Generalised Polygons Admitting a Point-Primitive Almost Simple Group of Suzuki or Ree Type

Luke Morgan,Tomasz Popiel

doi:10.37236/5510

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 $^2\mathrm{G}_2$, and that if $S$ is a Ree group of type $^2\mathrm{F}_4$, then $\Gamma$ is (up to point-line duality) the classical Ree-Tits generalised octagon.

Highlights

A generalised d-gon is a point–line incidence geometry Γ whose bipartite incidence graph has diameter d and girth 2d
The minimal normal subgroup S of G cannot be a Suzuki group or a Ree group of type 2G2, and that if S is a Ree group of type 2F4, Γ is the classical Ree–Tits generalised octagon
We are concerned with the cases d = 6, and d = 8

Summary

Introduction

A generalised d-gon is a point–line incidence geometry Γ whose bipartite incidence graph has diameter d and girth 2d. The action of the collineation group is primitive on both the points and the lines of Γ, and transitive on the flags of Γ, namely the incident point–line pairs. Buekenhout and Van Maldeghem [4] showed that point-distance-transitivity implies pointprimitivity for a thick finite generalised hexagon or octagon, and proved that there exist no point-distance-transitive examples other than the known classical examples. Schneider and Van Maldeghem [10] showed that a group G acting point-primitively, line-primitively, and flag-transitively on a thick finite generalised hexagon or octagon must be an almost simple group of Lie type. Let G be a point-primitive collineation group of a thick finite generalised hexagon or generalised octagon Γ, with S G Aut(S) for some nonabelian finite simple group S. Theorem 1 is proved in three sections: the Suzuki groups are considered in Section 3; the small and large Ree groups are dealt with in Sections 4 and 5, respectively

Preliminaries

Proof of Theorem 1: S a Suzuki group

Proof of Theorem 1: S a Ree group of type 2G2

Proof of Theorem 1: S a Ree group of type 2F4