Commutators in finite \(p\)-groups with powerful derived subgroup

Abstract

Let \(G\) be a finite \(p\)-group whose derived subgroup \(G'\) can be generated by \(2\) elements. If \(G'\) is abelian, Guralnick proved that every element of \(G'\) is a commutator. In this paper, we extend this result to the case when \(G'\) is powerful. Even more, we prove that every element of \(G'\) is a commutator of the form \([x,g]\) for a fixed \(x\in G\).