Part II - The local theory of fusion systems

Summary

Part II of the book is intended to be an introduction to the local theory of saturated fusion systems. By the “local theory of fusion systems” we mean an extension of some part of the local theory of finite groups to the setting of saturated fusion systems on finite p-groups.

Recall the notions of a p-local finite group and its associated linking system defined in Definitions III.4.1 and III.4.4. One can ask: Why deal with saturated fusion systems rather than p-local finite groups? There are two reasons for this choice. First, it is not known whether to each saturated fusion system there is associated a unique p-local finite group. Thus it remains possible that the class of saturated fusion systems is larger than the class of p-local finite groups. But more important, to date there is no accepted notion of a morphism of p-local finite groups, and hence no category of p-local groups. (The problem arises already for fusion systems and linking systems of groups, since a group homomorphism α from G to H need not send p-centric subgroups of G to p-centric subgroups of H, so it is not clear how to associate to α a map which would serve as a morphism of the p-local group of G with the p-local group of H.) The local theory of finite groups is inextricably tied to the notion of group homomorphism and factor group, so to extend the local theory of finite groups to a different category, we must at the least be dealing with an actual category.