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15 February 2022 Fusion systems with Benson–Solomon components
Ellen Henke, Justin Lynd
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Duke Math. J. 171(3): 673-737 (15 February 2022). DOI: 10.1215/00127094-2021-0031

Abstract

The Benson–Solomon systems comprise an infinite ascending 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 F having an involution centralizer with a component C isomorphic to a Benson–Solomon fusion system, and we show under rather general hypotheses that F cannot be simple. Furthermore, we prove that if F is almost simple with these properties, then 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 F is on the list of currently known quasisimple 2-fusion systems.

Citation

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Ellen Henke. Justin Lynd. "Fusion systems with Benson–Solomon components." Duke Math. J. 171 (3) 673 - 737, 15 February 2022. https://doi.org/10.1215/00127094-2021-0031

Information

Received: 27 February 2019; Revised: 28 March 2021; Published: 15 February 2022
First available in Project Euclid: 17 February 2022

Digital Object Identifier: 10.1215/00127094-2021-0031

Subjects:
Primary: 20D20
Secondary: 20D05 , 20D06 , 20G40 , 55R35

Keywords: component , finite simple group , Fusion system , involution centralizer

Rights: Copyright © 2022 Duke University Press

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Vol.171 • No. 3 • 15 February 2022
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