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

Abstract

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 L = T(O) has order (s, s2), with s odd, we prove that if L′ is a subquadrangle of order s of L, then L′ is the classical GQ Q(4, s) and either L is the classical GQ Q(5, s) or L′ is one of the s3 + s2 subquadrangles of order s containing the line π of L. Further, some characterizations of particular eggs are obtained. Finally, it is shown that if L is a flock GQ of order (s2, s), s odd, with base point (∞) and L′ is a subquadrangle of order s of L containing the point (∞), then L is a Kantor semifield flock GQ, L′ is isomorphic to the classical GQ W(s) and either L is isomorphic to the classical GQ H(3, s2) or L′ 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. © 2004 Elsevier Inc. All rights reserved.