Components and realizability of fusion systems

Abstract

For $p\in \{2,3\}$ it is known that a saturated $p$-fusion system is realizable if and only if each of its components is realizable by a finite simple group. For primes $p\ge 5$ this is false. Building on work of Broto, Møller, Oliver and Ruiz, we show however that a fusion system $\mathcal{F}$ is realizable if and only if for each of its components $\mathcal{C}$ there exists a realizable subnormal subsystem $\mathcal{E}$ of $\mathcal{F}$ with $O^{p^{\prime }}(\mathcal{E})=\mathcal{C}$.