subgroups of $\prod_p C_p$ 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?
 A: 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. 
A: Seems like all compact subgroups are indeed of the form $\prod_{p\in S}C_p$ for some subset $S$ of primes.
As mentioned in the comment, via Pontryagin duality this is equivalent to the following statement: any quotient of $\bigoplus_pC_p$ is of the form $\bigoplus_{p\in S}C_p$.
For an element $a=(a_p)_p$ of $\bigoplus_pC_p$ define its support $\operatorname{supp}(a):=\{p\mid a_p\ne0\}$; this is a finite set of primes. Clearly any $a$ is a generator of the subgroup $\bigoplus_{p\in\operatorname{supp}(a)}C_p\cong C_{\prod_{p\in\operatorname{supp}(a)}p}$.
Let now the quotient in question be by some subgroup $H<\bigoplus_pC_p$. For an element $h\in H$, the quotient map $\bigoplus_pC_p\twoheadrightarrow(\bigoplus_pC_p)/\langle h\rangle$, by the above, can be identified with the projection $\bigoplus_pC_p\twoheadrightarrow\bigoplus_{p\notin\operatorname{supp}(h)}C_p$. 
It then follows (I believe) that the quotient $\bigoplus_pC_p\twoheadrightarrow(\bigoplus_pC_p)/H$ is isomorphic to the projection $\bigoplus_pC_p\twoheadrightarrow\bigoplus_{p\notin\bigcup_{h\in H}\operatorname{supp}(h)}C_p$.
