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\(PSL(3,q)\) and line-transitive linear spaces. (English) Zbl 1169.51001

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.

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
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