Let $f\colon S^1 \rightarrow S^1$ given by $f(z)=z^p$, and think $S^1 = \{z\in \mathbb{C}\colon |z|=1\}$ as a multiplicative group, so $f$ is an homomorphism. Let $S_n=S^1$ and $f_n=f$ for all $n$, and consider the inverse limit

$$\mathcal{S}_p =\varprojlim(S_n, f_n)$$.

Note that $\mathcal{S}_p$ is a topological group.

I know that all closed subgroups of $S^1$ are $S^1$ itself and all finite cyclic subgroups.

I want to classify all closed subgroups of $\mathcal{S}_p$ using this fact. So far, I have proved that $\mathcal{S}_p$ and all finite cyclic groups $C_m$, where $m$ is the cardinality of the group with $(m,p)=1$, occur in that list. But those are not all subgroups of $\mathcal{S}_p$, so I ask you if you know what groups are missing and if you can sketch a proof for getting them, please.


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