Ascending chains of finitely generated subgroups

We show that a nonempty family of n-generated subgroups of a pro-p group has a maximal element. This suggests that ‘Noetherian Induction’ can be used to discover new features of finitely generated subgroups of pro-p groups. To demonstrate this, we show that in various pro-p groups Γ (e.g. free pro-p   groups, nonsolvable Demushkin groups) the commensurator of a finitely generated subgroup H≠1H≠1 is the greatest subgroup of Γ containing H as an open subgroup. We also show that an ascending chain of n-generated subgroups of a limit group must terminate (this extends the analogous result for free groups proved by Takahasi, Higman, and Kapovich–Myasnikov).