# Math Help - Existence of Normal Subgroup

1. ## Existence of Normal Subgroup

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

2. Originally Posted by Chandru1

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
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 $P$ is a Sylow subgroup of a group $G$ and $P \subseteq Z(N_G(P)),$ then there exists a normal subgroup $N$ of $G$ such that $P \cap N=\{1\}$ and $G=PN.$