We study three classes of subgroups of Polish groups: Borel subgroups, Polishable subgroups, and maximal divisible subgroups. The membership of a subgroup in each of these classes allows one to assign to it a rank, that is, a countable ordinal, measuring in a natural way complexity of the subgroup. We prove theorems comparing these three ranks and construct subgroups with prescribed ranks. In p...

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 ...