A characterization of powerful p-groups

Abstract

In [10] Benjamin Klopsch and Ilir Snopce recently posted the conjecture that for p ≥ 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 (|Ω1(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 ≥ 5.