[Paper Review] Searching for line transitive, point imprimitive, linear spaces

Gregory Cresp|ArXiv.org|Apr 25, 2006

Advanced Graph Theory Research3 references3 citations

TL;DR

This paper investigates the existence of line transitive, point imprimitive linear spaces using computational search methods. It develops a GAP and C-based algorithm to construct such spaces from group actions on 451 points, applying the Orbit Theorem and Delandtsheer-Doyen constraints to narrow the search space. Despite exhaustive testing, no new examples were found, but the method provides a robust framework for future searches in design theory.

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.

Motivation & Objective

• To determine whether point imprimitive, line transitive linear spaces other than projective planes are rare or under-identified due to search bias.
• To develop a systematic computational method for constructing such linear spaces from group actions on a point set.
• To test the feasibility of the search algorithm on two specific automorphism groups acting on 451 points, as identified by theoretical work of Praeger and Tuan.
• To eliminate certain group actions from further consideration by proving the non-existence of solutions under the given constraints.

Proposed method

• The search employs the Orbit Theorem to generate candidate designs from group actions, ensuring that block orbits satisfy symmetry and incidence conditions.
• A hybrid algorithm written in GAP and C systematically explores possible block systems by recursively building blocks while tracking used orbits and row coverage.
• Constraints are enforced via data structures tracking covered orbits and rows, with early pruning to avoid invalid configurations.
• The Delandtsheer-Doyen Theorem is applied to restrict possible parameter sets by analyzing invariant partitions on the point set.
• The algorithm checks two key conditions: the intercept vector condition (relating to orbit structure) and the orbit condition (ensuring consistent block incidence across orbits).
• The method is validated on a known example to confirm correctness before applying it to the two target groups on 451 points.

Experimental results

Research questions

• RQ1 Are there any line transitive, point imprimitive linear spaces with 451 points beyond the known projective plane examples?
• RQ2 Can the proposed algorithm successfully reconstruct a known example of such a space, confirming its validity?
• RQ3 Do the two specific automorphism groups on 451 points admit any line transitive, point imprimitive linear spaces?
• RQ4 What constraints from the Delandtsheer-Doyen Theorem and orbit structure effectively eliminate potential solutions?
• RQ5 Is the computational approach scalable and efficient enough to explore larger parameter sets in design theory?

Key findings

• No line transitive, point imprimitive linear spaces were found for either of the two automorphism groups acting on 451 points.
• The search algorithm successfully validated on a known example, confirming its correctness and reliability.
• The implementation of orbit and row tracking data structures enabled efficient pruning of invalid search paths.
• The application of the Delandtsheer-Doyen Theorem significantly reduced the search space by eliminating non-viable parameter sets.
• The algorithm is extendible to other groups and parameter sets, providing a reusable framework for future searches in design theory.
• The absence of solutions for the tested groups suggests these actions may not support such linear spaces, narrowing future search directions.