Coprime commutators in finite groups

Let $G$ be a finite group and let $k\ge 2$. We prove that the coprime subgroup ${\gamma }_k^{\ast }(G)$ is nilpotent if and only if $\mathrm{\mid }xy\mathrm{\mid }=\mathrm{\mid }x\mathrm{\mid }\mathrm{\mid }y\mathrm{\mid }$ for any ${\gamma }_k^{\ast }$-commutators $x,y\in G$ of coprime orders (Theorem A). Moreover, we show that the coprime subgroup ${\delta }_k^{\ast }(G)$ is nilpotent if and only if $\mathrm{\mid }ab\mathrm{\mid }=\mathrm{\mid }a\mathrm{\mid }\mathrm{\mid }b\mathrm{\mid }$ for any powers of ${\delta }_k^{\ast }$-commutators $a,b\in G$ of coprime orders (Theorem B).