Imprimitive partial linear spaces and groups of rank 3

A partial linear space (PLS) 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. We say that a PLS is 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 PLS occurs when the automorphism group G is transitive on ordered pairs of collinear points, and on ordered pairs of non-collinear points. In this case, G is a transitive rank 3 group on the points. While the primitive rank 3 PLSs 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. We construct several infinite families of examples and ten individual examples. The examples admit a rank 3 action of a linear or unitary group, and to our knowledge most of our examples have not appeared before in the literature. This is a joint work with Alice Devillers and Cheryl Praeger.