A characterization of powerful p-groups

2013/07/02 by Jon González‐Sánchez, Gonzalez-Sanchez, Jon, Amaia Zugadi-Reizabal +1
Mathematics · #Advanced Topology and Set Theory #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.1307.0613

openalex publication_date 2013/07/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In [10] Benjamin Klopsch and Ilir Snopce 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.

Related

• Quasi-powerful p-groups
• A characterisation of uniform pro-p groups
• Powerfully embedded subgroups of extensions of powerful pro-p groups
• On finite p-groups with powerful subgroups
• The powerful class of groups