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

A pro-$p$ group $G$ is called strongly Frattini-resistant if the function $H \mapsto Φ(H)$, 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 $p$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.

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