THE CLASSIFICATION OF FINITE GENERALIZED QUADRANGLES ADMITTING A GROUP ACTING TRANSITIVELY ON ORDERED PENTAGONS

Let L be a thick generalized quadrangle and let G be a group of automo rphisms of L. If G acts transitively on the set of non-degenerate orde red pentagons, then L is one of the classical generalized quadrangles W(q), Q(4,q), Q(5,q) or H(3,q(2)). The possibilities for G in each cas e are determined. We do not use the classification of the finite simpl e groups (from which this result also follows).