Generalized quadrangles of orrder (s, s2), I

J.A Thas
1994 Journal of combinatorial theory. Series A  
In this paper generalized quadrangles of order (s, s2), 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
more » ... ~). 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 ((oo), L).
doi:10.1016/0097-3165(94)90009-4 fatcat:febemdkbnvei5mmgysv5eb4cxq