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

Let Γ be a finite generalized quadrangle of order (q,q2 ), and suppose that it has a subquadrangle Δ isomorphic to Q(4,q ). We show that Γ is isomorphic to the classical generalized quadrangle Q(5,q ) if at least one of the following holds: (1) all linear collineations of Δ extend toΓ ; (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 } ofΔ is 3-regular in Γ and { x, y, z } ⊥ ⊥ ⊂ Δ, then Γ is classical. We also show that, if for every centric triad { x, y, z } of an ovoid O of the quadrangleΔ ∼ = Q(4, q), q odd, all points of { x,y , z } ⊥ ⊥ belong to O, then O is classical.