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

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

I've been able to prove the theorem holds for $$p=q$$ and $$p>q$$. If $$p the best I've been able to do is use Sylow to show $$p^n+p^{n-1}-1\leq q$$.

Yet I seem to be stuck. I would appreciate any help.

• – verret Jan 9 '20 at 4:11
• @Bach It's essentially the same: to show it's solvable, you show it's not simple and use induction... – verret Jan 11 '20 at 19:40

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?
• I used such fact when proving the case $p>q$, but I wouldn't know how to apply it to $p<q$. Also, technically the result you mention doesn't need Sylow Theory. – Leo Jan 10 '20 at 23:13