Author(s): Professor Robert T. Curtis
Series: Encyclopedia of Mathematics and its Applications
Publisher: CUP
Year: 2007
Language: English
Pages: 331
Contents ......Page 7
Preface ......Page 9
Acknowledgements ......Page 13
I MOTIVATION ......Page 15
Introduction to Part I ......Page 16
1.1 The combinatorial approach ......Page 17
1.2 The regular dodecahedron ......Page 21
1.3 The algebraic approach ......Page 23
1.4 Independent proofs ......Page 24
2.1 The combinatorial approach ......Page 29
2.2 The Klein map ......Page 32
2.3 The algebraic approach ......Page 39
2.4 Independent proofs ......Page 40
Conclusions to Part I ......Page 54
II INVOLUTORY SYMMETRIC GENERATORS ......Page 57
3.1 Free products of cyclic groups of order 2 ......Page 59
3.2 Semi-direct products and the progenitor P ......Page 60
3.3 The Cayley graph of P over N ......Page 64
3.5 Homomorphic images of P ......Page 68
3.6 The lemma ......Page 72
3.7 Further properties of the progenitor ......Page 73
3.8 Coxeter diagrams and Y-diagrams ......Page 76
3.9 Introduction to MAGMA and GAP ......Page 78
3.10 Algorithm for double coset enumeration ......Page 80
3.11 Systematic approach ......Page 88
4.1 The group PGL2(7) ......Page 103
4.2 Exceptional behaviour of Sn ......Page 111
4.3 The 11-point biplane and PGL2(11) ......Page 130
4.4 The group of the 28 bitangents ......Page 135
5.1 The Mathieu group M22 ......Page 141
5.2 The Janko group J1 ......Page 151
5.3 The Higman¨CSims group ......Page 161
5.4 The Hall¨CJanko group and the Suzuki chain ......Page 175
5.5 The Mathieu groups M12 and M24 ......Page 187
5.6 The Janko group J3 ......Page 188
5.7 The Mathieu group M24 as control sub group ......Page 191
5.8 The Fischer groups ......Page 240
5.9 Transitive extensions and the O?ˉNan group ......Page 247
5.10 Symmetric representation of groups ......Page 249
5.11 Appendix to Chapter 5 ......Page 252
III NON-INVOLUTORY SYMMETRIC GENERATORS ......Page 261
6.1 Monomial automorphisms ......Page 263
6.2 Monomial representations ......Page 264
6.3 Monomial action of a control subgroup ......Page 270
7.1 The Mathieu group M11 ......Page 277
7.2 The Mathieu group M23 ......Page 281
7.3 The Mathieu group M24 ......Page 285
7.4 Factoring out a ??classical?ˉ relator ......Page 287
7.5 The Suzuki chain and the Conway group ......Page 302
7.6 Systematic approach ......Page 306
7.7 Tabulated results ......Page 315
7.8 Some sporadic groups ......Page 322
References ......Page 323
Index ......Page 329
