A Generalization of the $Z^*$-Theorem

Glauberman's $Z^*$-theorem and analogous statements for odd primes show that, for any prime $p$ and any finite group $G$ with Sylow $p$- subgroup $S$, the centre of $G$ is determined by the fusion system $\mathcal{F}_S(G)$. Building on these results we show a statement that can be considered as a generalization: For any normal subgroup $N$ of $G$, the centralizer $C_S(N)$ is expressed in terms of the fusion system $\mathcal{F}_S(G)$ and its normal subsystem induced by $N$.