Orders of products of elements and nilpotency of terms in the lower central series and the derived series

Abstract

In this paper we prove that if G is a finite group, then the k -th term of the lower central series is nilpotent if and only if for every \gamma_{k} -values x,y \in G with coprime orders, either \pi(o(x)o(y))\subseteq \pi(o(xy)) or o(x)o(y) \leq o(xy) . We obtain an analogous version for the derived series of finite solvable groups, but replacing \gamma_{k} -values by \delta_{k} -values. We will also discuss the existence of normal Sylow subgroups in the derived subgroup in terms of the order of the product of certain elements.