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 ^ N =e and PN=G
this is just a special case of Burnside's theorem usually called Burnside's normal complement theorem or Burnside's transfer theorem. the proof can be found almost anywhere:
Burnside's theorem: if $\displaystyle P$ is a Sylow subgroup of a group $\displaystyle G$ and $\displaystyle P \subseteq Z(N_G(P)),$ then there exists a normal subgroup $\displaystyle N$ of $\displaystyle G$ such that $\displaystyle P \cap N=\{1\}$ and $\displaystyle G=PN.$