Enumeration of finite groups / Simon R. Blackburn, Peter M. Neumann, Geetha Venkataraman.

Format:

Book

Author/Creator:

Blackburn, Simon R.

Contributor:

Neumann, P. M.

Venkataraman, Geetha.

Series:

Cambridge tracts in mathematics ; 173.

Cambridge tracts in mathematics ; 173

Language:

English

Subjects (All):

Finite groups.

Physical Description:

xii, 281 pages : illustrations ; 24 cm.

Place of Publication:

Cambridge ; New York : Cambridge University Press, 2007.

Contents:

I Elementary Results 3

2 Some basic observations 5

II Groups of Prime Power Order 9

3.1 Tensor products and exterior squares of abelian groups 11

3.2 Commutators and nilpotent groups 12

3.3 The Frattini subgroup 17

3.4 Linear algebra 19

4 Enumerating p-groups: a lower bound 23

4.1 Relatively free groups 23

4.2 Proof of the lower bound 26

5 Enumerating p-groups: upper bounds 28

5.1 An elementary upper bound 28

5.2 An overview of the Sims approach 30

5.3 'Linearising' the problem 31

5.4 A small set of relations 35

5.5 Proof of the upper bound 40

III Pyber's Theorem 45

6 Some more preliminaries 47

6.1 Hall subgroups and Sylow systems 47

6.2 The Fitting subgroup 50

6.3 Permutations and primitivity 52

7 Group extensions and cohomology 60

7.1 Group extensions 60

7.2 Cohomology 67

7.3 Restriction and transfer 73

7.4 The McIver and Neumann bound 75

8 Some representation theory 78

8.1 Semisimple algebras 78

8.2 Clifford's theorem 80

8.3 The Skolem-Noether theorem 81

8.4 Every finite skew field is a field 85

9 Primitive soluble linear groups 88

9.1 Some basic structure theory 88

9.2 The subgroup B 90

10 The orders of groups 94

11 Conjugacy classes of maximal soluble subgroups of symmetric groups 98

12 Enumeration of finite groups with abelian Sylow subgroups 102

12.1 Counting soluble A-groups: an overview 103

12.2 Soluble A-subgroups of the general linear group and the symmetric groups 103

12.3 Maximal soluble p'-A-subgroups 108

12.4 Enumeration of soluble A-groups 109

13 Maximal soluble linear groups 113

13.1 The field K and a subfield of K 113

13.2 The quotient G/C and the algebra <C> 114

13.3 The quotient B/A 116

13.4 The subgroup B 119

13.5 Structure of G determined by B 125

14 Conjugacy classes of maximal soluble subgroups of the general linear groups 127

15 Pyber's theorem: the soluble case 132

15.1 Extensions and soluble subgroups 133

15.2 Pyber's theorem 135

16 Pyber's theorem: the general case 140

16.1 Three theorems on group generation 140

16.2 Universal central extensions and covering groups 146

16.3 The generalised Fitting subgroup 150

16.4 The general case of Pyber's theorem 154

IV Other Topics 161

17 Enumeration within varieties of abelian groups 163

17.1 Varieties of abelian groups 164

17.2 Enumerating partitions 167

17.3 Further results on abelian groups 173

18 Enumeration within small varieties of A-groups 174

18.1 A minimal variety of A-groups 175

18.2 The join of minimal varieties 184

19 Enumeration within small varieties of p-groups 187

19.1 Enumerating two small varieties 189

19.2 The ratio of two enumeration functions 191

20 Miscellanea 195

20.1 Enumerating d-generator groups 195

20.2 Groups with few non-abelian composition factors 206

20.3 Enumerating graded Lie rings 211

20.4 Groups of nilpotency class 3 216

21 Survey of other results 222

21.1 Graham Higman's PORC conjecture 222

21.2 Isoclinism classes of p-groups 224

21.3 Groups of square-free order 227

21.4 Groups of cube-free order 233

21.5 Groups of arithmetically small orders 236

21.6 Surjectivity of the enumeration function 238

21.7 Densities of certain sets of group orders 246

21.8 Enumerating perfect groups 256

22 Some open problems 259

Appendix A Maximising two functions 269.

Notes:

Includes bibliographical references (pages 275-279) and index.

ISBN:

0521882176

9780521882170

OCLC:

154682311