Author(s): Norman L. Biggs, A. T. White
Series: London Mathematical Society Lecture Note Series
Publisher: CUP
Year: 1979
Language: English
Pages: 148
Contents......Page 5
Introduction......Page 7
Note on the Projects......Page 8
1. 1 Preliminary definitions......Page 9
1. 2 Counting principles......Page 11
1. 3 Transitivity......Page 13
1. 4 Applications to group theory......Page 17
1. 5 Extensions of multiply transitive groups......Page 19
1. 6 Primitivity......Page 21
1. 7 Regular normal subgroups......Page 25
1. 8 Project: Proof of Sylow's theorem......Page 28
1. 9 Project: Some multiply transitive groups......Page 29
Notes and references......Page 31
2. 1 Introduction......Page 32
2. 2 Finite fields......Page 34
2. 3 Finite vector spaces......Page 36
2. 4 The structure of GL(V) and SL(V)......Page 38
2. 5 Projective spaces and their groups......Page 45
2. 6 More about projective spaces......Page 49
2. 7 The classical simple groups......Page 53
2. 8 Project: Near-fields and sharply 2-transitive groups......Page 57
2. 10 Project: A unitary polarity in PG(2, 9)......Page 59
Notes and references......Page 60
3. 1 Four fundamental problems......Page 62
3. 2 Designs......Page 63
3. 3 Symmetric designs......Page 67
3. 4 Automorphisms of designs......Page 73
3. 5 Extensions of designs......Page 74
3. 6 Mathieu groups and associated designs......Page 78
3. 7 Project: Hadamard matrices and designs......Page 82
3. 9 Project: Uniqueness of the 3 - (22, 6, 1) design......Page 84
Notes and references......Page 86
4. 1 Permutation groups and graphs......Page 88
4. 2 Automorphisms of graphs......Page 93
4. 3 Rank 3 groups and the associated graphs......Page 95
4. 4 Feasibility conditions for strongly regular graphs......Page 97
4. 5 The Higman-Sims group......Page 101
4. 6 Project: Some graphs and their automorphism groups......Page 107
4. 7 Project: Strongly regular graphs and biplanes......Page 108
Notes and references......Page 109
5. 1 Maps and surfaces......Page 111
5. 2 Automorphisms of maps......Page 117
5. 3 Cayley graphs and Cayley maps......Page 123
5. 4 Complete maps and a theorem of Frobenius......Page 130
5. 5 Symmetrical maps......Page 138
5. 6 Project: Generalized Cayley maps......Page 140
5. 7 Project: Paley maps......Page 142
5. 8 Project: Symmetrical Cayley maps......Page 143
Notes and references......Page 144
Index......Page 146
