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

For an arbitrary division algebra T we study the arrangement of subgroups of the special linear group Γ = SLn(T)(n ≥ 3) that contain the subgroup Δ = SDn(T) of diagonal matrices with Dieudonneʹs determinant (see [1]) equal to 1. We show that the description of these subgroups is standard in the following sense: For any subgroup H, Δ ≤ H ≤ Γ there exists a unique D-net σ such that Γ(σ) ≤ H ≤ NΓ(σ), where Γ(σ) is the D-net subgroup corresponding to the net σ and NΓ(σ) is the normalizer of Γ(σ) in Γ.