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

Control of Fusion and Solubility in Fusion Systems