Line-transitive point-imprimitive linear spaces with Fang-Li parameter gcd(k,r) at most 10

TL;DR

This paper classifies certain finite linear spaces with specific automorphism properties, proving that under given conditions, the space must be the Desarguesian projective plane PG(2,9).

Contribution

It establishes a classification result for line-transitive, point-imprimitive linear spaces with Fang-Li parameter gcd(k,r) of 9 or 10, identifying the Desarguesian plane as the unique case.

Findings

If gcd(k,r) is 9 or 10, the linear space is PG(2,9).

Automorphism group G is line-transitive and point-imprimitive.

Provides a classification for these linear spaces.

Abstract

This paper is a further contribution to the classification of line-transitive finite linear spaces. We prove that if S is a non-trivial finite linear space with the Fang-Li parameter gcd(k,r) is 9 or 10, the automorphism group G of S is line-transitive and point-imprimitive, then S is the Desarguesian projective plane PG(2,9).