The Nilpotency Criterion for the Derived Subgroup of a Finite Group

It is proved that the derived subgroup of a finite group is nilpotent if and only if $\mathrm{\mid }ab\mathrm{\mid }\ge \mathrm{\mid }a\mathrm{\mid }\mathrm{\mid }b\mathrm{\mid }$ for all primary commutators $a$ and $b$ of coprime orders.