turkmath.org

This week we'll start off by completing our discussion of the Alperin-Goldschmidt-Puig Fusion Theorem. We'll characterize the F-essential subgroups in terms of the fusion system F, which will lead to the important notions of F-centric, F-radical, and strongly p-embedded subgroups. The characterization of F-essential subgroups leads naturally to the orbit category of a fusion system, which roughly plays the role of the category of transitive G-sets in the theory of finite groups. The analogy is not perfect, however, and we will spend some time investigating the subtleties involved. We will finish by introducing the linking system associated to F and give some motivation for its importance from the point of view of homotopy theory.