No almost simple sporadic group acts primitively on the points of a generalised quadrangle

TL;DR

This paper proves that no almost simple sporadic group can act primitively on the points of a finite generalized quadrangle, advancing understanding of automorphism groups in finite geometry.

Contribution

It establishes a non-existence result for certain group actions and introduces algorithms and conjectures related to primitive group actions on generalized quadrangles.

Findings

No almost simple sporadic group acts primitively on generalized quadrangle points.

Introduces an algorithm to test primitive group actions on generalized quadrangles.

Proposes a conjecture on the line-orbit structure of primitive groups acting on these geometries.

Abstract

A generalised quadrangle is a point-line incidence geometry G such that: (i) any two points lie on at most one line, and (ii) given a line L and a point p not incident with L, there is a unique point on L collinear with p. They are a specific case of the generalised polygons introduced by Tits, and these structures and their automorphism groups are of some importance in finite geometry. An integral part of understanding the automorphism groups of finite generalised quadrangles is knowing which groups can act primitively on their points, and in particular, which almost simple groups arise as automorphism groups. We show that no almost simple sporadic group can act primitively on the points of a finite (thick) generalised quadrangle. We also present two new ideas contributing towards analysing point-primitive groups acting on generalised quadrangles. The first is the outline and implementation of an algorithm for determining whether a given group can act primitively on the points of some generalised quadrangle. The second is the discussion of a conjecture resulting from observations made in the course of this work: any group acting primitively on the points of a generalised quadrangle must either act transitively on lines or have exactly two line-orbits, each containing half of the lines.