Control of Fusion and Solubility in Fusion Systems

TL;DR

This paper advances the understanding of fusion systems by providing elementary proofs of key results, consolidating quotient theory, and characterizing p-soluble fusion systems, with implications for Thompson Factorization.

Contribution

It offers new elementary proofs of known results, unifies quotient theory in fusion systems, and characterizes p-soluble fusion systems, extending Thompson Factorization.

Findings

Elementary proofs of fusion system results

Unified quotient theory for fusion systems

Characterization of p-soluble fusion systems

Abstract

In this article, we consider the control of fusion in fusion systems, proving three previously known, non-trivial results in a new, largely elementary way. We then reprove a result of Aschbacher, that the product of two strongly closed subgroups is strongly closed; to do this, we consolidate the theory of quotients of fusion systems into a consistent theory. We move on considering p-soluble fusion systems, and prove that they are constrained, allowing us to effectively characterize fusion systems of p-soluble groups. This leads us to recast Thompson Factorization for Qd(p)-free fusion systems, and consider Thompson Factorization for more general fusion systems.