EMBEDDED THICK FINITE GENERALIZED HEXAGONS IN PROJECTIVE-SPACE

We show that every embedded finite thick generalized hexagon H of orde r (s, t) in PG(n,q) which satisfies the conditions (i) s = q, (ii) the set of all points of H generates PG(n, q), (iii) for any point x of H , the set of ail points collinear in H with x is contained in a plane of PG(n, q), (iv) for any point x of H, the set of all points of H not opposite x in H is contained in a hyperplane of PG(n, q), is necessar ily the standard representation of H(q) in PG(6, q) (on the quadric Q( 6, q)), the standard representation of H(q) for q even in PG(5, q) (in side a symplectic space). or the standard representation of H(q, cube- root q) in PG(7, q) (where the lines of H are the lines fixed by a tri ality on the quadric Q(+)(7, q)). This generalizes a result by Cameron and Kantor [3], which is used in our proof.