# Is there a Noetherian profinite group of infinite rank?

Is there a profinite group $G$ such that any closed subgroup $H \leq G$ is finitely generated, but there is no $n \in \mathbb{N}$ such that every closed subgroup of $G$ can be generated by at most $n$ elements?

• The question is well-known for pro-$p$ groups. You can find it under the form : is every noetherian pro-$p$ group analytique $p$-adic? (See for instance "New Horizons in pro-$p$ Groups", by du Sautoy et Al; Appendix, Problem 1. – Yassine Guerboussa Jul 21 '15 at 9:19