Focal subgroup theorem

This is a purely group-theoretic statement whose proof requires the use of some more advanced machinery, namely that from: representation theory

Statement

Let $P$ be a $p$-Sylow subgroup of a finite group $G$ and let:

$P_0=<x{y}^{-1}|x,y\in P,\mathrm{\exists }g\in G,gx{g}^{-1}=y>$

Then:

$P\cap G^{\prime }=P_0$