On subgroups in the special linear group over a division algebra that contain the subgroup of diagonal matrices

For an arbitrary division algebra T we study the arrangement of subgroups of the special linear group Gamma = SLn(T)(n greater than or equal to 3) that contain the subgroup Delta = SDn(T) of diagonal matrices with Dieudonne's determinant (see [1]) equal to 1. We show that the description of these subgroups is standard in the following sense: For any subgroup H, Delta less than or equal to H less than or equal to Gamma there exists a unique D-net sigma such that Gamma(sigma) less than or equal to> H less than or equal to N-Gamma(sigma), where Gamma(sigma) is the D-net subgroup corresponding to the net sigma and N-Gamma(sigma) is the normalizer of Gamma(sigma) in Gamma. (C) 1997 Elsevier Science B.V.