I'm having a problem with this following question:
Let $G$ be finite group of order $p^rm$ when $p$ is a prime number, $r\ge1$ and $\gcd(m,p)=1$. Prove that if $p^r\nmid(m-1)!$ then $G$ is not simple.
What I've tried so far: using Sylow's third theorem and come to conclusion that $G$ has a unique sylow-$p$ subgroup, and therefore $G$ is not simple.
Any hint would be appreciated.