A Sufficient Condition for Nilpotency of the Nilpotent Residual of a Finite Group

Let $G$ be a finite group with the property that if $a,b$ are powers of ${\delta }_1^{\ast }$-commutators such that $(\mathrm{\mid }a\mathrm{\mid },\mathrm{\mid }b\mathrm{\mid })=1$, then $\mathrm{\mid }ab\mathrm{\mid }=\mathrm{\mid }a\mathrm{\mid }\mathrm{\mid }b\mathrm{\mid }$. We show that ${\gamma }_{\mathrm{\infty }}(G)$ is nilpotent.