Extremely primitive groups and linear spaces

summary:A non-regular primitive permutation group is called extremely primitive if a point stabilizer acts primitively on each of its nontrivial orbits. Let $\mathcal S$ be a nontrivial finite regular linear space and $G\leq {\rm Aut}(\mathcal S).$ Suppose that $G$ is extremely primitive on points and let rank$(G)$ be the rank of $G$ on points. We prove that rank$(G)\geq 4$ with few exceptions. Moreover, we show that ${\rm Soc}(G)$ is neither a sporadic group nor an alternating group, and $G={\rm PSL}(2,q)$ with $q+1$ a Fermat prime if ${\rm Soc}(G)$ is a finite classical simple group