Lower central words in finite $p$-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 $γ_r$ and for $p\ge 5$, Guralnick showed that if $G$ is a finite $p$-group such that the verbal subgroup $γ_r(G)$ is abelian and 2-generator, then $γ_r(G)$ consists only of $γ_r$-values. In this paper we extend this result, showing that the assumption that $γ_r(G)$ 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.