If $G$ contains normal subgroups of prime orders $p$ and $q$, then $G$ contains an element of order $pq$.

If $G$ contains normal subgroups of prime orders $p$ and $q$, then $G$
contains an element of order $pq$.

Let $G$ be a group that contains normal subgroups of prime orders $p$ and
$q$, respectively. Prove that $G$ contains an element of order $pq$.
I tried using Lagrange's theorem but I'm not sure if it applies since it
isn't said that $G$ is finite. Can it be shown that this is the case? i.e.
Does a group with non-trivial subgroups of finite order necessarily have
finite order itself?
If it does hold, using Lagrange's theorem I could say that $pq$ divides
$|G|$, but I'm not sure what follows. Can I then somehow show that there
exists a subgroup in $G$ of order $pq$, and so there must be some element
in $G$ of order $pq$ (since the group operation for the subgroup is the
same as that of $G$)?