Centralizers of normal subgroups and the $Z^*$-Theorem

TL;DR

This paper generalizes the $Z^*$-theorem by relating the centralizers of normal subgroups in finite groups to their fusion systems, extending known results about the center of quotient groups.

Contribution

It introduces a new statement connecting the centralizer of a normal subgroup with its fusion system and normal subsystem, broadening the scope of the $Z^*$-theorem.

Findings

Centralizers of normal subgroups are determined by their fusion systems.

Extension of the $Z^*$-theorem to more general subgroup contexts.

Provides a framework for analyzing group structure via fusion systems.

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 $H$.