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Enumeration of finite groups. (English) Zbl 1141.20001
Cambridge Tracts in Mathematics 173. Cambridge: Cambridge University Press (ISBN 978-0-521-88217-0/hbk). xii, 281 p. (2007).
The book under review arises from a series of lectures given at Oxford in 1991/92 by Graham Higman, Simon R. Blackburn and Peter M. Neumann on the theme “how many groups of order \(n\) are there?” Considerable work has been done on the notes taken by Blackburn and Geetha Venkataraman at that time, so that the book is very much up-to-date. Only modest prerequisites in group theory are required, but the book contains a brief presentation of most of them, so that it is reasonably self-contained.
Part II (Chapters 3-5) is devoted to groups of prime power order, in particular to the Higman-Sims bound for their number. Higman proved that there are at least \[ p^{\frac{2}{27}m^2(m-6)} \] (pairwise non-isomorphic) groups of order \(p^m\). (Here \(p\) is a prime number, and \(m\) is a positive integer.) Charles C. Sims proved a corresponding upper bound. Using an unpublished approach of Mike Newman and Craig Seeley, which improves the error term in Sims’ formula, one has that there are at most \[ p^{\frac{2}{27}m^3+O(m^{5/2})} \] such groups.
Part III (Chapters 6-16) deals with Laci Pyber’s theorem for the number of groups of order \(n\). If \(n=p_1^{\alpha_1}\cdots p_k^{\alpha_k}\), where the \(p_i\) are distinct primes, let \(\mu=\max\{\alpha_i:1\leq i\leq k\}\). Then Pyber’s result states that there are at most \[ n^{(97/4)\mu+278\,852} \] groups of order \(n\), and at most \[ n^{8\mu+278\,833} \] soluble ones.
Part IV (Chapters 17-22) deals with “Other Topics”, it contains a considerable wealth of new material, in particular for varieties of groups, and ends with a substantial list of open problems.
The book is splendidly written. Each chapter and section begins with a sentence clearly describing their goals, so even the casual reader, which is not immediately interested in delving in all the technical details, has always a clear sense of aim and direction. The same applies to proofs. The first paragraph of the proof of Pyber’s theorem for soluble groups (Theorem 15.5, p. 135) is worth quoting: “Before we embark on the details of a proof of Pyber’s theorem, it is worthwhile to consider the main stages of the proof. We first present a summary of these.” The concluding Open Problems are presented in such a clear and informative way, that one can get a clear sense of what the current lines of research are.
One cannot praise this book high enough. It is a magnificent introduction to a beautiful and very active subject. It is particularly recommended for students that are approaching the subject, but group theorists at all levels of experience will benefit from it.

20-02 Research exposition (monographs, survey articles) pertaining to group theory
20Dxx Abstract finite groups
20Exx Structure and classification of infinite or finite groups
20D15 Finite nilpotent groups, \(p\)-groups
20D60 Arithmetic and combinatorial problems involving abstract finite groups
11N45 Asymptotic results on counting functions for algebraic and topological structures
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20E10 Quasivarieties and varieties of groups