Existence of a normal subgroup in a finite group.

Existence of a normal subgroup in a finite group.

Let $G$ be a finite group. If a Sylow $p$-subgroup $P$ of $G$ is contained
in the centre, then does there exist a normal subgroup $N$ of $G$, such
that $P \cap N = \{e\}$ and $PN=G$?
Thanks in advance.