Centralizers of normal subgroups and the $Z^{\ast }$-Theorem

Abstract

Glauberman's $Z^{\ast }$-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 $G/O_{p^{\prime }}(G)$ is determined by the fusion system ${\mathcal{F}}_S(G)$. Building on these results we show a statement that seems a priori more general: For any normal subgroup $H$ of $G$ with $O_{p^{\prime }}(H)=1$, the centralizer $C_S(H)$ is expressed in terms of the fusion system ${\mathcal{F}}_S(H)$ and its normal subsystem induced by $H$.

3 pages; to appear in the Journal of Algebra