Dimensional linear spaces whose automorphism group is (line, hyperplane)-flag transitive

Abstract

We classify the pairs (S, G) where S is a finite n-dimensional linear space with n ≥ 4 and G is an automorphism group of S acting transitively on the (line, hyperplane)-flags. We show in particular that S must be either a Desarguesian projective or affine space provided with its subspaces of dimension ≤ n - 1, or a Mathieu-Witt design provided with its blocks and its subsets of size ≤ n - 1. Our proof uses a recent classification of the flag transitive 2-(v, k, 1) designs, which in turn heavily depends on the classification of all finite simple groups. The case n = 3 has been settled in another paper.