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

TL;DR

This paper proves that in finite p-groups with a 2-generator derived subgroup, every element of the derived subgroup is a commutator, removing the previous restriction that the subgroup be abelian.

Contribution

It establishes that the abelian condition on the derived subgroup is unnecessary for all elements to be commutators, providing a stronger result in the structure of finite p-groups.

Findings

Every element of the derived subgroup is a commutator.

The commutator can be expressed as [x,g] for a fixed x in G.

The abelian condition on G' is not required.

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 prove that the condition that $G'$ should be abelian is not needed. Even more, we prove that every element of $G'$ is a commutator of the form $[x,g]$ for a fixed $x\in G$.