Subquadrangles of order s of generalized quadrangles of order , Part III

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 , 2 n , q ) is an egg in PG ( 4 n − 1 , q ) for which the translation dual O ∗ ( n , 2 n , q ) is good at the tangent space of O ( n , 2 n , q ) at its element π and if there is a pseudo-oval on O ( n , 2 n , q ) not containing π, then O ( n , 2 n , 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 ( ∞ ) .