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

In this paper, we investigate subquadrangles of order s of flock generalized quadrangles S of order (s2,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,s2), 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 (s2,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(∞).