Enumeration of finite groups

Simon R. Blackburn, Peter M. Neumann OBE, Geetha Venkataraman0521882176, 9780521882170, 9780511355899

How many groups of order n are there? This is a natural question for anyone studying group theory, and this Tract provides an exhaustive and up-to-date account of research into this question spanning almost fifty years. The authors presuppose an undergraduate knowledge of group theory, up to and including Sylow’s Theorems, a little knowledge of how a group may be presented by generators and relations, a very little representation theory from the perspective of module theory, and a very little cohomology theory – but most of the basics are expounded here and the book is more or less self-contained. Although it is principally devoted to a connected exposition of an agreeable theory, the book does also contain some material that has not hitherto been published. It is designed to be used as a graduate text but also as a handbook for established research workers in group theory.

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Table of contents :
Half-title……Page 3
Title……Page 5
Copyright……Page 6
Dedication……Page 7
Contents……Page 9
how many groups of order n are there?……Page 13
1 Introduction……Page 15
I Elementary results……Page 17
2 Some basic observations……Page 19
II Groups of prime power order……Page 23
3.1 Tensor products and exterior squares of abelian groups……Page 25
3.2 Commutators and nilpotent groups……Page 26
3.3 The Frattini subgroup……Page 31
3.4 Linear algebra……Page 33
4.1 Relatively free groups……Page 37
4.2 Proof of the lower bound……Page 40
5.1 An elementary upper bound……Page 42
5.2 An overview of the Sims approach……Page 44
5.3 ‘Linearising’ the problem……Page 45
5.4 A small set of relations……Page 49
5.5 Proof of the upper bound……Page 54
III Pyber’s theorem……Page 59
6.1 Hall subgroups and Sylow systems……Page 61
6.2 The Fitting subgroup……Page 64
6.3 Permutations and primitivity……Page 66
7.1 Group extensions……Page 74
7.2 Cohomology……Page 81
7.3 Restriction and transfer……Page 87
7.4 The McIver and Neumann bound……Page 89
8.1 Semisimple algebras……Page 92
8.2 Clifford’s theorem……Page 94
8.3 The Skolem–Noether theorem……Page 95
8.4 Every finite skew field is a field……Page 99
9.1 Some basic structure theory……Page 102
9.2 The subgroup B……Page 104
10 The orders of groups……Page 108
11 Conjugacy classes of maximal soluble subgroups of symmetric groups……Page 112
12 Enumeration of finite groups with abelian Sylow subgroups……Page 116
12.2 Soluble A-subgroups of the general linear group and the symmetric group……Page 117
12.3 Maximal soluble p’-A-subgroups……Page 122
12.4 Enumeration of soluble A-groups……Page 123
13.1 The field and a subfield of……Page 127
13.2 The quotient G/C and the algebra……Page 128
13.3 The quotient B/A……Page 130
13.4 The subgroup B……Page 133
13.5 Structure of G determined by B……Page 139
14 Conjugacy classes of maximal soluble subgroups of the general linear group……Page 141
15 Pyber’s theorem: the soluble case……Page 146
15.1 Extensions and soluble subgroups……Page 147
15.2 Pyber’s theorem……Page 149
16.1 Three theorems on group generation……Page 154
16.2 Universal central extensions and covering groups……Page 160
16.3 The generalised Fitting subgroup……Page 164
16.4 The general case of Pyber’s theorem……Page 168
IV Other topics……Page 175
17 Enumeration within varieties of abelian groups……Page 177
17.1 Varieties of abelian groups……Page 178
17.2 Enumerating partitions……Page 181
17.3 Further results on abelian groups……Page 187
18 Enumeration within small varieties of A-groups……Page 188
18.1 A minimal variety of A-groups……Page 189
18.2 The join of minimal varieties……Page 198
19 Enumeration within small varieties of p-groups……Page 201
19.1 Enumerating two small varieties……Page 203
19.2 The ratio of two enumeration functions……Page 205
20.1 Enumerating d-generator groups……Page 209
20.2 Groups with few non-abelian composition factors……Page 220
20.3 Enumerating graded Lie rings……Page 225
20.4 Groups of nilpotency class 3……Page 230
21.1 Graham Higman’s PORC conjecture……Page 236
21.2 Isoclinism classes of p-groups……Page 238
21.3 Groups of square-free order……Page 241
21.4 Groups of cube-free order……Page 247
21.5 Groups of arithmetically small orders……Page 250
21.6 Surjectivity of the enumeration function……Page 252
21.7 Densities of certain sets of group orders……Page 260
21.8 Enumerating perfect groups……Page 270
22 Some open problems……Page 273
Appendix A: Maximising two functions……Page 283
References……Page 289
Index……Page 294