Abstract

If x is a regular point of a generalized quadrangle S = (P, B, I) of order (s, t), s ≠ 1, then x defines a dual net with t + 1 points on any line and s lines through every point. If s ≠ t, s > 1, t > 1, then S is isomorphic to a T3(O) of Tits if and only if S has a coregular point x such that for each line L incident with x the corresponding dual net satisfies the Axiom of Veblen. As a corollary we obtain some elegant characterizations of the classical generalized quadrangles Q(5, s). Further we consider the translation generalized quadrangles S(p) of order (s, s2), s ≠ 1, with base point p for which the dual net defined by L, with p I L, satisfies the Axiom of Veblen. Next there is a section on Property (G) and the Axiom of Veblen, and a section on flock generalized quadrangles and the Axiom of Veblen. This last section contains a characterization of the TGQ of Kantor in terms of the Axiom of Veblen. Finally, we prove that the dual net defined by a regular point of S, where the order of S is (s, t) with s ≠ t and s ≠ 1 ≠ t, satisfies the Axiom of Veblen if and only if S admits a certain set of proper subquadrangles.