Embedded Thick Finite Generalized Hexagons in Projective Space

Abstract

We show that every embedded finite thick generalized hexagon ℋ of order (s, t) in PG(n,q) which satisfies the conditions s = q the set of all points of ℋ generates PG(n, q) for any point x of ℋ, the set of all points collinear in ℋ with x is contained in a plane of PG(n, q) 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 necessarily 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) (inside a symplectic space), or the standard representation of H(q, q 3 ) in PG(7, q) (where the lines of ℋ are the lines fixed by a triality on the quadric Q+(7, q)). This generalizes a result by Cameron and Kantor [3], which is used in our proof.