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 $o(x)o(y)\le o(xy)$. 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.