A characterization of the split Cayley generalized hexagon H(q) using one subhexagon of order (1,q)

In this paper, we prove that, if a generalized hexagon Γ of order q contains a subhexagon Γ′ (of order (1,q)) isomorphic to the incidence graph of the Desarguesian plane PG(2,q), and if the automorphism group of Γ stabilizing Γ′ induces all elations in PG(2,q), then Γ must be isomorphic to the split Cayley hexagon H(q).