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 $\prod_{p} G(p)$...