On the enumeration of finite groups
Let \(G(n)\) denote the number of non-isomorphic groups of order \(n\). Using methods of analytic number theory and an explicit algebraic equation of Hölder, the authors derive interesting asymptotic information about \(G(n)\) when \(n\) is square-free. The main results are: \[ G(n) = \Omega(n^{1-\varepsilon})\text{ for every \(\varepsilon > 0\) when \(n\) is square-free,} \tag{i} \] \[ \log G(n) = (1+o(1))\;\log\log n\sum_{p\mid n}(\log p)/(p-1) \tag{ii} \] for almost all square-free \(n\). In addition, they derive asymptotic estimates for \(F_k(x)\) where \(F_k(x)=\text{card}\{n\leq x: G(n)=k\}\): \[ F_k(x) = (c(a)+o(1)) x/(\log\log\log x)^{a+1}\text{ for } k = 2^a, \] \[ F_k(x) = O(x/(\log\log x)^{1-\varepsilon})\text{ for } k\neq 2^a, \] where \(c(a)\) is an appropriate constant.
scientific article; zbMATH DE number 3989466