Centralizers of normal subgroups and the $Z^*$Theorem
Abstract
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 $G/O_{p^\prime}(G)$ 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 $H$.
 Publication:

arXiv eprints
 Pub Date:
 November 2014
 arXiv:
 arXiv:1411.1932
 Bibcode:
 2014arXiv1411.1932H
 Keywords:

 Mathematics  Group Theory;
 20D20
 EPrint:
 3 pages