A Finite Generalized Hexagon Admitting a Group Acting Transitively on Ordered Heptagons is Classical

LetΓbe a thick finite generalized hexagon and letGbe a group of automorphisms ofΓ. IfGacts transitively on the set of non-degenerate ordered heptagons, thenΓis one of the Moufang hexagonsH(q) or3H(q) associated to the Chevalley groupsG2(q) or3D4(q) respectively, or their duals; andGcontains the corresponding Chevalley group. Moreover, we show that no thick generalized octagon admitting a group acting transitively on the set of ordered nonagons (enneagons) can exist. This completes the determination of all finite thick generalizedn-gons,n≥3, with a group acting transitively on the set of ordered (n+1)-gons with elementary methods. Because we do not use the classification of the finite simple groups, from which these results also follow.