On a characterization of the finite twisted triality hexagons using one classical ideal split Cayley subhexagon

Abstract

Let $\mathrm{\Delta }$ be a generalized hexagon of order $\left(q^3,q\right)$, for some prime power $q$ not divisible by $3$. Suppose that $\mathrm{\Delta }$ contains a subhexagon $\mathrm{\Gamma }$ of order $\left(q,q\right)$ isomorphic to a split Cayley hexagon (associated to Dickson’s group ${\mathsf{G}}_2\left(q\right)$), and suppose that every axial elation (long root elation) in $\mathrm{\Gamma }$ is induced by Aut${\left(\mathrm{\Delta }\right)}_{\mathrm{\Gamma }}$. Then we show that $\mathrm{\Delta }$ is isomorphic to the twisted triality hexagon $\mathsf{T}\left(q^3,q\right)$ associated to the group ${}^3{\mathsf{D}}_4\left(q\right)$.