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?
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1$\begingroup$ 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. $\endgroup$– Yassine GuerboussaJul 21, 2015 at 9:19
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