On large cyclic subgroups of finite groups

Abstract:

It is known that for each (composite) $n$ every group of order $n$ contains a proper subgroup of order greater than ${n^{1/3}}$. We prove that given $0 < \delta < 1$, for almost all $n \leqslant x$, as $x \to \infty$, every group $G$ of order $n$ contains a characteristic cyclic subgroup of square-free order $> {n^{1 - 1/{{(\log n)}^{1 - \delta }}}}$, and provide an upper bound to the number of exceptional $n$. This leads immediately to a like density result for a lower bound to the number of conjugacy classes in $G$.