If $G$ is a finite abelian group, then we have a decomposition
$$G\cong \prod\_{p} G(p)$$
where $G(p)$ is the $p$-Sylow subgroup of $G$. This product makes sense as for all but finitely many primes $p$, we have $G\_p=\{0\}$. This is proven by showing that the cardinality of $G$ and $\prod\_{p} G(p)$ agree. If we now assume that $P$ is a profinite abelian group, there still exists the notion of a $p$-Sylow subgroup $P(p)$ which is now a pro-$p$-group. I'm curious if there exists an isomorphism
$$P\cong \prod\_{p} P(p).$$



Sylow subgroups of abelian profinite groups
This is Proposition 2.3.8 of [Ribes and Zaleskii - Profinite groups (second edition)](https://doi.org/10.1007/978-3-642-01642-4). (I originally gave references specifically for the finer structure of profinite Abelian groups, but assuming finite generation, in Section 4.3 of the same book.)


