Normal Subsystems of Fusion Systems

TL;DR

This paper proves that in saturated fusion systems, the smallest weakly normal subsystem on a strongly closed subgroup is actually normal, and explores related structural properties and concepts.

Contribution

It establishes the equivalence of weakly normal and normal subsystems in saturated fusion systems and develops a theory of weakly normal maps and related concepts.

Findings

Weakly normal subsystems are actually normal in saturated fusion systems.

The notion of simplicity is independent of using weakly normal or normal subsystems.

Developed a theory of weakly normal maps and studied intersections, products, and the hypercentre.

Abstract

In this article we prove that for any saturated fusion system, that the (unique) smallest weakly normal subsystem of it on a given strongly closed subgroup is actually normal. This has a variety of corollaries, such as the statement that the notion of a simple fusion system is independent of whether one uses weakly normal or normal subsystems. We also develop a theory of weakly normal maps, consider intersections and products of weakly normal subsystems, and the hypercentre of a fusion system.