A characterization of powerful p-groups

TL;DR

This paper confirms that for primes p ≥ 5, a finite p-group is powerful if and only if its minimal number of generators equals the logarithm base p of the size of its first omega subgroup.

Contribution

It provides a positive answer to the conjecture linking the structure of finite p-groups to their generators for primes p ≥ 5.

Findings

For p ≥ 5, the characterization of powerful p-groups is confirmed.

The minimal number of generators equals the log of the omega subgroup size in these groups.

Supports the conjecture relating group structure and generator count for specific primes.

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 $d(G)=\log_p(|\Omega_1(G)|)$ 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 $p\geq 5$.