Point-primitive generalised hexagons and octagons

TL;DR

This paper proves that point-primitivity alone ensures that the automorphism group of a generalized hexagon or octagon is almost simple of Lie type, refining previous results that required additional conditions.

Contribution

It shows point-primitivity suffices for the classification of automorphism groups of generalized polygons, reducing the conditions needed compared to earlier work.

Findings

Point-primitivity implies the automorphism group is almost simple of Lie type.

Narrowed the classification of generalized hexagons and octagons with primitive collineation groups.

Classical examples are the only known cases with such symmetry groups.

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.