Description of subgroups of the full linear group over a semilocal ring that contain the group of diagonal matrices

Abstract

It has been proved (Ref. Zh. Mat., 1978, 9A237) that for a semilocal ring Λ in which each residue field of the center contains at least seven elements we have the following description of subgroups of the full linear group GL(n,Λ) that contain the group of diagonal matrices: for each such subgroup H there is a uniquely definedD -net of idealsS (Ref. Zh. Mat., 1977, 2A288) such that G(S)≤H≤N(S), ,whereN(S) is the normalizer of theS -net subgroup G(S). It is noted that this result is also true under the following weaker assumption: a decomposition of a quotient ring of the ring Λ into a direct sum of full matrix rings over skew fields does not contain skew fields with centers of less than seven elements or the ring of second-order matrices over the field of two elements.