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.

How many groups of order n are there? This is a natural question for anyone studying group theory, and this Tract provides an exhaustive, connected, and up-to-date account of research into this question spanning almost fifty years. The authors presuppose little beyond an undergraduate knowledge of group theory but where more advanced theory is needed, the basics are expounded and the book is more or less self-contained. 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.