On Fusion Systems of Component Type

Abstract

Introduction. This series of lectures involves the interplay between local group theory and the theory of fusion systems, with the focus of interest the possibility of using fusion systems to simplify part of the proof of the theorem classifying the finite simple groups. For our purposes, the classification of the finite simple groups begins with the GorensteinWalter Dichotomy Theorem (cf. [ALSS]) which says that each finite group G of 2-rank at least 3 is either of component type or of characteristic 2-type. This supplies a partition of the finite groups into groups of odd and even characteristic, from the point of view of their 2-local structure. We will be concerned almost exclusively with the groups of odd characteristic: the groups of component type. However Ulrich Meierfrankenfeld’s lectures can be thought of as being concerned with the groups of even characteristic. In the case of a saturated fusion system F , the situation vis-a-vis the GorensteinWalter dichotomy is nicer: F is either of characteristic p-type or component type, irrespective of rank. Further the Dichotomy Theorem for saturated fusion systems is much easier to prove than the theorem for groups; indeed once the notion of the generalized Fitting subsystem F ∗(F) of a saturated fusion system F is put in place, and suitable properties of F ∗(F) are established, including E-balance, the proof of the Dichotomy Theorem for fusion systems is easy. But of more importance, it seems easier to work with 2-fusion systems of component type than with groups of component type. This is because in a group G of component type, a 2-local subgroup H of G may have a nontrivial core, where the core of H is the largest normal subgroup O(H) of H of odd order. The existence of these cores introduces big problems into the analysis of groups of component type. These problems can be minimized if one can prove the B-Conjecture, which says that, in a simple group,