“How many groups of order n are there?”
Quite a few! On page 1 we learn immediately that there are more than 244 groups of order 211. Unfortunately, we also see quickly that there is no clear, obvious pattern. While we can sometimes find an exact number (we are told that there are exactly 49,487,365,422 groups of order 210), in general we must content ourselves with finding asymptotic estimates for upper and lower bounds. Naturally, by “asymptotic estimates” we mean big-O notation.
This text, developed from lecture notes, is divided into four parts. The first section, which is very short, first estimates the number of binary tables, then passes through estimates for semigroups and Latin squares to give a rough estimate for groups. Part two focuses on p-groups; that is, groups whose order is a prime power. Part three leads the reader to Pyber’s Theorem, whose upper bound for the number of non-isomorphic groups is derived from the prime factorization of n. (As one might expect, Sylow subgroups are involved.) The fourth part discusses other subtopics within this problem; the preface informs the reader that it is contains previously unpublished results. This part concludes with a review of Open Questions, 37 in all.
The book is targeted at researchers and graduate-level study. It is not a text for the faint of heart: prerequisites include a fair amount of algebra that goes beyond basic, undergraduate group theory. That said, the authors take time to review much of this material. I had never studied cohomology before reading this book, and I found the included material to be more than adequate, as well as clear. Nevertheless, readers with a weak background (such as myself) will have to spend substantial time engaging the material.
I volunteered to review this monograph in the hope of learning a little about a field that has aroused my curiosity since my first undergraduate algebra class. In fact, I’ve learned a lot about the field, and I’ve only scratched the surface of this excellent text. I hope to return to it one day when I have more time, and engage it more thoroughly.
John Perry is an Assistant Professor of Mathematics at the University of Southern Mississippi.
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.