Extensions of homomorphisms between localities

TL;DR

This paper proves that the automorphism group of a linking system depends only on the underlying fusion system when the object set is invariant, extending known results to more general linking systems and localities.

Contribution

It generalizes the dependence of automorphism groups to broader classes of linking systems and localities, and introduces an extension lemma for homomorphisms between localities.

Findings

Automorphism group depends only on the fusion system for invariant object sets.

Extension lemma for homomorphisms between localities established.

Reproves a bijection between partial normal subgroups of linking localities.

Abstract

We show that the automorphism group of a linking system associated to a saturated fusion system $\mathcal{F}$ depends only on $\mathcal{F}$ as long as the object set of the linking system is $\mathrm{Aut}(\mathcal{F})$-invariant. This was known to be true for linking systems in Oliver's definition, but we demonstrate that the result holds also for linking systems in the considerably more general definition introduced previously by the author of this paper. A similar result is proved for linking localities, which are group-like structures corresponding to linking systems. Our argument builds on a general lemma about the existence of an extension of a homomorphism between localities. This lemma is also used to reprove a theorem of Chermak showing that there is a natural bijection between the sets of partial normal subgroups of two possibly different linking localities over the same fusion system.