Splitting the center of a Sylow subgroup

Abstract

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 an analogous result for all groups except in some cases where $p=2$ and $G$ is not solvable, where we have counterexamples. We also extend this result to fusion systems.