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

N. A. Vavilov

Zapiski Nauchnykh Seminarov POMI, Algebraic numbers and finite groups, Tome 86 (1979), pp. 30-33

Résumé

It has been proved (Ref. Zh. Mat., 1978, 9A237) that for a semilocal ring $\Lambda$ 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 $GL(n,\Lambda)$ that contain the group of diagonal matrices: for each such subgroup $H$ there is a uniquely defined $D$-net of ideals $\sigma$ (Ref. Zh. Mat., 1977, 2A288) such that $G(\sigma)\leqslant h\leqslant N(\sigma)$, ,where $N(\sigma)$ is the normalizer of the $\sigma$-net subgroup $G(\sigma)$. It is noted that this result is also true under the following weaker assumption: a decomposition of a quotient ring of the ring $\Lambda$ 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.