Reductions to simple fusion systems

TL;DR

The paper demonstrates how complex fusion systems can be simplified to basic ones through a series of reductions, aiding the analysis of involution centralizers and contributing to the classification of finite simple groups.

Contribution

It establishes a method to reduce saturated fusion systems to simpler subsystems under specific conditions, answering a question by Aschbacher and aiding group classification efforts.

Findings

Fusion systems can be reduced via normal subsystems of p-power or prime-to-p index.

Reductions are possible when certain automorphism groups are p-solvable.

Applicable to simple fusion systems realized by known simple groups.

Abstract

We prove that if $\mathcal{E}\trianglelefteq\mathcal{F}$ are saturated fusion systems over $p$-groups $T\trianglelefteq S$, such that $C_S(\mathcal{E})\le T$, and either $Aut_{\mathcal{F}}(T)/Aut_{\mathcal{E}}(T)$ or $Out(\mathcal{E})$ is $p$-solvable, then $\mathcal{F}$ can be "reduced" to $\mathcal{E}$ by alternately taking normal subsystems of $p$-power index or of index prime to $p$. In particular, this is the case whenever $\mathcal{E}$ is simple and "tamely realized" by a known simple group. This answers a question posed by Michael Aschbacher, and is useful when analyzing involution centralizers in simple fusion systems, in connection with his program for reproving parts of the classification of finite simple groups by classifying certain 2-fusion systems.