Almost simple groups with socle Ree(q) acting on finite linear spaces

Abstract

This article is part of a project set up to classify groups and linear spaces where the group acts transitively on the lines of the space. Let G be an automorphism group of a linear space. We know that the study of line-transitive finite linear spaces can be reduced to three cases, distinguishable by means of properties of the action on the point-set: that in which G is of affine type in the sense that it has an elementary abelian transitive normal subgroup; that in which G has an intransitive minimal normal subgroup; and that in which G is almost simple, in the sense that there is a simple transitive normal subgroup T in G whose centraliser is trivial, so that T ⊴ G ≤ Aut ( T ) . In this paper we treat almost simple groups in which T is a Ree group and obtain the following theorem: Let T ⊴ G ≤ Aut ( T ) , and let S be a finite linear space on which G acts as a line-transitive automorphism group. If T is isomorphic to G 2 2 ( q ) , then T is line-transitive and S is a Ree unitary space.