Line-transitive Automorphism Groups of Linear Spaces

Abstract

In this paper we prove the following theorem.

Let $\cal S$ be a linear space. Assume that $\cal S$ has an automorphism group $G$ which is line-transitive and point-imprimitive with $k < 9$. Then $\cal S$ is one of the following:-
(a) A projective plane of order $4$ or $7$,
(b) One of $2$ linear spaces with $v=91$ and $k=6$,
(c) One of $467$ linear spaces with $v=729$ and $k=8$.

In all cases the full automorphism group Aut(${\cal S} \!$) is known.