This mostly self-contained volume is a welcome and well-written addition to the theory of finite groups. The authors present in a unified style, which is accessible to graduate students, an up-to-date account of research concerning the question “how many groups of order n are there?”. The book opens with a brief detour into enumeration of semigroups and loops that clearly demonstrates the power of the combination of the associative law and inverses. An investigation of the function f(n) that counts the number of groups of order n up to isomorphism starts in earnest in Part II of the book.
The authors first present Higman's proof of f(pm) ≥ p(2/27)(m-6))m^2 and then move onto Sims, Newman and Seeley's upper bound f(pm) ≤ p(2/27)m^3+O(m^(5/2)), where p is a prime. The highlight of the book is the 80 page treatment of Pyber's theorem on the enumeration of solvable and general groups. It is shown that the number of solvable groups of order n with fixed Sylow subgroups P1 ,…, Pk does not exceed n8μ(n)+278833, where μ(n) is the highest power to which any prime divides n. From this estimate it is then not hard to conclude that the number of solvable A-groups (that is, solvable groups whose nilpotent subgroups are Abelian) of order n does not exceed n8μ(n)+278834 . The current proof of the general case of Pyber's theorem depends on the classification of finite simple groups. With this caveat, it is shown that the number of groups of order n with fixed Sylow subgroups P1,…,Pk does not exceed n(97/4)μ(n)+278852. From this it then follows that f(n) ≤ n(2/27)μ(n)^2+O(μ(n)^(3/2)) and that there are at most n(97/4)μ(n)+278853 A-groups of order n.
The book concludes with estimates on the number of groups in certain varieties. Enumeration of Abelian groups (necessarily combinatorial in nature due to the Fundamental theorem for finitely generated Abelian groups) is followed by enumeration in small varieties of A-groups, d-generator groups, groups with few non-Abelian composition factors, groups of nilpotency class 3 and other results. The array of techniques needed at various points in the book is impressive and the authors are to be commended for presenting them in a concise yet clear manner. The effort to elucidate ideas behind some of the proofs is impressive, as is the occasional heuristic explanation of results (for instance, why 2/27 appears in the exponent of f(n)). Several graduate courses can be designed based on this book, despite the principled decision to avoid exercises.