Subquadrangles of order s of generalized quadrangles of order (s,s2), Part I

In this paper, subquadrangles of order s of generalized quadrangles (GQs) of order (s,s2) are investigated. In the case where S is a dual flock GQ of order (s,s2), s even, the classification of subquadrangles of order s is completed. In the case where E is an egg good at an element and the corresponding translation generalized quadrangle T(E) of order (s,s2), s even, has base point (∞) it is proved that if T(E) has a subquadrangle of order s not containing (∞), then T(E) is the classical GQ Q(5,s). Also a new strong characterization of the classical GQ Q(5,s), s even or odd, is obtained. In Part II, the case s odd is considered.