Minimal characteristic bisets for fusion systems

Abstract

We show that every saturated fusion system $\mathcal{F}$ has a unique minimal $\mathcal{F}$-characteristic biset $Λ_\mathcal{F}$. We examine the relationship of $Λ_\mathcal{F}$ with other concepts in $p$-local finite group theory: In the case of a constrained fusion system, the model for the fusion system is the minimal $\mathcal{F}$-characteristic biset, and more generally, any centric linking system can be identified with the $\mathcal{F}$-centric part of $Λ_\mathcal{F}$ as bisets. We explore the grouplike properties of $Λ_\mathcal{F}$, and conjecture an identification of normalizer subsystems of $\mathcal{F}$ with subbisets of $Λ_\mathcal{F}$.