If $G$ is a residually $p$ group and $G_i$ ANY filtration (i.e. $G_i\subset G_{i-1}$ and $\cap G_i=e$) of normal
$p$-power index subgroups, is the corresponding filtration the usual pro-$p$ filtration?
Put differently, if $H$ is any normal $p$-power index subgroup, does it contain one of the $G_i$'s?
The answer to the corresponding question for finite filtrations is of course negative, e.g. the filtration $p^i\Bbb{Z}$ of $\Bbb{Z}$ does not give the profinite topology of $\Bbb{Z}$. On the other hand for abelian groups the answer to the above question is positive.
Best Answer
Here are three properties a topological group $G$ might have.
(1) The group is topologically finitely generated.
(2) All abstract subgroups with finite index are open.
(3) There are finitely many abstract subgroups with each index.
In 1975, Serre showed that (1) implies (2) for pro-$p$ groups (a proof is in Sect. 4.3.4 of J. S. Wilson, Profinite Groups, Oxford Univ. Press, Oxford, 1998) and he conjectured that (1) implies (2) for all profinite groups. This conjecture was proved in 2007 by N. Nikolov and D. Segal in 2007 using the classification of finite simple groups. (N. Nikolov and D. Segal, On finitely generated profinite groups. I. Strong completeness and uniform bounds, Ann. of Math. 165 (2007), 171--238 and On finitely generated profinite groups. II. Products in quasisimple groups, Ann. of Math. 165 (2007), 239--273.
H. L. Peterson (Discontinuous characters and subgroups of finite index, Pacific J. Math. 44 (1973), 683--691) showed (2) implies (3) for compact Hausdorff groups and M. P. Anderson (Subgroups of finite index in profinite groups, Pacific J. Math. 62 (1976), 19--28) showed (3) implies (2) for pro-solvable groups. Therefore (2) and (3) are equivalent for pro-solvable groups, thus in particular for pro-$p$ groups. That (3) implies (2) for all compact Hausdorff groups (thus all profinite groups) is due to J. S. Wilson (Lemma 6 in Groups satisfying the maximal condition for normal subgroups, Math. Z. 118 (1970), 107--114), while M. G. Smith and J. S. Wilson (On subgroups of finite index in compact Hausdorff groups, Arch. Math. 80 (2003), 123--129) reproved the equivalence of (2) and (3) for compact Hausdorff groups.
At this point we see that (1) implies (2) for profinite groups and (2) and (3) are equivalent for compact Hausdorff groups. To complete the circuit, it is natural to guess that (3) implies (1) for profinite groups (and perhaps even all compact Hausdorff groups). We will check (3) implies (1) for pro-$p$ groups, so (1), (2), and (3) are equivalent when $G$ is a pro-$p$ group.
Let $G$ be a pro-$p$ group satisfying (3). A maximal open subgroup of a pro-$p$ group has index $p$, so (3) implies there are finitely many maximal open subgroups of $G$. The intersection of finitely many open subgroups is open, so the intersection of the maximal open subgroups of $G$ is open. This intersection is the Frattini subgroup $\Phi(G)$, so $\Phi(G)$ is open in $G$. It follows from Prop. 2.8.10 in L. Ribes and P. Zalesskii, Profinite Groups, Springer--Verlag, Berlin, 2000 that $G$ is topologically finitely generated.