Prove that a group is non-cyclic

Question

I have some problems with an exercise of group theory. It sounds like this:

Let $G$ be a finite group of order $p^2q^2$ with $p$, $q$ prime numbers, $p$ is odd, $p<q$. I know that $G$ doesn't contain a normal subgroup of order $q$ but contains a normal Sylow $q$-subgroup $H$. I must prove that $H$ is non-cyclic.

Answer 1

If $H$ is cyclic, then $H$ contains a unique (hence characteristic) subgroup of order $q$. This implies that the subgroup of order $q$ is normal in $G$, a contradiction.