Characterizations of translation generalized quadrangles
If x is a regular point of the generalized quadrangle S of order (s,t), s not equal1 not equal t, then x defines a dual net N-x*. If 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 N-x* satisfies certain transitivity properties, then S is a translation generalized quadrangle. This result has many applications. We give one example. If s=t not equal 1, then N-x* is a dual affine plane. Let S be a generalized quadrangle of order s,s odd and s not equal 1, which contains a line L of regular points. If for at least one point x on L the plane N-x* is Desarguesian, then S is isomorphic to the classical generalized quadrangle W(s).
translation generalized quadrangle net incidence geometry