Coprime commutators in finite groups

Abstract

Let $G$ be a finite group and let $k \geq 2$. We prove that the coprime subgroup $γ_k^*(G)$ is nilpotent if and only if $|xy|=|x||y|$ for any $γ_k^*$-commutators $x,y \in G$ of coprime orders (Theorem A). Moreover, we show that the coprime subgroup $δ_k^*(G)$ is nilpotent if and only if $|ab|=|a||b|$ for any powers of $δ_k^*$-commutators $a,b\in G$ of coprime orders (Theorem B).