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I saw that a topological group $G$ is profinite if and only if it is compact, Hausdorff and totally disconnected. In such groups, an open subgroup is closed but not vice-versa. From this, I came to a question as a curiosity:

In a profinite group $G$, if every subgroup is closed, can we say about structure of $G$ in terms of topology, or algebraic property?

The examples of groups with the property in question I know are discrete topological groups; but beside such groups, are there other examples of topological groups in which every subgroup is closed?

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Every such group is finite (and therefore discrete). The reason being, that every infinite profinite group is uncountable (in fact of cardinality at least continuum). Since every infinite group contains some countable subgroup, there can be no infinite example.

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Such a group is necessarily torsion. If a subgroup $H$ of a profinite group $G$ is closed, then $H$ itself is profinite. If any element of $G$ has infinite order, then it generates a subgroup of $G$ isomorphic to $\mathbb{Z}$, but $\mathbb{Z}$ is not profinite (under any topology, but we can check that this copy of $\mathbb{Z}$ has the discrete topology).

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