Frattini-resistant direct products of pro-p groups

2022/06/16 by Ilir Snopce, Snopce, Ilir, Slobodan Tanushevski +1
Mathematics · #12F10 #20E18 #22E20 #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2206.07863

openalex publication_date 2022/06/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A pro-p group G is called strongly Frattini-resistant if the function H ↦ Φ(H), from the poset of all closed subgroups of G into itself, is a poset embedding. Frattini-resistant pro-p groups appear naturally in Galois theory. Indeed, every maximal pro-p Galois group over a field that contains a primitive pth root of unity (and also contains √(-1) if p=2) is strongly Frattini-resistant. Let G₁ and G₂ be non-trivial pro-p groups. We prove that G₁ × G₂ is strongly Frattini-resistant if and only if one of the direct factors G₁ or G₂ is torsion-free abelian and the other one has the property that all of its closed subgroups have torsion-free abelianization. As a corollary we obtain a group theoretic proof of a result of Koenigsmann on maximal pro-p Galois groups that admit a non-trivial decomposition as a direct product. In addition, we give an example of a group that is not strongly Frattini-resistant, but has the property that its Frattini-function defines an order self-embedding of the poset of all topologically finitely generated subgroups.

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