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 Zap. Nauchn. Sem. LOMI, 1979, Volume 94, Pages 13–20 (Mi znsl1800)

Subgroups of the full linear group over a Dedekind ring

Z. I. Borevich, N. A. Vavilov, V. Narkevich

Abstract: We study the subgroups of the full linear group $GL(n,R)$ over a Dedekind ring $R$ that contain the group of quasidiagonal matrices of fixed type with diagonal blocks of at least third order, each of which is generated by elementary matrices. For any such subgroup $H$ there exists a unique $D$-net $\sigma$of ideals of $R$ such that, where $E(\sigma)$ is the subgroup generated by all transvections of the net subgroup $G(\sigma)$. and is the normalizer of $G(\sigma)$. The subgroup $E(\sigma)$ is normal in. To study the factor group we introduce an intermediate subgroup $F(\sigma)$, $E(\sigma)\leqslant F(\sigma)\leqslant G(\sigma)$. The group is finite and is connected with permutations in the symmetric group. The factor group $G(\sigma)/F(\sigma)$ is Abelian – these are the values of a certain “determinant”. In the calculation of $F(\sigma)/E(\sigma)$ appears the $SK_1$-functor. Results are stated without proof.

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English version:
Journal of Soviet Mathematics, 1982, 19:1, 982–987

Bibliographic databases:

UDC: 519.46

Citation: Z. I. Borevich, N. A. Vavilov, V. Narkevich, “Subgroups of the full linear group over a Dedekind ring”, Rings and modules. Part 2, Zap. Nauchn. Sem. LOMI, 94, "Nauka", Leningrad. Otdel., Leningrad, 1979, 13–20; J. Soviet Math., 19:1 (1982), 982–987

Citation in format AMSBIB
\Bibitem{BorVavNar79} \by Z.~I.~Borevich, N.~A.~Vavilov, V.~Narkevich \paper Subgroups of the full linear group over a Dedekind ring \inbook Rings and modules. Part~2 \serial Zap. Nauchn. Sem. LOMI \yr 1979 \vol 94 \pages 13--20 \publ "Nauka", Leningrad. Otdel. \publaddr Leningrad \mathnet{http://mi.mathnet.ru/znsl1800} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=571511} \zmath{https://zbmath.org/?q=an:0445.20028} \transl \jour J. Soviet Math. \yr 1982 \vol 19 \issue 1 \pages 982--987 \crossref{https://doi.org/10.1007/BF01476109} 

• http://mi.mathnet.ru/eng/znsl1800
• http://mi.mathnet.ru/eng/znsl/v94/p13

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. N. A. Vavilov, “On subgroups of symplectic group containing a subsystem subgroup”, J. Math. Sci. (N. Y.), 151:3 (2008), 2937–2948
2. A. V. Alexandrov, N. A. Vavilov, “Parabolic subgroups of $\mathrm{SL}_n$ and $\mathrm{Sp}_{2l}$ over a Dedekind ring of arithmetic type”, J. Math. Sci. (N. Y.), 171:3 (2010), 307–316
3. K. O. Batalkin, N. A. Vavilov, “Parabolic subgroups of $\mathrm{SO}_{2l}$ over a Dedekind ring of arithmetic type”, J. Math. Sci. (N. Y.), 192:2 (2013), 154–163
4. 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