Size of the intersection of a $p$-Sylow subgroup and a normal subgroup.

Question

Assume $G$ is a group whose size is $(p^r)*m$, where $p$ is prime and $p$ doesn't divide $m$. Let $P$ be a $P$-Sylow subgroup of $G$, and $H$ a normal subgroup of $G$. Lagrange's theorem gives us that the size of $H$ must be $(p^s)*k$, $s \le r$ and $k\mid m$. I need to show that the size of $P\cap H$; the intersection of P and H is $p^s$, and nothing I tried works. Any ideas?