A metanilpotency criterion for a finite solvable group

Denote by $|x|$ the order of an element $x$ of a group. An element of a group is called primary if its order is a nonnegative integer power of a prime. If $a$ and $b$ are primary elements of coprime orders of a group, then the commutator $a^{-1}b^{-1}ab$ is called a $\star$-commutator. The intersection of all normal subgroups of a group such that the quotient groups by them are nilpotent is called the nilpotent residual of the group. It is established that the nilpotent residual of a finite group is generated by commutators of primary elements of coprime orders. It is proved that the nilpotent residual of a finite solvable group is nilpotent if and only if $|ab|\ge|a||b|$ for any $\star$-commutators of $a$ and $b$ of coprime orders.