Generalized polygons with highly transitive collineation groups

Abstract

Abstract. It will be proved that the compact connected topological generalized quadrangles which admit a collineation group that acts transitively on ordered pentagons are precisely the real or complex orthogonal quadrangles, up to duality. Classitieations (1991): 51E 12, 51H 15, 51H20. Mathematics SubjectIt is natural to characterize classes of geometries which have a lot of symmetry. Usually this symmetry is expressed in terms of collineation groups. So we want to examine geometries -here we are interested in generalized polygons -with large collineation groups. There are several ways to make precise what 'large' means. In this paper we investigate the situation where the collineation group of generalized n-gon acts transitively on the set of ordered (n + 1)-gons. This condition is of special interest because it is related to questions of coordinatization (cp. Hughes and Piper [12, §V.3]