Characterizations of Translation Generalized Quadrangles

Abstract

If x is a regular point of the generalizedquadrangle $\mathcal{S}$ of order (s,t), s ≠ 1≠ t, then x defines a dual net $\mathcal{N}_x^*$. If$\mathcal{S}$ contains a line L of regularpoints and if for at least one point x on Lthe automorphism group of the dual net$\mathcal{N}_x^*$ satisfies certain transitivityproperties, then$\mathcal{S}$ is a translation generalized quadrangle. Thisresult has many applications. We give one example. Ifs=t ≠ 1, then $\mathcal{N}_x^*$is a dual affine plane. Let $\mathcal{S}$ be a generalizedquadrangle of orders,s odd and s ≠ 1, which contains a lineL of regular points. If for at least one pointx on L the plane$\mathcal{N}_x^*$ is Desarguesian, then$\mathcal{S}$ is isomorphic to the classical generalizedquadrangleW(s).