Rigid automorphisms of linking systems

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 $p^{\prime }$-order modulo the group of inner automorphisms, provided $G$ has no nontrivial normal $p^{\prime }$-subgroups. We present two applications of this last result, one to tame fusion systems.

21 pages; v2: minor corrections and improvements; v3: no mathematical changes

References in corpus (2)

• Centers of Sylow Subgroups and Automorphisms
• Finite Localities III

Cited by in corpus (1)

• Extensions of homomorphisms between localities