Normalizers and centralizers of $p$-subgroups in normal subsystems of fusion systems

TL;DR

This paper generalizes the concept of normalizers and centralizers of $p$-subgroups within fusion systems using locality theory, extending previous results to broader contexts.

Contribution

It introduces a generalized approach to normalizers and centralizers in fusion systems through locality theory, expanding on Aschbacher's earlier work.

Findings

Defined a normal subsystem $C_{E}(X)$ for fully $F$-centralized subgroups.

Extended the notion of normalizers to a more general setting.

Provided a unified framework for studying subgroups in fusion systems.

Abstract

Suppose $\mathcal{E}$ is a normal subsystem of a saturated fusion system $\mathcal{F}$ over $S$. If $X\leq S$ is fully $\mathcal{F}$-normalized, then Aschbacher defined a normal subsystem $N_{\mathcal{E}}(X)$ of $N_{\mathcal{F}}(X)$. In this short note we revisit and generalize this result using the theory of localities. Our more general approach leads in particular to a normal subsystem $C_{\mathcal{E}}(X)$ of $C_{\mathcal{F}}(X)$ for every $X\leq S$ which is fully $\mathcal{F}$-centralized.