Large dimensional classical groups and linear spaces

Abstract

Suppose that a group $G$ has socle $L$ a simple large-rank classical group. Suppose furthermore that $G$ acts transitively on the set of lines of a linear space $\mathcal{S}$. We prove that, provided $L$ has dimension at least $25$, then $G$ acts transitively on the set of flags of $\mathcal{S}$ and hence the action is known. For particular families of classical groups our results hold for dimension smaller than $25$. The group theoretic methods used to prove the result (described in Section 3) are robust and general and are likely to have wider application in the study of almost simple groups acting on finite linear spaces.