A characterization of powerful p-groups

Abstract

In [10] Benjamin Klopsch and Ilir Snopce posted the conjecture that for $p\geq 3$ and $G$ a torsion-free pro-$p$ group $d(G)=\dim (G)$ is a sufficient and necessary condition for the pro-$p$ group $G$ to be uniform. They pointed out that this follows from the more general question of whether for a finite $p$-group $d(G)=\log_p(|Ω_1(G)|)$ is a sufficient and necessary condition for the group $G$ to be powerful. In this short note we will give a positive answer to this question for $p\geq 5$.