Let $G$ be a group of order $pm$ with $p$ a prime that does not divide $m$. Suppose there exists a $p$-Sylow subgroup $P\leq G$ contained in the center of the group $Z$. Prove that there is a normal subgroup $N\trianglelefteq G$ that satisfies $G = PN$ and $P\cap N = \langle e\rangle$ (in other words, $G$ is the internal direct product of $P$ and $N$).
I know this is a special case of Burnside's transfer theorem, but I am searching for an elementary proof which only uses the tools proved in a first course on group theory.