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 Zap. Nauchn. Sem. POMI: Year: Volume: Issue: Page: Find

 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 \publ "Nauka", Leningrad. Otdel. \publaddr Leningrad \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}