Commutators in finite $p$-groups with $2$-generator derived subgroup

Abstract

Let $G$ be a finite $p$-group whose derived subgroup $G^{\prime }$ can be generated by $2$ elements. If $G^{\prime }$ is abelian, Guralnick proved that every element of $G^{\prime }$ is a commutator. In this paper, we prove that the condition that $G^{\prime }$ should be abelian is not needed. Even more, we prove that every element of $G^{\prime }$ is a commutator of the form $[x,g]$ for a fixed $x\in G$.

11 pages