Proving that a group of order $p^nq$ for primes $p$ and $q$ is not simple. 
Prove that a group of order $p^nq$ for primes $p$ and $q$ is not simple.

I've been able to prove the theorem holds for $p=q$ and $p>q$. If $p<q$ the best I've been able to do is use Sylow to show: $$p^n+p^{n-1}-1\leq q$$
Yet I seem to be stuck. I would appreciate any help.

A consequence of one of Sylow's theorems is that if there is exactly one $p$-Sylow subgroup $H$ of $G$, then is it normal. Can you do the rest?
