A coordinatization structure for generalized quadrangles with a regular spread

We shall coordinatize generalized quadrangles with a regular spread by means of a Steiner system (P,L)(P,L), a set XX and a certain nice map Δ:P×P→Sym(X). We shall then show how this coordinatization method can be used to improve a result independently obtained by Kantor [W.M. Kantor, Note on span-symmetrical generalized quadrangles, Adv. Geom. 2 (2) (2002) 197–200] and Thas [K. Thas, Classification of span-symmetric generalized quadrangles of order ss, Adv. Geom. 2 (2) (2002) 189–196] stating that a generalized quadrangle of order s≥2s≥2 is isomorphic to W(s)W(s) if it has a hyperbolic line all of whose points are centres of symmetry. We shall show that if a generalized quadrangle QQ of order s≥2s≥2 has a hyperbolic line containing only regular points, then all these points are also centres of symmetry. Combining this with the above-mentioned result independently obtained by Kantor and Thas, we then obtain that QQ is isomorphic to the symplectic quadrangle W(s)W(s).