A criterion for nilpotency in finite groups (Q6189005)

It is a well-known fact by \textit{B. Baumslag} and \textit{J. Wiegold} [``A sufficient condition for nilpotency in a finite group'', Preprint, \url{arXiv:1411.2877}] that a finite group is nilpotent if and only if for every \( a, b \in G\) with coprime orders, one has \(|a||b|=|ab|\). Let \(\pi(n)\) denote the set of prime divisors of $n$. The authors prove a criterion of nilpotence of finite groups extending Baumslag-Wiegold's result [loc. cit.] as follows: Theorem. A finite group \(G\) is nilpotent if and only if \(\pi(|a||b|)=\pi(|ab|)\) for every \(a,b \in G\). The proof uses Ito's criterion of \(p\)-nilpotency as well as the Baer-Suzuki theorem and Burnside's \(pq\)-theorem.