On the number of groups of a given order
The authors obtain some new estimates for the total number \(G(n)\) of non-isomorphic groups of order \(n\). Their first main theorem states that \(G(n)\leq \prod_{p| n}(p-1,n)\leq \varphi(n)\) for square-free \(n\), and depends on the result that groups of square-free order are supersolvable. One corollary is that groups of square-free order are ``scarce'', i.e. have zero density amongst all finite groups. This is derived with the aid of the theorem of Higman and Sims on enumerating \(p\)-groups [see \textit{C. C. Sims}, Proc. Lond. Math. Soc. (3) 15, 151--166 (1965; Zbl 0133.28401)]. Another corollary, depending on heavier number-theoretical machinery, is that \(G(n)<(\log n)^ 2\) for a set of square-free numbers \(n\) of positive density. Lastly, the authors discuss estimates (when \(k=2\) or 3) for the number of \(F_ k(x)\) of integers \(n\leq x\) for which \(G(n)=k\). An asymptotic formula for \(F_ 1(x)\) was previously known as a corollary of a number-theoretical result of \textit{P. Erdős} [J. Indian Math. Soc., II. Ser. 12, 75--78 (1948; Zbl 0041.36807)]. The interesting problem of finding an asymptotic formula for \(F_ k(x)\) in general is raised, and the reviewer would like to take this opportunity to draw attention to the related unsolved problem of finding such a formula for the average \(x^{-1}\sum_{n\leq x}G(n).\)
scientific article; zbMATH DE number 3841986