Control of Fusion and Solubility in Fusion Systems

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.