Point-primitive generalised hexagons and octagons

The only known examples of finite generalised hexagons and octagons arise from the finite almost simple groups of Lie type G2, D43, and F42. These groups act transitively on flags, primitively on points, and primitively on lines. The best converse result prior to the writing of this paper was that of Schneider and Van Maldeghem (2008): if a group G acts flag-transitively, point-primitively, and line-primitively on a finite generalised hexagon or octagon, then G is an almost simple group of Lie type. We strengthen this result by showing that the same conclusion holds under the sole assumption of point-primitivity.