Fusion action systems by Matthew J.K. Gelvin.

Abstract

The study of fusion first arose in the local theory of finite groups. Puig abstracted the fusion data of a finite group to the notion of fusion system, an object that reflects local data in more abstract algebraic settings, such as the block theory of finite groups. Martino and Priddy conjectured that the algebraic data of a fusion system of a finite group should have a topological interpretation, which result was proved by Oliver using the notion of p-local finite group introduced by the team of Broto, Levi, and Oliver. The study of fusion systems and p-local finite groups thus provides a bridge between algebraic fields related to local group theory and algebraic topology. In this thesis we generalize the notion of abstract fusion system to model the local structure of a group action on a finite set. The resulting fusion action systems can be seen as a generalization of the notion of abstract fusion system, though we describe other possible interpretations as well. We also develop the notion of a p-local finite group action, which allows for connections between fusion action system theory and algebraic topology..