Frattini-resistant direct products of pro-$p$ groups

TL;DR

This paper characterizes when direct products of pro-p groups are strongly Frattini-resistant, linking group-theoretic properties to Galois theory, and provides examples illustrating the concept's boundaries.

Contribution

It establishes a precise criterion for strong Frattini-resistance in direct products of pro-p groups and offers a new proof related to Galois groups, plus an illustrative example.

Findings

G_1 × G_2 is strongly Frattini-resistant iff one factor is torsion-free abelian and the other has all subgroups with torsion-free abelianization.

Provides a group-theoretic proof of Koenigsmann's result on maximal pro-p Galois groups.

Constructs an example of a group with a self-embedding Frattini-function that is not strongly Frattini-resistant.

Abstract

A pro-$p$ group $G$ is called strongly Frattini-resistant if the function $H \mapsto \Phi(H)$, 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 $p$th root of unity (and also contains $\sqrt{-1}$ if $p=2$) is strongly Frattini-resistant. Let $G_1$ and $G_2$ be non-trivial pro-$p$ groups. We prove that $G_1 \times G_2$ is strongly Frattini-resistant if and only if one of the direct factors $G_1$ or $G_2$ 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.