RWPRI and (2T)1 Flag-transitive Linear Spaces

Abstract

Abstract. The classification of finite flag-transitive linear spaces is almost complete. For the thick case, this result was announced by Buekenhout, Delandtsheer, Doyen, Kleidman, Liebeck and Saxl, and in the thin case (where the lines have 2 points), it amounts to the classification of 2-transitive groups, which is generally considered to follow from the classification of finite simple groups. These two classifications actually leave an open case, which is the so-called 1-dimensional case. In this paper, we work with two additional assumptions. These two conditions, namely (2T)1 and RWPri, are taken from another field of study in Incidence Geometry and allow us to obtain a complete classification, which we present at the end of this paper. In particular, for the 1-dimensional case, we show that the only (2T)1 flag-transitive linear spaces are AG(2, 2) and AG(2, 4), with AΓL(1, 4) and AΓL(1, 16) as respective automorphism groups. 1