4.3.3;The alternating groups are simple;32
4.4;Outer automorphisms;33
4.4.1;Automorphisms of alternating groups;33
4.4.2;The outer automorphism of S6;34
4.5;Subgroups of Sn;34
4.5.1;Intransitive subgroups;35
4.5.2;Transitive imprimitive subgroups;35
4.5.3;Primitive wreath products;36
4.5.4;Affine subgroups;36
4.5.5;Subgroups of diagonal type;37
4.5.6;Almost simple groups;37
4.6;The O'Nan--Scott Theorem;38
4.6.1;General results;39
4.6.2;The proof of the O'Nan--Scott Theorem;41
4.7;Covering groups;42
4.7.1;The Schur multiplier;42
4.7.2;The double covers of An and Sn;43
4.7.3;The triple cover of A6;44
4.7.4;The triple cover of A7;45
4.8;Coxeter groups;46
4.8.1;A presentation of Sn;46
4.8.2;Real reflection groups;47
4.8.3;Roots, root systems, and root lattices;48
4.8.4;Weyl groups;49
4.9;Further reading;50
4.10;Exercises;50
5;The classical groups;55
5.1;Introduction;55
5.2;Finite fields;56
5.3;General linear groups;57
5.3.1;The orders of the linear groups;58
5.3.2;Simplicity of PSLn(q);59
5.3.3;Subgroups of the linear groups;60
5.3.4;Outer automorphisms;62
5.3.5;The projective line and some exceptional isomorphisms;64
5.3.6;Covering groups;67
5.4;Bilinear, sesquilinear and quadratic forms;67
5.4.1;Definitions;68
5.4.2;Vectors and subspaces;69
5.4.3;Isometries and similarities;70
5.4.4;Classification of alternating bilinear forms;70
5.4.5;Classification of sesquilinear forms;71
5.4.6;Classification of symmetric bilinear forms;71
5.4.7;Classification of quadratic forms in characteristic 2;72
5.4.8;Witt's Lemma;73
5.5;Symplectic groups;74
5.5.1;Symplectic transvections;75
5.5.2;Simplicity of PSp2m(q);75
5.5.3;Subgroups of symplectic groups;76
5.5.4;Subspaces of a symplectic space;77
5.5.5;Covers and automorphisms;78
5.5.6;The generalised quadrangle;78
5.6;Unitary groups;79
5.6.1;Simplicity of unitary groups;80
5.6.2;Subgroups of unitary groups;81
5.6.3;Outer automorphisms;82
5.6.4;Generalised quadrangles;82
5.6.5;Exceptional behaviour;83
5.7;Orthogonal groups in odd characteristic;83
5.7.1;Determinants and spinor norms;84
5.7.2;Orders of orthogonal groups;85
5.7.3;Simplicity of Pn(q);86
5.7.4;Subgroups of orthogonal groups;88
5.7.5;Outer automorphisms;89
5.8;Orthogonal groups in characteristic 2;90
5.8.1;The quasideterminant and the structure of the groups;90
5.8.2;Properties of orthogonal groups in characteristic 2;91
5.9;Clifford algebras and spin groups;92
5.9.1;The Clifford algebra;93
5.9.2;The Clifford group and the spin group;93
5.9.3;The spin representation;94
5.10;Maximal subgroups of classical groups;95
5.10.1;Tensor products;96
5.10.2;Extraspecial groups;97
5.10.3;The Aschbacher--Dynkin theorem for linear groups;99
5.10.4;The Aschbacher--Dynkin theorem for classical groups;100
5.10.5;Tensor products of spaces with forms;101
5.10.6;Extending the field on spaces with forms;103
5.10.7;Restricting the field on spaces with forms;104
5.10.8;Maximal subgroups of symplectic groups;106
5.10.9;Maximal subgroups of unitary groups;107
5.10.10;Maximal subgroups of orthogonal groups;108
5.11;Generic isomorphisms;110
5.11.1;Low-dimensional orthogonal groups;110
5.11.2;The Klein correspondence;111
5.12;Exceptional covers and isomorphisms;113
5.12.1;Isomorphisms using the Klein correspondence;113
5.12.2;Covering groups of PSU4(3);114
5.12.3;Covering groups of PSL3(4);115
5.12.4;The exceptional Weyl groups;117
5.13;Further reading;119
5.14;Exercises;120
6;The exceptional groups ;124
6.1;Introduction;124
6.2;The Suzuki groups;126
6.2.1;Motivation and definition;126
6.2.2;Generators for Sz(q);128
6.2.3;Subgroups;130
6.2.4;Covers and automorphisms;131
6.3;Octonions and groups of type G2 ;131
6.3.1;Quaternions;131
6.3.2;Octonions;132
6.3.3;The order of G2(q);134
6.3.4;Another basis for the octonions;135
6.3.5;The parabolic subgroups of G2(q);136
6.3.6;Other subgroups of G2(q);138
6.3.7;Simplicity of G2(q);139
6.3.8;The generalised hexagon;141
6.3.9;Automorphisms and covers;141
6.4;Integral octonions ;142
6.4.1;Quaternions in characteristic 2;142
6.4.2;Integral octonions;142
6.4.3;Octonions in characteristic 2;144
6.4.4;The isomorphism between G2(2) and PSU3(3):2;145
