Generalised polygons admitting a point-primitive almost simple group of Suzuki or Ree type

TL;DR

This paper investigates the symmetry groups of finite generalized polygons, proving restrictions on the types of groups that can act primitively, and classifying certain cases involving Ree groups.

Contribution

It extends previous results by excluding Suzuki and Ree groups of certain types as minimal normal subgroups in these symmetries, and classifies the Ree--Tits octagon case.

Findings

Suzuki groups cannot be minimal normal subgroups.

Ree groups of type ^2G_2 are excluded.

Ree--Tits octagon is characterized when Ree group of type ^2F_4 is involved.

Abstract

Let $G$ be a collineation group of a thick finite generalised hexagon or generalised octagon $\Gamma$. If $G$ acts primitively on the points of $\Gamma$, then a recent result of Bamberg et al. shows that $G$ must be an almost simple group of Lie type. We show that, furthermore, the minimal normal subgroup $S$ of $G$ cannot be a Suzuki group or a Ree group of type $^2\text{G}_2$, and that if $S$ is a Ree group of type $^2\text{F}_4$, then $\Gamma$ is (up to point--line duality) the classical Ree--Tits generalised octagon.