# Lower central words in finite $p$-groups

## TL;DR

This paper extends known results about the structure of verbal subgroups generated by lower central words in finite and pro-$p$ groups, removing the abelian assumption and including the case p=3.

## Contribution

It proves that the verbal subgroup generated by a lower central word in finite p-groups is composed solely of that word's values, even without the abelian assumption.

## Key findings

- Verbal subgroup equals the set of lower central word values in finite p-groups.
- Result holds for p=3, extending previous work for p≥5.
- The analogous statement is true for pro-p groups.

## Abstract

It is well known that the set of values of a lower central word in a group $G$ need not be a subgroup. For a fixed lower central word $\gamma_r$ and for $p\ge 5$, Guralnick showed that if $G$ is a finite $p$-group such that the verbal subgroup $\gamma_r(G)$ is abelian and 2-generator, then $\gamma_r(G)$ consists only of $\gamma_r$-values. In this paper we extend this result, showing that the assumption that $\gamma_r(G)$ is abelian can be dropped. Moreover, we show that the result remains true even if $p=3$. Finally, we prove that the analogous result for pro-$p$ groups is true.