Is $G/pG$ is a $p$-group? Jack is trying to prove:

Let $G$ be an abelian group, and $n\in\Bbb Z$. Denote $nG = \{ng \mid g\in G\}$.
(1) Show that $nG$ is a subgroup in $G$.
(2) Show that if $G$ is a finitely generated abelian group, and $p$ is prime,
   then $G/pG$ is a $p$-group (a group whose order is a power of $p$).

I think $G/pG$ is a $p$-group because it is a direct sum of cyclic groups of order $p$.
But I cannot give a detailed proof.         

Since $G/pG$ is a finitely generated torsion group, it is finite.
Let $q$ be a prime number which divides $|G/pG|$.
Then it has an element of order $q$ by the theorem of Cauchy.
Hence $q = p$.
Hence $G/pG$ is a $p$-group.
