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 Zap. Nauchn. Sem. LOMI, 1978, Volume 75, Pages 32–34 (Mi znsl3783)

Subgroups of the full linear group over a semilocal ring

Z. I. Borevich, N. A. Vavilov

Abstract: 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.

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English version:
Journal of Soviet Mathematics, 1987, 37:2, 935–937

Bibliographic databases:

Document Type: Article
UDC: 519.46

Citation: Z. I. Borevich, N. A. Vavilov, “Subgroups of the full linear group over a semilocal ring”, Rings and linear groups, Zap. Nauchn. Sem. LOMI, 75, "Nauka", Leningrad. Otdel., Leningrad, 1978, 32–34; J. Soviet Math., 37:2 (1987), 935–937

Citation in format AMSBIB
\Bibitem{BorVav78}
\by Z.~I.~Borevich, N.~A.~Vavilov
\paper Subgroups of the full linear group over a~semilocal ring
\inbook Rings and linear groups
\serial Zap. Nauchn. Sem. LOMI
\yr 1978
\vol 75
\pages 32--34
\mathnet{http://mi.mathnet.ru/znsl3783}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=0507043}
\zmath{https://zbmath.org/?q=an:0612.20027|0443.20040}
\transl
\jour J. Soviet Math.
\yr 1987
\vol 37
\issue 2
\pages 935--937
\crossref{https://doi.org/10.1007/BF01089084}