Paracyclic and quasi-cyclic groups
Let G be a finite group. The authors define the multiplicity \(\gamma_ G\) of G to be the maximum of \(| (x\in G:\) \(x^ d=1)| /d\) taken over all divisors d of \(| G|\), and the cocyclicity \(\iota_ G\) as the least index in G of any cyclic subgroup of G. In earlier work [\textit{R. Dvornicich} and \textit{M. Forti}, J. Algebra 91, 499-519 and 520- 535 (1984; Zbl 0581.20022 and Zbl 0558.20018)] the authors have shown that the ratio \(\iota_ G/\gamma_ G\) is unbounded but grows no faster than \(\gamma_ G^{\epsilon}\) for any \(\epsilon >0\). The main result of the present paper is that this ratio is at most 1 when G is metacyclic. Results are also obtained in the case where G has at most one noncyclic Sylow subgroup.
scientific article; zbMATH DE number 4002366