Point-primitive, line-transitive generalised quadrangles of holomorph type

TL;DR

This paper investigates the symmetry groups of finite generalized quadrangles, proving restrictions on their primitive actions and reducing the classification problem to cases with a unique non-Abelian minimal normal subgroup.

Contribution

It shows that certain primitive actions of collineation groups cannot be of holomorph simple or compound type, narrowing the classification of these geometries.

Findings

Primitive action cannot be of holomorph simple type.

Primitive action cannot be of holomorph compound type.

Classification reduces to cases with a unique non-Abelian minimal normal subgroup.

Abstract

Let $G$ be a group of collineations of a finite thick generalised quadrangle $\Gamma$. Suppose that $G$ acts primitively on the point set $\mathcal{P}$ of $\Gamma$, and transitively on the lines of $\Gamma$. We show that the primitive action of $G$ on $\mathcal{P}$ cannot be of holomorph simple or holomorph compound type. In joint work with Glasby, we have previously classified the examples $\Gamma$ for which the action of $G$ on $\mathcal{P}$ is of affine type. The problem of classifying generalised quadrangles with a point-primitive, line-transitive collineation group is therefore reduced to the case where there is a unique minimal normal subgroup $M$ and $M$ is non-Abelian.