The geometry of k-transvection groups

Let k be a (commutative) field and G a group, then a conjugacy class of Abelian subgroups of G is called a class of k-transvection subgroups in G if and only if it generates G and any two elements of the class either commute or are full unipotent subgroups of the group they generate and which is isomorphic to (P)SL2(k). In this paper we study the geometry of k-transvection groups. Given a class of k-transvection groups S, we consider a partial linear space whose points are the elements of S, and whose lines correspond to the groups generated by two noncommuting elements from S. We derive several properties of this partial linear space. These properties are used to give a characterization of the geometries of k-transvection groups and provide a classification of groups generated by k-transvection subgroups.