Let $\Delta$ be a set, each element of which is a profinite group (2 distinct elements of $\Delta$ may be isomorphic). Under what conditions on $\Delta$, there exists a profinite group $G$ which has $\Delta$ as the family of its normal open subgroups? By that I mean, there is a bijection $u$ from $\Delta$ onto the set of normal open subgroups of $G$, such that $H$ is topologically isomorphic to $u(H)$ for every $H\in\Delta$?
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$\begingroup$ Certainly they need to form a directed system, in the sense that, for each $N, N' \in \Delta$, there is $N'' \in \Delta$ that injects into both with normal, cofinite index. Do you mean $\Delta$ to consist of the proper normal open subgroups? Otherwise it seems possible that directed + has a (unique) maximal element is enough. $\endgroup$– LSpiceCommented Dec 30, 2019 at 16:12
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2$\begingroup$ I think OP means that the data is the a set $\Delta$ of isomorphism classes of profinite groups, and the question is whether there exists a profinite groups whose set of isomorphism classes of open normal subgroups is exactly $\Delta$. (In this case, the answer is plainly: if and only if there is some $G\in\Delta$ such that every $H\in\Delta$ is isomorphic to some open normal subgroup of $G$, and such that every open normal subgroup of $G$ is isomorphic to some element of $\Delta$.) $\endgroup$– YCorCommented Dec 30, 2019 at 17:39
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$\begingroup$ @YCor Thanks for your useful comment. But really I want $\Delta$ to be all of my open normal subgroups, not only the isomorphism classes. $\endgroup$– MSMalekanCommented Dec 31, 2019 at 5:24
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$\begingroup$ So, if I understand correctly $\Delta$ is a set of profinite groups, and you're asking whether there is an isomorphism-preserving bijection from $\Delta$ to the set of normal open subgroups of some profinite group. $\endgroup$– YCorCommented Dec 31, 2019 at 9:58
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$\begingroup$ @YCor Yes, that's exactly what I mean. $\endgroup$– MSMalekanCommented Dec 31, 2019 at 11:12
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