Rigid automorphisms of linking systems

TL;DR

This paper investigates the structure of rigid automorphisms in linking systems, revealing new properties at the prime 2 and implications for automorphisms of finite groups, with applications to fusion systems.

Contribution

It proves that the group of rigid outer automorphisms at prime 2 is elementary abelian and splits over rigid inner automorphisms, extending known results at odd primes.

Findings

Rigid outer automorphisms at prime 2 form an elementary abelian group.

The group of rigid automorphisms splits over rigid inner automorphisms.

Automorphisms fixing the centric linking system are of p'-order modulo inner automorphisms under certain conditions.

Abstract

A rigid automorphism of a linking system is an automorphism which 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.