A FINITE GENERALIZED HEXAGON ADMITTING A GROUP ACTING TRANSITIVELY ONORDERED HEPTAGONS IS CLASSICAL

Let Gamma be a thick finite generalized hexagon and let G be a group o f automorphisms of Gamma. If G acts transitively on the set of non-deg enerate ordered heptagons, then Gamma is one of the Moufang hexagons H (q) or H-3(q) associated to the Chevalley groups G(2)(q) or D-3(4)(q) respectively, or their duals; and G contains the corresponding Chevall ey group. Moreover, we show that no thick generalized octagon admittin g a group acting transitively on the set of ordered nonagons (enneagon s) can exist. This completes the determination of all finite thick gen eralized n-gons, n greater than or equal to 3, with a group acting tra nsitively on the set of ordered (n+1)-gons with elementary methods. Be cause we do not use the classification of the finite simple groups, fr om which these results also follow. (C) 1996 Academic Press, Inc.