Abstract: | A generalized quadrangle is a point-line incidence geometry such that any two points lie on at most one line and, given a line and a point not incident with , there is a unique point of collinear with . We study the structure of groups acting regularly on the point set of a generalized quadrangle. In particular, we provide a characterization of the generalized quadrangles with a group of automorphisms acting regularly on both the point set and the line set and show that such a thick generalized quadrangle does not admit a polarity. Moreover, we prove that a group acting regularly on the point set of a generalized quadrangle of order or , where is odd and is coprime to 3, cannot have any nonabelian minimal normal subgroups. |