Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 44 (2003), No. 1, 25-46.

RWPRI and (2T)_{1} Flag-transitive inear Spaces F. Buekenhout P.-O. Dehaye^{∗} D. Leemans^{†} Département de Mathématique, Université Libre de Bruxelles C.P. 216 - Géométrie, Boulevard du Triomphe, B-1050 Bruxelles, Belgium Abstract. The classification of finite Ωag-transitive linear spaces is almost com- lete. For the thick case, this result was announced by Buekenhout, Delandtsheer, oyen, Kleidman, Liebeck and Saxl, and in the thin case (where the lines have 2 oints), it amounts to the classification of 2-transitive groups, which is generally onsidered to follow from the classification of finite simple groups. These two clas- ifications actually leave an open case, which is the so-called 1-dimensional case. n this paper, we work with two additional assumptions. These two conditions, amely (2T)_{1} and RWPri, are taken from another field of study in Incidence Ge- metry and allow us to obtain a complete classification, which we present at the nd of this paper. In particular, for the 1-dimensional case, we show that the only 2T)_{1} Ωag-transitive linear spaces are AG(2, 2) and AG(2, 4), with AΓL(1, 4) and AΓL(1, 16) as respective automorphism groups. 1. Introduction Flag-transitive linear spaces^{1} are very common objects appearing in the theory of Incidence Geometry [5]. A major goal of Incidence Geometry is to generalize J. Tits’ theory of Buildings, which was itself developed in order to acquire a better understanding of the simple groups of Lie type.

0138-4821/93 $ 2.50 \begin{matrix}c \\ ⊗ \end{matrix} 2003 Heldermann Verlag

∗ The address of this author is now: Department of Mathematics, Building 380, Stanford University, 94305-2125 Stanford, CA, U.S.A.

† e-mail: dleemans@ulb.ac.be

1 See Section 2 for the definitions.