Subgroups of the full linear group over a~semilocal ring

Z. I. Borevich ; N. A. Vavilov

Zapiski Nauchnykh Seminarov POMI, Rings and linear groups, Tome 75 (1978), pp. 32-34

Résumé

Let $\Lambda$ be a semilocal ring (a factor ring with respect to the Jacobson–Artin radical) for which the residue field $C/m$ of its center $C$ with respect to each maximal ideal $m\subset C$ contains no fewer than seven elements. The structure of subgroups $H$ in the full linear group $\mathrm{GL}(n,\Lambda)$ containing the group of diagonal matrices is considered. The main theorem: for any subgroup $H$ there is a uniquely determined $D$-net of ideals $\sigma$ such that $G(\sigma)\le H\le N(\sigma)$, where $N(\sigma)$ is the normalizer of the $D$-net subgroup $G(\sigma)$. A transparent classification of subgroups $\mathrm{GL}(n,\Lambda)$ normalizable by diagonal matrices is thus obtained. Further, the factor group $N(\sigma)/G(\sigma)$ is studied. Bibl. 4 titles.