The saturated fusion systems on a Sylow 2-subgroup of ${\Omega }_8^{+}(2)$

Abstract

We consider saturated fusion systems $\mathcal{F}$ on a Sylow $2$-subgroup of ${\Omega }_8^{+}(2)$ with $O_2(\mathcal{F})=1$. Examples for this are the $2$-fusion systems of ${\Omega }_8^{+}(2)$, ${\Omega }_8^{+}(2):3$, $P{\Omega }_8^{+}(3)$ and $P{\Omega }_8^{+}(3):3$