Characterizations of Translation GeneralizedQuadrangles

Abstract

If x is a regular point of the generalized quadrangle \mathcal{S} of order (s,t), s\ne 1 \ne t, then x defines a dual net \mathcal{N}^\ast_x. If \mathcal{S} contains a line L of regular points and if for at least one point x on L the automorphism group of the dual net \mathcal{N}_x^\ast satisfies certain transitivity properties, then \mathcal{S} is a translation generalized quadrangle. This result has many applications. We give one example. If s=t\ne 1, then \mathcal{N}^\ast_x is a dual affine plane. Let \mathcal{S} be a generalized quadrangle of order s,s odd and s\ne 1, which contains a line L of regular points. If for at least one point x on L the plane \mathcal{N}^\ast_x is Desarguesian, then \mathcal{S} is isomorphic to the classical generalized quadrangle W(s).