Some extensions and embeddings of near polygons

Let (P, L, *) be a near polygon having s + 1 points per line, s > 1, and suppose k is a field. Let V k be the k-vector space with basis { vp |p Î P}vppP Then the subspace generated by the vectors v1 = Sp*1 vpv1=p1vp, where l ÎL, has codimension at least 2 in V k. This observation is used in two ways. First we derive the existence of certain diagram geometries with flag transitive automorphism group, and secondly, we show that any finite near polygon with 3 points per line can be embedded in an affine GF(3)-space.