Fusion systems and localities

Abstract

We introduce objective partial groups, of which the linking systems and p-local finite groups of Broto, Levi, and Oliver, the transporter systems of Oliver and Ventura, and the $\mathcal{F}$-localities of Puig are examples, as are groups in the ordinary sense. As an application we show that if $\mathcal{F}$ is a saturated fusion system over a finite p-group then there exists a centric linking system $\mathcal{L}$ having $\mathcal{F}$ as its fusion system, and that $\mathcal{L}$ is unique up to isomorphism. The proof relies on the classification of the finite simple groups in an indirect and—for that reason—perhaps ultimately removable way.