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.

• The first book devoted to this exciting and vigorous area of modern group-theoretic research • Written by leading specialists in the field; contains hitherto unpublished material • Includes many open problems - ideal for graduate students in group theory

### 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.

### Review

'… a welcome and well-written addition to the theory of finite groups.' EMS Newsletter