Commutators in finite p-groups with 3-generator derived subgroup

Abstract

It is well known that, in general, the set of commutators of a group $G$ may not be a subgroup. Guralnick showed that if $G$ is a finite $p$-group with $p\ge 5$ such that $G'$ is abelian and $3$-generator, then all the elements of the derived subgroup are commutators. In this paper, we extend Guralnick's result by showing that the condition of $G'$ to be abelian is not needed. In this way, we complete the study of this property in finite $p$-groups in terms of the number of generators of the derived subgroup. We will also see that the same result is true when the action of $G$ on $G'$ is uniserial modulo $(G')^p$ and $|G':(G')^p|$ does not exceed $p^{p-1}$. Finally, we will prove that analogous results are satisfied when working with pro-$p$ groups.