Frattini-resistant direct products of pro-p groups

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