turkmath.org

The fusion system of a finite group is an organization framework for understanding the p-local structure of that group. The data are encoded in a category with objects the subgroups of a fixed p-group S, and morphisms certain group injections that mimic conjugation homomorphisms induced by an absent ambient group. It can be seen as an algebraic structure in its own right, one which serves as a bridge between the fields of local group theory, modular representation theory, and algebraic topology. In this talk I will discuss joint work with Sune Reeh, in which we parameterize the characteristic bisets of a saturated fusion system. A characteristic biset can be thought of a collection of abstract group elements that conjugate subgroups of S in such a way to recreate the original fusion system. It turns out that there is a simple parameterization of these bisets, and as a consequence we discover the existence of a unique minimal characteristic biset for any saturated fusion system.