M ar 2 00 6 PSL ( 3 , q ) and line - transitive linear spaces Nick

We present a partial classification of those finite linear spaces S on which an almost simple group G with socle PSL(3, q) acts line-transitively. A linear space S is an incidence structure consisting of a set of points Π and a set of lines Λ in the power set of Π such that any two points are incident with exactly one line. The linear space is called non-trivial if every line contains at least three points and there are at least two lines. Write v = |Λ| and b = |Π|. The investigation of those finite linear spaces which admit an almost simple group that is transitive upon lines is already underway [4, 7] motivated largely by the theorem of Cameron and Praeger [5]. We continue this investigation by considering the situation when the socle of a line-transitive automorphism group is PSL(3, q). The statement of our theorem is as follows: Theorem A. Suppose that PSL(3, q) G ≤ AutPSL(3, q) and that G acts line-transitively on a finite linear space S. Then one of the following holds: • S = PG(2, q), the Desarguesian projective plane, and G acts 2-transitively on points; • PSL(3, q) is point-transitive but not line-transitive on S. Furthermore, if Gα is a point-stabilizer in G then Gα ∩ PSL(3, q) ∼= PSL(3, q0) where q = q 0 for some integer a. The proof of Theorem A will depend heavily upon an unpublished result of Camina, Neumann and Praeger which classifies the line-transitive actions of PSL(2, q) (a weaker version of this result has appeared in the literature, see [16]): Theorem 1. Let G = PSL(2, q), q ≥ 4 and suppose that G acts line-transitively on a linear space S. Then one of the following holds: • G = PSL(2, 2), a ≥ 3 acting transitively on S, a Witt-Bose-Shrikhande space. Here Π is the set of dihedral subgroups of G of order 2(q + 1) and Λ is the set of involutions t ∈ G with the incidence relation being inclusion. ∗This paper contains results from the author’s PhD thesis. I would like to thank my supervisor, Professor Jan Saxl.