I remember reading (without proof) that for $\Gamma$ a profinite, pro-$p$ group, the following are equivalent:


1) Every open subgroup $\Gamma\_0$ is topologically finitely generated.


2) The abelianization of every open subgroup $\Gamma\_0$ is finitely generated (as a $\mathbb Z\_p$ module .


3) Every open subgroup $\Gamma\_0$ only admits finitely many maps $\Gamma\_0 \to \mathbb F\_p$.


Clearly, $1) \implies 2) \implies 3)$ but how do you show equivalence?



For a pro-p, profinite group, abelianization being finitely generated is the same as being topologically finitely generated
Following TKe's reference, here is the solution:


Suppose $\mathscr I$ is a set of elements in $G$, a pro-$p$ group.


Denote by $G^\*$ the group generated by $G^p$ and $[G,G]$ together. Note that $H^1(G,\mathbb F\_p) = \operatorname{Hom}(G,\mathbb F\_p) = \operatorname{Hom}(G/G^\*,\mathbb Q\_p/\mathbb Z\_p)$. Then, the claim is that $\mathscr I$ generates $\Gamma$ (topologically) if and only if $\mathscr I$ generates $G/G^\*$ as a $\mathbb F\_p$ vector space.


This is clearly sufficient to show that $3) \implies 1).$ The crucial idea is the following:



> 
> **Lemma:** Any maximal closed subgroup $H \subset G$ is normal of index $p$.
> 
> 
> **Proof:** There exists an open subgroup $U$ such that $HU > U$ (otherwise $H=G)$. But then, by Sylow theory, the image of $H$ in $G/U$ is a maximal proper subgroup and is therefore of index $p$ and normal. Moreover, the pullback of this subgroup is in fact exactly $H$ since $H$ is maximal and we are done.
> 
> 
> 


Finally, we have:



> 
> **Lemma:** If $G,G'$ are two $pro-p$ groups, then a map $f: G \to G'$ is surjective if and only if $f^\*: H^1(G',\mathbb F\_p) \to H^1(G,\mathbb F\_p)$ is injective.
> 
> 
> **Proof:** Clearly, if $f$ is surjective, $f^\*$ is injective. Conversely, suppose $f$ is not surjective and let $H = f(G)$. Then, $H$ is closed (by compactness+hausdorff) and hence is contained in a maximal normal subgroup $H'$. $G/H'$ is then isomorphic to $\mathbb F\_p$ by the previous lemma and hence gives an element in $H^1(G',\mathbb F\_p)$ whose restriction along $f^\*$ is $0$ and hence $f^\*$ is not injective.
> 
> 
> 


To complete the proof, consider a pro- $p$ group $G$ such that $H^1(G,\mathbb F\_p)$ is finite dimensional and let $H$ be a finitely generated subgroup whose image in $G/G^\*$ generates $G/G^\*$. Then:


$H \to G$ is surjective $\iff$ $H^1(G,\mathbb F\_p) \to H^1(H,\mathbb F\_p)$ is injective $\iff$ $H/H^\* \to G/G^\*$ is surjective where the final equivalence is by Poincare duality.


