Partial linear spaces with a rank 3 affine primitive group of automorphisms

TL;DR

This paper classifies finite proper partial linear spaces with rank 3 affine primitive automorphism groups, completing the classification except for some small group families, and provides detailed insights into rank 3 affine primitive permutation groups.

Contribution

It offers a comprehensive classification of such partial linear spaces and enhances understanding of rank 3 affine primitive permutation groups.

Findings

Complete classification of proper partial linear spaces with rank 3 affine primitive automorphism groups.

Identification of exceptions involving small groups, including subgroups of AΓL_1(q).

Detailed analysis of rank 3 affine primitive permutation groups.

Abstract

A partial linear space is a pair $(\mathcal{P},\mathcal{L})$ where $\mathcal{P}$ is a non-empty set of points and $\mathcal{L}$ is a collection of subsets of $\mathcal{P}$ called lines such that any two distinct points are contained in at most one line, and every line contains at least two points. A partial linear space is proper when it is not a linear space or a graph. A group of automorphisms $G$ of a proper partial linear space acts transitively on ordered pairs of distinct collinear points and ordered pairs of distinct non-collinear points precisely when $G$ is transitive of rank 3 on points. In this paper, we classify the finite proper partial linear spaces that admit rank 3 affine primitive automorphism groups, except for certain families of small groups, including subgroups of $A\Gamma L_1(q)$. Up to these exceptions, this completes the classification of the finite proper partial linear spaces admitting rank 3 primitive automorphism groups. We also provide a more detailed version of the classification of the rank 3 affine primitive permutation groups, which may be of independent interest.