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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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The Used, Rental and eBook copies of this book are not guaranteed to include any supplemental materials. Typically, only the book itself is included. This is true even if the title states it includes any access cards, study guides, lab manuals, CDs, etc.