Abstract

Abstract A rigid automorphism of a linking system is an automorphism that restricts to the identity on the Sylow subgroup. A rigid inner automorphism is conjugation by an element in the center of the Sylow subgroup. At odd primes, it is known that each rigid automorphism of a centric linking system is inner. We prove that the group of rigid outer automorphisms of a linking system at the prime $2$ is elementary abelian and that it splits over the subgroup of rigid inner automorphisms. In a second result, we show that if an automorphism of a finite group G restricts to the identity on the centric linking system for G, then it is of $p'$ -order modulo the group of inner automorphisms, provided G has no nontrivial normal $p'$ -subgroups. We present two applications of this last result, one to tame fusion systems.

Highlights

  • A saturated fusion system F is a category in which the objects are the subgroups of a fixed finite group and the morphisms are injective group homomorphisms between subgroups, which are subject to axioms first outlined by Puig [Pui06, AKO11]

  • From a group theoretic point of view, centric linking systems or, more generally, the transporter systems of Oliver-Ventura [OV07] and the localities of Chermak [Che13], provide finer approximations to -local structure. They abstract the transporter categories of finite groups and form structures appearing in recent new approaches to revising the classification of finite simple groups

  • We study here in more detail the comparison maps between automorphism groups of finite groups, linking systems and fusion systems

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Summary

Introduction

A saturated fusion system F is a category in which the objects are the subgroups of a fixed finite group and the morphisms are injective group homomorphisms between subgroups, which are subject to axioms first outlined by Puig [Pui, AKO11]. From a group theoretic point of view, centric linking systems or, more generally, the transporter systems of Oliver-Ventura [OV07] and the localities of Chermak [Che13], provide finer approximations to -local structure. They abstract the transporter categories of finite groups and form structures appearing in recent new approaches to revising the classification of finite simple groups. We study here in more detail the comparison maps between automorphism groups of finite groups, linking systems and fusion systems. When L is a centric linking system associated to the fusion system F, there are groups of automorphisms Aut(L) and Aut(F) and a map : Aut(L) → Aut(F) given essentially by restriction to the Sylow group.

George Glauberman and Justin Lynd of maps
Transporter systems
Localities
Equivalence between transporter systems and localities
Linking systems and linking localities
Rigid outer automorphisms of centric linking systems
Extending to larger object sets
Comparing automorphisms of groups and linking systems
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