GENERALIZED QUADRANGLES OF ORDER (S,S2),I

In this paper generalized quadrangles of order (s, s2), s > 1, satisfy ing property (G) at a line, at a pair of points, or at a flag, are stu died. Property (G) was introduced by S. E. Payne (Geom. Dedicata 32 (1 989), 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 (S2, S), s > 1, arising from a flock of a quadratic cone, has property (G) at its point (infinity). In particular translation generalized quadrangle s satisfying property (G) are considered here. As an application it is proved that the Roman generalized quadrangles of Payne contain at lea st S3 + S2 classical subquadrangles Q(4, s). Also, as a by-product, se veral 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 trans lation generalized quadrangles satisfying property (G) at some flag (( infinity), L). (C) 1994 Academic Press, Inc.