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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 Aschbachers 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 Walters Theorem for fusion systems, and our first result is specifically tailored for use in the proof of that step. We then apply Walters 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.
The Benson-Solomon systems comprise the only known family of simple saturated fusion systems at the prime two that do not arise as the fusion system of any finite group. We determine the automorphism groups and the possible almost simple extensions o
Linking systems were introduced to provide algebraic models for $p$-completed classifying spaces of fusion systems. Every linking system over a saturated fusion system $mathcal{F}$ corresponds to a group-like structure called a locality. Given such a
The theory of saturated fusion systems resembles in many parts the theory of finite groups. However, some concepts from finite group theory are difficult to translate to fusion systems. For example, products of normal subsystems with other subsystems
Suppose $mathcal{E}$ is a normal subsystem of a saturated fusion system $mathcal{F}$ over $S$. If $Xleq S$ 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
We prove that an isomorphism between saturated fusion systems over the same finite p-group is detected on the elementary abelian subgroups of the hyperfocal subgroup if p is odd, and on the abelian subgroups of the hyperfocal subgroup of exponent at