Linear spaces with significant characteristic prime

Abstract

Let $G$ be a group with socle a simple group of Lie type defined over the finite field with $q$ elements where $q$ is a power of the prime $p$. Suppose that $G$ acts transitively upon the lines of a linear space $\mathcal{S}$. We show that if $p$ is {\it significant} then $G$ acts flag-transitively on $\mathcal{S}$ and all examples are known.