Generalized quadrangles of orrder ( s, s2), I

Abstract

In this paper generalized quadrangles of order ( s, s 2), s > 1, satisfying property (G) at a line, at a pair of points, or at a flag, are studied. Property (G) was introduced by S. E. Payne ( Geom. Dedicata 32 (1989) , 93–118) and is weaker than 3-regularity (see S. E. Payne and J. A. Thas, “Finite Generalized Quadrangles,” Pitman, London, 1984). It was shown by Payne that each generalized quadrangle of order ( s 2, s), s > 1, arising from a flock of a quadratic cone, has property (G) at its point (∞). In particular translation generalized quadrangles satisfying property (G) are considered here. As an application it is proved that the Roman generalized quadrangles of Payne contain at least s 3 + s 2 classical subquadrangles Q(4, s). Also, as a by-product, several classes of ovoids of Q(4, s), s odd, are obtained; one of these classes is new. The goal of Part II is the classification of all translation generalized quadrangles satisfying property (G) at some flag ((∞), L).