Splitting the center of a Sylow subgroup

George Glauberman, Justin Lynd

Suppose $p$ is a prime and $S$ is a Sylow $p$-subgroup of a finite group $G$. If $S$ is normal in $G$, then $Z(S)$ is the direct product of $S\cap Z(G)$ with $[Z(S),G]$. We prove…