Abelian group as direct product of its p-Sylow subgroups.

If the number $|G|$ has the arithmetic factorisation $$|G|=p_1^{n_1}p_2^{n_2}\cdots p_r^{n_r},$$ then by Sylow-1 it has, for each $i$, a subgroup $H_i<G$ with $p_i^{n_i}$ elements.

With this subgroups you can prove that $$G=H_1H_2\cdots H_r.$$