Enumeration of finite groups
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Bibliographic Information
Enumeration of finite groups
（Cambridge tracts in mathematics, 173）
Cambridge University Press, 2007
- : hbk.
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Note
Includes bibliographical references and index
Description and Table of Contents
Description
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.
Table of Contents
- 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.
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