6.5;The small Ree groups;147
6.5.1;The outer automorphism of G2(3);147
6.5.2;The Borel subgroup of 2G2(q);148
6.5.3;Other subgroups;150
6.5.4;The isomorphism 2G2(3).5-.5.5-.5.5-.5.5-.5PL2(8);151
6.6;Twisted groups of type 3D4;153
6.6.1;Twisted octonion algebras;153
6.6.2;The order of 3D4(q);153
6.6.3;Simplicity;155
6.6.4;The generalised hexagon;156
6.6.5;Maximal subgroups of 3D4(q);156
6.7;Triality;158
6.7.1;Isotopies;159
6.7.2;The triality automorphism of P8+(q);160
6.7.3;The Klein correspondence revisited;161
6.8;Albert algebras and groups of type F4;161
6.8.1;Jordan algebras;161
6.8.2;A cubic form;162
6.8.3;The automorphism groups of the Albert algebras;163
6.8.4;Another basis for the Albert algebra;164
6.8.5;The normaliser of a maximal torus;166
6.8.6;Parabolic subgroups of F4(q);168
6.8.7;Simplicity of F4(q);170
6.8.8;Primitive idempotents;170
6.8.9;Other subgroups of F4(q);172
6.8.10;Automorphisms and covers of F4(q);174
6.8.11;An integral Albert algebra;175
6.9;The large Ree groups ;176
6.9.1;The outer automorphism of F4(2);176
6.9.2;Generators for the large Ree groups;177
6.9.3;Subgroups of the large Ree groups;178
6.9.4;Simplicity of the large Ree groups;179
6.10;Trilinear forms and groups of type E6;180
6.10.1;The determinant;180
6.10.2;Dickson's construction;182
6.10.3;The normaliser of a maximal torus;183
6.10.4;Parabolic subgroups of E6(q);183
6.10.5;The rank 3 action;184
6.10.6;Covers and automorphisms;185
6.11;Twisted groups of type 2E6;185
6.12;Groups of type E7 and E8;186
6.12.1;Lie algebras;187
6.12.2;Subgroups of E8(q);188
6.12.3;E7(q);190
6.13;Further reading;190
6.14;Exercises;191
7;The sporadic groups;196
7.1;Introduction;196
7.2;The large Mathieu groups;197
7.2.1;The hexacode ;197
7.2.2;The binary Golay code;198
7.2.3;The group M24;200
7.2.4;Uniqueness of the Steiner system S(5,8,24);201
7.2.5;Simplicity of M24;203
7.2.6;Subgroups of M24;203
7.2.7;A presentation of M24;204
7.2.8;The group M23;205
7.2.9;The group M22;206
7.2.10;The double cover of M22;207
7.3;The small Mathieu groups;208
7.3.1;The group M12;208
7.3.2;The Steiner system S(5,6,12);209
7.3.3;Uniqueness of S(5,6,12);210
7.3.4;Simplicity of M12;212
7.3.5;The ternary Golay code;212
7.3.6;The outer automorphism of M12;214
7.3.7;Subgroups of M12;214
7.3.8;The group M11;215
7.4;The Leech lattice and the Conway group;216
7.4.1;The Leech lattice;216
7.4.2;The Conway group Co1;218
7.4.3;Simplicity of Co1;219
7.4.4;The small Conway groups;219
7.4.5;The Leech lattice modulo 2;221
7.5;Sublattice groups;223
7.5.1;The Higman--Sims group HS;223
7.5.2;The McLaughlin group McL;227
7.5.3;The group Co3;229
7.5.4;The group Co2;230
7.6;The Suzuki chain;232
7.6.1;The Hall--Janko group J2;233
7.6.2;The icosians;233
7.6.3;The icosian Leech lattice;234
7.6.4;Properties of the Hall--Janko group;235
7.6.5;Identification with the Leech lattice;236
7.6.6;J2 as a permutation group;236
7.6.7;Subgroups of J2;237
7.6.8;The exceptional double cover of G2(4);237
7.6.9;The map onto G2(4);239
7.6.10;The complex Leech lattice;240
7.6.11;The Suzuki group;242
7.6.12;An octonion Leech lattice;243
7.7;The Fischer groups;247
7.7.1;A graph on 3510 vertices;248
7.7.2;The group Fi22;250
7.7.3;Conway's description of Fi22;254
7.7.4;Covering groups of Fi22;255
7.7.5;Subgroups of Fi22;256
7.7.6;The group Fi23;256
7.7.7;Subgroups of Fi23;259
7.7.8;The group Fi24;259
7.7.9;Parker's loop;260
7.7.10;The triple cover of Fi24';261
7.7.11;Subgroups of Fi24;263
7.8;The Monster and subgroups of the Monster;263
7.8.1;The Monster;264
7.8.2;The Griess algebra;268
7.8.3;6-transpositions;269
7.8.4;Monstralisers and other subgroups;269
7.8.5;The Y-group presentations;270
7.8.6;The Baby Monster;272
7.8.7;The Thompson group;273
7.8.8;The Harada--Norton group;275
7.8.9;The Held group;276
7.8.10;Ryba's algebra;277
7.9;Pariahs;278
7.9.1;The first Janko group J1;280
7.9.2;The third Janko group J3;281
7.9.3;The Rudvalis group;283
7.9.4;The O'Nan group;285
7.9.5;The Lyons group;287
7.9.6;The largest Janko group J4;289
7.10;Further reading;291
7.11;Exercises;292
8;References;295
9;Index;303