Enumeration of Finite Groups: 173 (Cambridge Tracts in Mathematics, Series Number 173)
Cambridge University Press, 10/18/2007
EAN 9780521882170, ISBN10: 0521882176
Hardcover, 294 pages, 23.6 x 16 x 2 cm
Language: English
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.
1. Introduction
Part I. Elementary Results
2. Some basic observations
Part II. Groups of Prime Power Order
3. Preliminaries
4. Enumerating p-groups
a lower bound
5. Enumerating p-groups
upper bounds
Part III. Pyber's Theorem
6. Some more preliminaries
7. Group extensions and cohomology
8. Some representation theory
9. Primitive soluble linear groups
10. The orders of groups
11. Conjugacy classes of maximal soluble subgroups of symmetric groups
12. Enumeration of finite groups with abelian Sylow subgroups
13. Maximal soluble linear groups
14. Conjugacy classes of maximal soluble subgroups of the general linear group
15. Pyber's theorem
the soluble case
16. Pyber's theorem
the general case
Part IV. Other Topics
17. Enumeration within varieties of abelian groups
18. Enumeration within small varieties of A-groups
19. Enumeration within small varieties of p-groups
20. Miscellanea
21. Survey of other results
22. Some open problems
Appendix A. Maximising two equations.
"The book is a valuable contribution to this particular area of group theory. In addition to covering key results and techniques, it points towards a broad range of recent results and open problems." - Benjamin Klopsch