Searching for line transitive, point imprimitive, linear spaces

TL;DR

This paper explores the rare existence and construction methods of line transitive, point imprimitive linear spaces with specific automorphism groups, using algorithms implemented in GAP and C.

Contribution

It develops algorithms to construct such linear spaces with given automorphism groups, extending the understanding of their rarity and properties.

Findings

Constructed examples over 451 points using new algorithms.

Demonstrated methods to generate line transitive, point imprimitive spaces.

Extended algorithm applicability to broader group classes.

Abstract

A finite linear space is a finite set of points and lines, where any two points lie on a unique line. Well known examples include projective planes. This project focuses on linear spaces which admit certain types of symmetries. Symmetries of the space which preserve the line structure are called automorphisms. A group of these is called an automorphism group of the linear space. Two interesting properties of linear spaces are point imprimitivity and line transitivity. Point imprimitive spaces admit a second structure on the points aside from the lines, which is also preserved by an automorphism group. In line transitive spaces, given any two lines, an automorphism can be found that maps one line to the other. Very few point imprimitive, line transitive linear spaces, apart from projective planes, are known. Such spaces that have been found have been surprising. One point of interest is whether such spaces are rare and the known ones are in some sense exceptional, or if there are many such spaces, but mathematicians have been looking in the wrong places. Here we investigate methods to construct a line transitive, point imprimitive linear space over a given point set and automorphism group. We employ these methods on two given automorphism groups, both on a set of 451 points. This was an exceptional situation identified in theoretical work of Praeger and Tuan. Included in this is the development of an algorithm, written in GAP, an algebraic programming system, and C, to perform these constructions. This algorithm is extendible to a wider class of groups.