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

Abstract

For an arbitrary division algebra T we study the arrangement of subgroups of the special linear group Γ = SL n ( T)( n ≥ 3) that contain the subgroup Δ = SD n ( 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, Δ ≤ H ≤ Γ there exists a unique D-net σ such that Γ( σ) ≤ H ≤ N Γ ( σ), where Γ(σ) is the D-net subgroup corresponding to the net σ and N Γ ( σ) is the normalizer of Γ(σ) in Γ.