Existence of a normal subgroup in a finite group.

Question

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.

Answer 1

The answer is yes, by Schur--Zassenhaus. The point is that $P$ is a $p$-Sylow of $G$ which is also central in $G$, and thus is normal in $G$, while $G/P$ is of coprime order in $G$ (because $P$ is a $p$-Sylow).

(As Mikko Korhonen noted above in comments, the Burnside transfer theorem can be used instead.)