If $G$ is a finite group whose $p$-Sylow subgroup $P$ lies in its center, then there is a normal subgroup $N$ of $G$ with $P\cap N=\{e\}$ and $PN=G$

Question

If $G$ is a finite group and its $p$-sylow subgroup $P$ lies in the center of $G$, prove that there exists a normal subgroup $N$ of $G$ with $P\cap N=\{e\}$ and $PN=G$

Answer 1

This follows from the Schur-Zassenhaus Theorem - in your case $P$ is a normal Hall subgroup (that is gcd(index$[G:P],|P|)=1$). This implies that there is a subgroup $H$ of $G$ with $G=HP$ and $H \cap P =1$. But $P \subseteq Z(G)$, so $H$ is normalized by $P$ and of course by $H$, whence $H$ is normal in $G$. Hence $G \cong H \times P$.