Block-Exoticity of a Family of Exotic Fusion Systems

Abstract

We prove that each exotic fusion system $\mathcal{F}$ on a Sylow $p$-subgroup of $G_2(p)$ for an odd prime $p$ with ${\mathcal{O}}_p(\mathcal{F})=1$ is block-exotic. This gives evidence for the conjecture that each exotic fusion system is block-exotic. We prove two reduction theorems for block-realisable fusion systems.