[Paper Review] Automorphisms of fusion systems of finite simple groups of Lie type

Carles Broto, Jesper M. Møller+1 more|arXiv (Cornell University)|Jan 18, 2016

Finite Group Theory Research49 references16 citations

TL;DR

This paper classifies automorphisms of fusion and linking systems for finite simple groups of Lie type at a prime $p$, showing that when $p$ is the defining characteristic, the automorphism groups of $G$, its fusion system, and its linking system are isomorphic—except for a small list of exceptions. When $p$ differs from the defining characteristic, the automorphism structure is more complex but remains reducible to a split surjection from $\mathrm{Out}(G)$ to the outer automorphisms of the fusion system.

ABSTRACT

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 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 the p-completion of the classifying space BG in terms of Out(G).

Motivation & Objective

• To compare the automorphism groups of finite simple groups of Lie type $G$ with those of their fusion and linking systems at a prime $p$.
• To understand the structure of $\mathrm{Out}(BG^\wedge_p)$, the group of self homotopy equivalences of the $p$-completed classifying space of $G$, via automorphisms of fusion systems.
• To determine when the natural map $\mathrm{Out}(G) \to \mathrm{Out}_{\mathrm{typ}}(\mathcal{L}_S^c(G))$ is an isomorphism or split surjection.
• To resolve the tameness and realization properties of fusion systems associated with groups of Lie type.
• To extend the local structure of $G$ by automorphisms and describe the resulting homotopy-theoretic invariants.

Proposed method

• Use the natural homomorphisms $\kappa_G: \mathrm{Out}(G) \to \mathrm{Out}_{\mathrm{typ}}(\mathcal{L}_S^c(G))$ and $\mu_G: \mathrm{Out}_{\mathrm{typ}}(\mathcal{L}_S^c(G)) \to \mathrm{Out}(S, \mathcal{F}_S(G))$ to compare automorphism groups.
• Apply results from Aschbacher’s program on tameness of fusion systems over $2$-groups to analyze the structure of $\mathcal{F}_S(G)$.
• Leverage the isomorphism $\mathrm{Out}_{\mathrm{typ}}(\mathcal{L}_S^c(G)) \cong \mathrm{Out}(BG^\wedge_p)$ to relate group automorphisms to homotopy equivalences.
• Analyze centralizers and fixed-point structures in classical groups (e.g., $\mathrm{Sp}_8(\mathbb{K})$, $\mathrm{SO}_8(\mathbb{K})$) to bound automorphism group actions.
• Use group-theoretic bounds such as $|C_{H/Z}(P)/C_H(P)/Z| \leq |P/\mathrm{Fr}(P)|$ to constrain possible conjugacy classes.
• Apply Proposition A.4 to rule out $\overset{\sim}{G}$-conjugacy of certain subgroups $E$ to elements in $\widehat{\mathcal{Z}}$ via non-transitive action on components of centralizers.

Experimental results

Research questions

• RQ1 When is the map $\mathrm{Out}(G) \to \mathrm{Out}_{\mathrm{typ}}(\mathcal{L}_S^c(G))$ an isomorphism for $G$ a finite simple group of Lie type?
• RQ2 How does the structure of $\mathrm{Out}(S, \mathcal{F}_S(G))$ relate to $\mathrm{Out}(G)$ when $p$ is not the defining characteristic of $G$?
• RQ3 Under what conditions is the fusion system $\mathcal{F}_S(G)$ tamely realized by $G$?
• RQ4 When does the action of $\mathrm{Aut}_{\overset{\sim}{G}}(E)$ on $\pi_0(C_{\overset{\sim}{G}}(E))$ fail to be transitive, and what does this imply for conjugacy in $\widehat{\mathcal{Z}}$?
• RQ5 What is the precise relationship between $\mathrm{Out}(BG^\wedge_p)$ and $\mathrm{Out}(G)$ for $G$ of Lie type?

Key findings

• For $G \in \mathfrak{Lie}(p)$ of universal or adjoint type and $G \not\cong \mathrm{PSL}_3(2)$, the composite map $\kappa_G \circ \mu_G: \mathrm{Out}(G) \to \mathrm{Out}(S, \mathcal{F}_S(G))$ is an isomorphism when $p$ is the defining characteristic.
• The maps $\kappa_G$ and $\mu_G$ are isomorphisms in the equicharacteristic case except when $G \cong \mathrm{PSL}_3(2)$, where $\mu_G$ is not an isomorphism.
• When $p$ differs from the defining characteristic of $G$, the map $\mathrm{Out}(G) \to \mathrm{Out}_{\mathrm{typ}}(\mathcal{L}_S^c(G))$ is split surjective.
• For $G = \mathrm{PSL}_3(2)$, the map $\mu_G$ is not an isomorphism, indicating a deviation from the general equicharacteristic behavior.
• In the case of $G = \mathrm{Sp}_8(\mathbb{K})$ or $\mathrm{SO}_8(\mathbb{K})$, the centralizer dimensions of $F_i\langle\theta\rangle\langle g\rangle$ are computed as $20, 24, 16$ for $g$ of type 2A or 2B, supporting the non-transitivity of automorphism group actions.
• The action of $\mathrm{Aut}_{\overset{\sim}{G}}(E)$ on $\pi_0(C_{\overset{\sim}{G}}(E)) \cong C_2^5$ is not transitive due to the absence of fixed points in $\theta \overset{\sim}{T}$, so $E$ is not $\overset{\sim}{G}$-conjugate to any $E^* \in \widehat{\mathcal{Z}}$.