\(PSL(3,q)\) and line-transitive linear spaces (Q2381812)

This paper is part of an ongoing program to classify all line-transitive automorphism groups of finite linear spaces. Let \(\mathcal S\) be a linear space which admits a line-transitive automorphism group \(G\) with \(\mathrm{PSL}(3,q) \trianglelefteq G \leq \mathrm{Aut}\mathrm{PSL}(3,q)\). Then either \(\mathcal S\) is isomorphic to the desarguesian projective plane of order \(q\) and \(G\) acts 2-transitively on the points of \(\mathcal S\), or \(\mathrm{PSL}(3,q)\) acts transitively on the points but not on the lines of \(\mathcal S\). Moreover, for each point of \(\mathcal S\) the intersection of the point-stabilizer with \(\mathrm{PSL}(3,q)\) is isomorphic to \(\mathrm{PSL}(3,q_ 0)\), where \(q = q_ 0^ a\) for some integer \(a\). No example for the second possibility is known and it is very unlikely to occur.