Consider the group $G:=\prod\_p C\_p$ where the product is taken over all primes, endowed with the product topology.


I'm trying to classify the compact subgroups of $G$. Is there any subgroups of $G$ besides direct products of finite groups?



subgroups of $\prod_p C_p$
Let $G$ be your group and $C$ a compact subgroup. Both $G$ and $C$ are profinite groups. The Sylow subgroups of $G$ are $C\_p$ and any Sylow subgroup of $C$ is contained in a Sylow subgroup of $G$. Therefore, it is either trivial or $C\_p$. Since $G$ is abelian so is $C$ and thus, $C$ is a direct product of its Sylow subgroups. 


