On the geometry of k-transvection groups

More than 20 years ago, B. Fischer [5,6] started his pioneering work on groups generated by a conjugacy class of 3-transpositions, i.e., a conjugacy class of involutions D, such that for all d, e ∈ D, we have that [d, e] = 1 or 〈d, e〉 ' SL2(2). Fischer classified all finite groups G generated by a class of 3-transpositions under the additional assumptions that G′′ = G′ and O2(G) = O3(G) = Z(G) = 1. Fischer’s work has been generalized in two directions. On one hand, J.I. Hall and the author [4] have obtained an almost complete classification of all centerfree 3-transposition groups allowing infinite groups and nontrivial normal 2-or 3-groups. See also [8,9]. On the other hand, Aschbacher [1] and more recently Timmesfeld [7] have extended Fischer’s work to groups generated by a class of k-transvection groups, for arbitrary fields k, where k-transvection groups are defined as follows. Suppose k is a field, G a group and Σ a conjugacy class of abelian subgroups of G such that