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

## TL;DR

This paper extends previous results by showing that in finite p-groups with a 3-generator derived subgroup, all elements are commutators, removing the need for the derived subgroup to be abelian.

## Contribution

It generalizes Guralnick's theorem by removing the abelian condition and explores similar properties in pro-p groups, completing the understanding of commutators in these groups.

## Key findings

- All elements of the derived subgroup are commutators without the abelian condition.
- The result holds when the action on G' is uniserial modulo (G')^p and |G':(G')^p| ≤ p^{p-1}.
- Analogous results are valid for pro-p groups.

## 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.