A criterion for metanilpotency of a finite group

Raimundo Bastos, Carmine Monetta, Pavel Shumyatsky

We prove that the $k$th term of the lower central series of a finite group $G$ is nilpotent if and only if $|ab|=|a||b|$ for any ${\gamma }_k$-commutators $a,b\in G$ of coprime orders.