Fusion systems with Benson-Solomon components

TL;DR

This paper investigates fusion systems containing Benson-Solomon components, providing evidence that these are the only simple exotic 2-fusion systems and exploring their extensions within the framework of the classification of finite simple groups.

Contribution

The paper proves that fusion systems with Benson-Solomon components are not simple and classifies almost simple cases, advancing the understanding of exotic 2-fusion systems.

Findings

Fusion systems with Benson-Solomon components are not simple.

Almost simple fusion systems with Benson-Solomon components are classified as extended Benson-Solomon systems.

Results support the conjecture that Benson-Solomon systems are the only simple exotic 2-fusion systems.

Abstract

The Benson-Solomon systems comprise a one-parameter family of simple exotic fusion systems at the prime $2$. The results we prove give significant additional evidence that these are the only simple exotic $2$-fusion systems, as conjectured by Solomon. We consider a saturated fusion system $\mathcal{F}$ having an involution centralizer with a component $\mathcal{C}$ isomorphic to a Benson-Solomon fusion system, and we show under rather general hypotheses that $\mathcal{F}$ cannot be simple. Furthermore, we prove that if $\mathcal{F}$ is almost simple with these properties, then $\mathcal{F}$ is isomorphic to the next larger Benson-Solomon system extended by a group of field automorphisms. Our results are situated within Aschbacher's program to provide a new proof of a major part of the classification of finite simple groups via fusion systems. One of the most important steps in this program is a proof of Walter's Theorem for fusion systems, and our first result is specifically tailored for use in the proof of that step. We then apply Walter's Theorem to treat the general Benson-Solomon component problem under the assumption that each component of an involution centralizer in $\mathcal{F}$ is on the list of currently known quasisimple $2$-fusion systems.