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

Abstract

In this paper, we investigate subquadrangles of order s of flock generalized quadrangles S of order (s 2 , s), s odd, with base point (∞), where the subquadrangle does not contain the point (∞). We prove that if the flock generalized quadrangle has such a subquadrangle, then S is classical. This solves "Remaining case (a)" in Brown and Thas [M.R. Brown, J.A. Thas, Subquadrangles of order s of generalized quadrangles of order (s, s 2 ), II, J. Combin. Theory Ser. A 106 (2004) 33-48] ("Remaining case (b)" was already handled in K. Thas [K. Thas, Symmetrieën in eindige veralgemeende vierhoeken (Symmetries in finite generalized quadrangles), Master thesis, Ghent University, Ghent, 1999, 186 p.]). As a corollary we have: if O(n, 2n, q)is an egg in PG(4n − 1, q) for which the translation dual O * (n, 2n, q) is good at the tangent space of O(n, 2n, q) at its element π and if there is a pseudo-oval on O(n, 2n, q) not containing π , then O(n, 2n, q) is classical. As a byproduct we prove that if the flock GQ S of order (s 2 , s), s odd, has a regular point x collinear with, but distinct from, the base point (∞), then S is a translation generalized quadrangle with base line x(∞).