This book has grown out of a series of lectures given in the Advanced Algebra Class at Oxford in Michaelmas Term 1991 and Hilary Term 1992, that is to say from October 1991 to March 1992. The focus was—and is—the big question

how many groups of order n are there?

Two of the lectures were given by Professor Graham Higman, FRS, two by Simon R. Blackburn and the rest by Peter M. Neumann. Notes were written up week by week by Simon Blackburn and Geetha Venkataraman and those notes formed the original basis of this work. They have, however, been re-worked and updated to include recent developments.

The lectures were designed for graduate students in algebra and the book has been drafted with a similar readership in mind. It presupposes 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 should therefore be found to be more or less self-contained. Although it remains a work principally devoted to connected exposition of an agreeable theory, it does also contain some material that has not hitherto been published, particularly in Part IV.