Rigid automorphisms of linking systems of finite groups of Lie type

2025/09/25 by Jonathon Villareal, Villareal, Jonathon
Mathematics · #Advanced Algebra and Geometry #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Group Theory (math.GR)

paper · pdf · doi:10.48550/arxiv.2509.21139

openalex publication_date 2025/09/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let L be a centric linking system associated to a saturated fusion system on a finite p-group S. An automorphism of L is said to be rigid if it restricts to the identity on the fusion system. An inner rigid automorphism is conjugation by some element of the center of S. If L is the centric linking system of a finite group G, then rigid automorphisms of L are closely related to automorphisms of G that centralize S. For odd primes, all rigid automorphisms are known to be inner, but this fails for the prime 2. We determine which known quasisimple linking systems at the prime 2 have a noninner rigid automorphism. Based on previous results, this reduces to handling the case of the linking systems at the prime 2 of finite simple groups of Lie type in odd characteristic. These have no noninner rigid automorphisms with two families of exceptions: the 2-dimensional projective special linear groups and even-dimensional orthogonal groups for quadratic forms of nonsquare discriminant.

Citations

• Realizability and tameness of fusion systems
• Automorphisms of fusion systems of finite simple groups of Lie type
• Fusion systems and localities
• Equivalences of classifying spaces completed at odd primes

Related

• Automorphisms of fusion systems of finite simple groups of Lie type
• Fusion systems and localities
• Centric linking systems of locally finite groups
• Characterizing inner automorphisms and realizing outer automorphisms
• Rigidity in automorphic representations and local systems