CONTENTS
Preface = ⅸ
PART Ⅰ
1. Preliminary Results = 1
1. Abstract representations = 1
2. Permutation representations = 2
3. Graphs = 4
4. Geometries and complexes = 6
5. The general linear group and its projective geometry = 13
6. Fiber products of groups = 15
2. 2-Structure in Finite Group = 18
7. Involutions = 18
8. Extraspecial groups = 21
3. Algebras, Codes, and Forms = 35
9. Forms and algebras = 36
10. Codes = 40
11. Derived forms = 41
4. Symplectic 2-Loops = 46
12. Symplectic 2-loops = 47
13. Moufang symplectic 2-loops = 54
14. Constructing a 2-local from a loop = 57
5. The Discovery, Existence, and Uniqueness of the Sporadics = 65
15. History and discovery = 65
16. Existence of the sporadics = 70
17. Uniqueness of the sporadics = 74
PART Ⅱ
6. The Mathieu Groups, Their Steiner Systems, and the Golay Code = 77
18. Steiner systems for the Mathieu groups = 78
19. The Golay and Todd modules = 85
7. The Geometry and Structure of M2 4 = 96
20. The geometry of M2 4 = 96
21. The local structure of M2 4 = 103
8. The Conway Groups and the Leech Lattice = 108
22. The Leech lattice and ·0 = 108
23. The Leech lattice mod 2 = 116
9. Subgroups of ·0 = 124
24. The groups Co3 , Mc, and HS = 125
25. The groups Co1 , Co2 , Suz, and J-2 = 132
26. Some local subgroups of Co1 = 135
10. The Griess Algebra and the Monster = 142
27. The subgroups C and N of the Monster = 143
28. The Griess algebra = 151
29. The action of N on B = 154
30. N preserves the Griess algebra = 162
31. The automorphism group of the Griess algebra = 167
11. Subgroups of Groups of Monster Type = 172
32. Subgroups of groups of Monster type = 173
PART Ⅲ
12. Coverings of Graphs and Simplicial Complexes = 175
33. The fundamental groupoid = 176
34. Triangulation = 182
35. Coverings of graphs and simplicial complexes = 184
13. The Geometry of Amalgams = 194
36. Amalgams = 195
37. Uniqueness systems = 198
38. The uniqueness systems of a string geometry = 204
14. The Uniqueness of Groups of Type M2 4 , He, and L5 (2) = 212
39. Some 2-local subgroups in L5 (2), M2 4 , and He = 213
40. Groups of type L5 (2), M2 4 , and He = 216
41. Groups of type L5 (2) and M2 4 = 219
42. Groups of type He = 223
43. The root 4-group graph for He = 230
44. The uniqueness of groups of type He = 236
15. The Group U4 (3) = 241
45. U4 (3) = 242
16. Groups of Conway, Suzuki, and Hall-Janko Type = 250
46. Groups of type Co1 , Suz, J2 , and J-3 = 250
47. Groups of type J2 = 259
48. Groups of type Suz = 267
49. Groups of type Co1 = 279
17. Subgroups of Prime Order in Five Sporadic Groups = 293
50. Subgroups of Suz of prime order = 293
51. Subgroups of Co1 of prime order = 295
52. Subgroups of prime order in He = 299
Symbols = 304
Bibliography = 306
Index = 311
