Enumeration of Finite Groups

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.

Preface	p. xi
Introduction	p. 1
Elementary Results	p. 3
Some basic observations	p. 5
Groups of Prime Power Order	p. 9
Preliminaries	p. 11
Tensor products and exterior squares of abelian groups	p. 11
Commutators and nilpotent groups	p. 12
The Frattini subgroup	p. 17
Linear algebra	p. 19
Enumerating p-groups: a lower bound	p. 23
Relatively free groups	p. 23
Proof of the lower bound	p. 26
Enumerating p-groups: upper bounds	p. 28
An elementary upper bound	p. 28
An overview of the Sims approach	p. 30
'Linearising' the problem	p. 31
A small set of relations	p. 35
Proof of the upper bound	p. 40
Pyber's Theorem	p. 45
Some more preliminaries	p. 47
Hall subgroups and Sylow systems	p. 47
The Fitting subgroup	p. 50
Permutations and primitivity	p. 52
Group extensions and cohomology	p. 60
Group extensions	p. 60
Cohomology	p. 67
Restriction and transfer	p. 73
The McIver and Neumann bound	p. 75
Some representation theory	p. 78
Semisimple algebras	p. 78
Clifford's theorem	p. 80
The Skolem-Noether theorem	p. 81
Every finite skew field is a field	p. 85
Primitive soluble linear groups	p. 88
Some basic structure theory	p. 88
The subgroup B	p. 90
The orders of groups	p. 94
Conjugacy classes of maximal soluble subgroups of symmetric groups	p. 98
Enumeration of finite groups with abelian Sylow subgroups	p. 102
Counting soluble A-groups: an overview	p. 103
Soluble A-subgroups of the general linear group and the symmetric groups	p. 103
Maximal soluble p'-A-subgroups	p. 108
Enumeration of soluble A-groups	p. 109
Maximal soluble linear groups	p. 113
The field K and a subfield of K	p. 113
The quotient G/C and the algebra	p. 114
The quotient B/A	p. 116
The subgroup B	p. 119
Structure of G determined by B	p. 125
Conjugacy classes of maximal soluble subgroups of the general linear groups	p. 127
Pyber's theorem: the soluble case	p. 132
Extensions and soluble subgroups	p. 133
Pyber's theorem	p. 135
Pyber's theorem: the general case	p. 140
Three theorems on group generation	p. 140
Universal central extensions and covering groups	p. 146
The generalised Fitting subgroup	p. 150
The general case of Pyber's theorem	p. 154
Other Topics	p. 161
Enumeration within varieties of abelian groups	p. 163
Varieties of abelian groups	p. 164
Enumerating partitions	p. 167
Further results on abelian groups	p. 173
Enumeration within small varieties of A-groups	p. 174
A minimal variety of A-groups	p. 175
The join of minimal varieties	p. 184
Enumeration within small varieties of p-groups	p. 187
Enumerating two small varieties	p. 189
The ratio of two enumeration functions	p. 191
Miscellanea	p. 195
Enumerating d-generator groups	p. 195
Groups with few non-abelian composition factors	p. 206
Enumerating graded Lie rings	p. 211
Groups of nilpotency class 3	p. 216
Survey of other results	p. 222
Graham Higman's PORC conjecture	p. 222
Isoclinism classes of p-groups	p. 224
Groups of square-free order	p. 227
Groups of cube-free order	p. 233
Groups of arithmetically small orders	p. 236
Surjectivity of the enumeration function	p. 238
Densities of certain sets of group orders	p. 246
Enumerating perfect groups	p. 256
Some open problems	p. 259
Maximising two functions	p. 269
References	p. 275
Index	p. 280
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