On fusion systems of component type - Aschbacher, Michael, 1944- (Memoirs of the American Mathematical Society, 2019)
This memoir begins a program to classify a large subclass of the class of simple saturated 2-fusion systems of component type. Such a classification would be of great interest in its own right, but in addition it should lead to a significant simplification of the proof of the theorem classifying the finite simple groups.Why should such a simplification be possible? Part of the answer lies in the fact that there are advantages to be gained by working with fusion systems rather than groups. In particular one can hope to avoid a proof of the B-Conjecture, a important but difficult result in finite group theory, established only with great effort.But in addition, the program involves a reorganization of the treatment of âĂĲgroups of component typeâĂİ, or perhaps more accurately, of âĂĲfusion systems of component typeâĂİ. The groups of component type should be viewed as âĂĲoddâĂİ groups, in that most examples are groups of Lie type over fields of odd order. The remaining simple groups should be viewed as âĂĲevenâĂİ groups, since most of the examples in this class are of Lie type over fields of even order. There are corresponding notions of âĂĲoddâĂİ and âĂĲevenâĂİ 2-fusion systems.In our program the class of odd groups, and/or fusion systems, is contracted in a carefully chosen manner, so as to avoid difficulties associated to certain âĂĲstandard form problemsâĂİ. This has the effect of greatly simplifying the treatment of the odd 2-fusion systems, and then also the treatment of the odd simple groups. Of course the flip side of such a reorganization is to enlarge the class of even objects, so that the approach may make it more difficult to treat that class. But it is our sense that the trade off should lead to a net simplification.