Proper partial linear spaces affording imprimitive rank 3 automorphism groups

TL;DR

This paper classifies a new family of imprimitive rank 3 automorphism groups acting on proper partial linear spaces, expanding understanding beyond the previously classified primitive cases.

Contribution

It provides the first substantial classification of imprimitive rank 3 automorphism groups on proper partial linear spaces, including new infinite families and specific examples.

Findings

Classified all imprimitive rank 3 proper partial linear spaces with semiprimitive automorphism groups.

Constructed several infinite families of examples with linear or unitary group actions.

Identified new examples not previously documented in literature.

Abstract

A partial linear space is a point--line incidence structure such that each line is incident with at least two points and each pair of points is incident with at most one line. It is said to be proper if there exists at least one non-collinear point pair, and at least one line incident with more than two points. The highest degree of symmetry for a proper partial linear space occurs when the automorphism group $G$ is transitive on ordered pairs of collinear points, and on ordered pairs of non-collinear points, that is to say, $G$ is a transitive rank $3$ group on the points. While the primitive rank 3 partial linear spaces are essentially classified, we present the first substantial classification of a family of imprimitive rank $3$ examples. We classify all imprimitive rank $3$ proper partial linear spaces such that the rank $3$ group is semiprimitive. In particular, this includes all partial linear spaces with a rank 3 imprimitive automorphism group that is innately transitive or quasiprimitive. We construct several infinite families of examples and ten individual examples. The examples in the infinite families admit a rank $3$ action of a linear or unitary group, and to our knowledge most of these examples have not appeared before in the literature.