A Characterization ofQ(5, q) Using One SubquadrangleQ(4, q )

Let 0 be a finite generalized quadrangle of order (q, q2), and suppose that it has a subquadrangle 1 isomorphic to Q(4, q). We show that 0 is isomorphic to the classical generalized quadrangle Q(5, q) if at least one of the following holds: (1) all linear collineations of 1 extend to 0; (2) all subtended ovoids are classical (and we present a uniform proof independent of the characteristic). Further, for q odd, we prove that if every triad {x, y, z} of 1 is 3-regular in 0 and {x, y, z} ⊂ 1, then 0 is classical. We also show that, if for every centric triad {x, y, z} of an ovoid O of the quadrangle