Large cyclic subgroups of finite groups (Q1068196)

\textit{M. Hausman} and \textit{H. N. Shapiro} [Commun. Pure Appl. Math. 33, 635-649 (1980)] have shown that there exists a function I such that whenever G is a finite group and \[ \gamma_ G=\max imum| \{x\in G| \quad x^ d=1\}| /d \] (where the maximum is over all divisor d of \(| G|)\) then \(\gamma \geq \gamma_ G\) implies that G has a cyclic subgroup of index \(\leq I(\gamma)\). Continuing the work which they begun in a previous paper [J. Algebra 91, 520-535 (1984; Zbl 0558.20018)], the authors find estimates for the (best possible) function I. In particular they show that for \(\gamma >2,\) \(I(\gamma)<\gamma^ 2\) and \(I(\gamma)<\exp ((1+6/\log\log\gamma) \log\gamma).\) It follows that \(I(\gamma)\ll \gamma^{1+\epsilon}\) (for \(\epsilon >0)\) and in the opposite direction the authors show that \(I(\gamma)/\gamma\) is unbounded. They also obtain an estimate for the corresponding function when attention is restricted to groups whose orders have at most a given number of prime divisors.