On groups of squarefree order
Continuing their interesting investigations of enumeration problems in finite group theory [see J. Number Theory 17, 29--36 (1983; Zbl 0516.10040), and the preceding review], the authors derive an exact formula for the number \(C(n)\) of groups of order \(n\) whose Sylow subgroups are all cyclic. This is used to prove a formula of Hölder for the total number \(G(n)\) of groups of square-free order \(n\), and then to derive the asymptotic formulae \[ \sum_{n\leq x}\log C(n) \sim Ax \log \log x;\quad \sum_{n\leq x}\mu^ 2(n) \log G(n) \sim Cx \log \log x \] as \(x\to \infty\), where \(A,C\) are explicitly stated constants. As may be expected, the arguments used involve both detailed group theory, and methods and results of analytic number theory.
scientific article; zbMATH DE number 3841987