Minimal characteristic bisets for fusion systems

Abstract

We show that every saturated fusion system $\mathcal{F}$ has a unique minimal $\mathcal{F}$-characteristic biset ${\Lambda }_{\mathcal{F}}$. We examine the relationship of ${\Lambda }_{\mathcal{F}}$ 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 ${\Lambda }_{\mathcal{F}}$ as bisets. We explore the grouplike properties of ${\Lambda }_{\mathcal{F}}$, and conjecture an identification of normalizer subsystems of $\mathcal{F}$ with subbisets of ${\Lambda }_{\mathcal{F}}$.

arXiv comment26 pages