On Some Automorphisms of Orthogonal Groups in Odd Characteristic

In this paper, it is proved that the simple orthogonal groups $O_{2n+1}(q)$ and $O_{2n}^\pm (q)$ (where $q$ is odd) cannot be automorphism groups of finite left distributive quasigroups. This is a particular case of the conjecture stating that the automorphism group of a left distributive quasigroup is solvable. To complete the proof of the conjecture, one must test all finite groups.