Control of fusion and solubility in fusion systems

In this article, we consider control of fusion, quotients, and p-soluble fusion systems. For control of fusion, we prove the three main theorems in the literature in a new, largely elementary way, significantly shortening their proofs. To prove one of these, and a theorem of Aschbacher that the product of strongly closed subgroups is strongly closed, we produce a consolidated treatment of quotients, collating and expanding the constructions previously available; we include analogues of the isomorphism theorems for fusion systems. We move on to 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 it for more general fusion systems.