Realizability of fusion systems by discrete groups: II

TL;DR

This paper investigates the conditions under which fusion systems over discrete p-toral groups can be realized by discrete groups, providing criteria, constructions, and extending classical theorems to this setting.

Contribution

It introduces new criteria for the realizability of fusion systems by discrete groups and constructs explicit examples, extending the theory to locally finite groups and discrete p-toral groups.

Findings

Fusion systems over discrete p-toral groups can be saturated under certain topological closure conditions.

Explicit constructions of saturated fusion systems and linking systems are provided for locally finite groups.

A version of the Cartan-Eilenberg stable elements theorem is proved for locally finite groups.

Abstract

We compare four different types of realizability for saturated fusion systems over discrete $p$-toral groups. For example, when $G$ is a locally finite group all of whose $p$-subgroups are artinian (hence discrete $p$-toral), we show that it has ``weakly Sylow'' $p$-subgroups and give explicit constructions of saturated fusion systems and associated linking systems associated to $G$. We also show that a fusion system over a discrete $p$-toral group $S$ is saturated if its set of morphisms is closed under a certain topology and the finite subgroups of $S$ satisfy the saturation axioms, and prove a version of the Cartan-Eilenberg stable elements theorem for locally finite groups.