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Automorphisms of fusion systems of finite simple groups of Lie type and Automorphisms of fusion systems of sporadic simple groups

About this Title

Carles Broto, Departament de Matemàtiques, Universitat Autònoma de Barcelona, E–08193 Bellaterra, Spain, Jesper M. Møller, Matematisk Institut, Universitetsparken 5, DK–2100 København, Denmark and Bob Oliver, Université Paris 13, Sorbonne Paris Cité, LAGA, UMR 7539 du CNRS, 99, Av. J.-B. Clément, 93430 Villetaneuse, France.

Publication: Memoirs of the American Mathematical Society
Publication Year: 2019; Volume 262, Number 1267
ISBNs: 978-1-4704-3772-5 (print); 978-1-4704-5507-1 (online)
DOI: https://doi.org/10.1090/memo/1267
Published electronically: December 19, 2019
Keywords: [Automorphisms of Fusion Systems of Finite Simple Groups of Lie Type] Groups of Lie type, fusion systems, automorphisms, classifying spaces; [Automorphisms of Fusion Systems of Sporadic Simple Groups] Fusion systems, sporadic groups, Sylow subgroups, finite simple groups
MSC: Primary [Automorphisms, of, Fusion, Systems, of, Finite, Simple, Groups, of, Lie, Type], Primary, 20D06.; Secondary 20D20, 20D45, 20E42, 55R35, [Automorphisms, of, Fusion, Systems, of, Sporadic, Simple, Groups], Primary:, 20E25, 20D08.

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Table of Contents

Automorphisms of Fusion Systems of Finite Simple Groups of Lie Type by Carles Broto, Jesper M. Møller, and Bob Oliver

• Introduction
• 1. Tame and reduced fusion systems
• 2. Background on finite groups of Lie type
• 3. Automorphisms of groups of Lie type
• 4. The equicharacteristic case
• 5. The cross characteristic case: I
• 6. The cross characteristic case: II
• 7. Injectivity of $\mu _G$ by Bob Oliver

Automorphisms of Fusion Systems of Sporadic Simple Groups by Bob Oliver

• Introduction
• 8. Automorphism groups of fusion systems: Generalities
• 9. Automorphisms of $2$-fusion systems of sporadic groups
• 10. Tameness at odd primes
• 11. Tools for comparing automorphisms of fusion and linking systems
• 12. Injectivity of $\mu _G$

Abstract

Automorphisms of Fusion Systems of Finite Simple Groups of Lie Type by Carles Broto, Jesper M. Møller, and Bob Oliver

For a finite group $G$ of Lie type and a prime $p$, we compare the automorphism groups of the fusion and linking systems of $G$ at $p$ with the automorphism group of $G$ itself. When $p$ is the defining characteristic of $G$, they are all isomorphic, with a very short list of exceptions. When $p$ is different from the defining characteristic, the situation is much more complex, but can always be reduced to a case where the natural map from $\mathrm {Out}(G)$ to outer automorphisms of the fusion or linking system is split surjective. This work is motivated in part by questions involving extending the local structure of a group by a group of automorphisms, and in part by wanting to describe self homotopy equivalences of $BG{}^\wedge _p$ in terms of $\mathrm {Out}(G)$.

Automorphisms of Fusion Systems of Sporadic Simple Groups by Bob Oliver

We prove here that with a very small number of exceptions, when $G$ is a sporadic simple group and $p$ is a prime such that the Sylow $p$-subgroups of $G$ are nonabelian, then $\mathrm {Out} (G)$ is isomorphic to the outer automorphism groups of the fusion and linking systems of $G$. In particular, the $p$-fusion system of $G$ is tame in the sense of [Kasper K. S. Andersen, Bob Oliver, and Joana Ventura, Reduced, tame and exotic fusion systems, Proc. Lond. Math. Soc. (3) 105 (2012), no. 1, 87–152], and is tamely realized by $G$ itself except when $G\cong M_{11}$ and $p=2$. From the point of view of homotopy theory, these results also imply that $\mathrm {Out} (G)\cong \mathrm {Out} (BG^\wedge _p )$ in many (but not all) cases.