Gill, Nick \(PSL(3,q)\) and line-transitive linear spaces. (English) Zbl 1169.51001 Beitr. Algebra Geom. 48, No. 2, 591-620 (2007). 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. Reviewer: Norbert Knarr (Gießen) Cited in 7 Documents MSC: 51A10 Homomorphism, automorphism and dualities in linear incidence geometry 51A05 General theory of linear incidence geometry and projective geometries 05B05 Combinatorial aspects of block designs 20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures Keywords:linear space; line-transitive automorphism group PDFBibTeX XMLCite \textit{N. Gill}, Beitr. Algebra Geom. 48, No. 2, 591--620 (2007; Zbl 1169.51001) Full Text: arXiv EuDML EMIS