Point-primitive generalised hexagons and octagons

Abstract

In 2008, Schneider and Van Maldeghem proved that if a group acts flag-transitively, point-primitively, and line-primitively on a generalised hexagon or generalised octagon, then it is an almost simple group of Lie type. We show that point-primitivity is sufficient for the same conclusion, regardless of the action on lines or flags. This result narrows the search for generalised hexagons or octagons with point- or line-primitive collineation groups beyond the classical examples, namely the two generalised hexagons and one generalised octagon admitting the Lie type groups ${\mathsf{G}}_2(q)$, ${}^3{\mathsf{D}}_4(q)$, and ${}^2{\mathsf{F}}_4(q)$, respectively.