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 Zap. Nauchn. Sem. LOMI, 1979, Volume 86, Pages 30–33 (Mi znsl1819)

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

N. A. Vavilov

Abstract: 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.

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English version:
Journal of Soviet Mathematics, 1981, 17:4, 1960–1963

Bibliographic databases:

UDC: 519.46

Citation: N. A. Vavilov, “Description of subgroups of the full linear group over a semilocal ring that contain the group of diagonal matrices”, Algebraic numbers and finite groups, Zap. Nauchn. Sem. LOMI, 86, "Nauka", Leningrad. Otdel., Leningrad, 1979, 30–33; J. Soviet Math., 17:4 (1981), 1960–1963

Citation in format AMSBIB
\Bibitem{Vav79} \by N.~A.~Vavilov \paper Description of subgroups of the full linear group over a semilocal ring that contain the group of diagonal matrices \inbook Algebraic numbers and finite groups \serial Zap. Nauchn. Sem. LOMI \yr 1979 \vol 86 \pages 30--33 \publ "Nauka", Leningrad. Otdel. \publaddr Leningrad \mathnet{http://mi.mathnet.ru/znsl1819} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=535477} \zmath{https://zbmath.org/?q=an:0462.20042|0418.20042} \transl \jour J. Soviet Math. \yr 1981 \vol 17 \issue 4 \pages 1960--1963 \crossref{https://doi.org/10.1007/BF01465452} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. A. A. Panin, A. V. Yakovlev, “The Galois theory for a class of modular lattices”, J. Math. Sci. (New York), 95:2 (1999), 2126–2135
2. A. V. Shchegolev, “Overgroups of elementary block-diagonal subgroups in hyperbolic unitary groups over quasi-finite rings: main results”, J. Math. Sci. (N. Y.), 222:4 (2017), 516–523