Classification of point-primitive linear spaces with $2pq$ points

Abstract

This paper is a further contribution to the classification of point-primitive finite linear spaces. Let $ p,q$ be two primes. We prove that if $\mathcal{S}$ is a non-trivial finite linear space with $2pq$ points, and $G\leq Aut(\mathcal{S})$ is point-primitive, then $G $ is line-transitive and $\mathcal{S}$ is the Ree unital $U_R(3), $ or the Hermitian unital $U_H(s). $