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

In this paper, subquadrangles of order s of generalized quadrangles (GQ) of order (s,s2), with s odd, are investigated. The even case was considered in Part I. In the case where O is an egg good at an element π and the translation generalized quadrangle S=T(O) has order (s,s2), with s odd, we prove that if S′ is a subquadrangle of order s of S, then S′ is the classical GQ Q(4,s) and either S is the classical GQ Q(5,s) or S′ is one of the s3+s2 subquadrangles of order s containing the line π of S. Further, some characterizations of particular eggs are obtained. Finally, it is shown that if S is a flock GQ of order (s2,s), s odd, with base point (∞) and S′ is a subquadrangle of order s of S containing the point (∞), then S is a Kantor semifield flock GQ, S′ is isomorphic to the classical GQ W(s) and either S is isomorphic to the classical GQ H(3,s2) or S′ is one of the s3+s2 subquadrangles of order s containing the point (∞). As an application there is a characterization of the Kantor semifield flock GQ in terms of the net defined by the regular point (∞) of the flock GQ.