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Zap. Nauchn. Sem. POMI, 2009, Volume 365, Pages 5–28 (Mi znsl3463)  

This article is cited in 4 scientific papers (total in 4 papers)

Overgroups of $E(m,R)\otimes E(n,R)$

A. S. Ananievskiy, N. A. Vavilov, S. S. Sinchuk

St.-Petersburg State University

Abstract: In the present paper we study subgroups $E(m,R)\otimes E(n,R)\le H\le G=\operatorname{GL}(mn,R)$, under assumption that the ring $R$ is commutative, and $m,n\ge3$. We define the group $\operatorname{GL}_m\otimes\operatorname{GL}_n$ by equations, calculate the normaliser of the group $E(m,R)\otimes E(n,R)$ and associate to each intermediate subgroup $H$ a uniquely determined lower level $(A,B,C)$, where $A,B,C$ are ideals in $R$ such that $mA,A^2\le B\le A$ and $nA,A^2\le C\le A$. Lower level specifies the largest elementary subgroup such that $E(m,n,R,A,B,C)\le H$. The standard answer to this problem asserts that $H$ is contained in the normaliser $N_G(E(m,n,R,A,B,C))$. Bibl. – 46 titles.

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English version:
Journal of Mathematical Sciences (New York), 2009, 161:4, 461–473

UDC: 512.5+512.6+512.7+512.8
Received: 10.06.2000

Citation: A. S. Ananievskiy, N. A. Vavilov, S. S. Sinchuk, “Overgroups of $E(m,R)\otimes E(n,R)$”, Problems in the theory of representations of algebras and groups. Part 18, Zap. Nauchn. Sem. POMI, 365, POMI, St. Petersburg, 2009, 5–28; J. Math. Sci. (N. Y.), 161:4 (2009), 461–473

Citation in format AMSBIB
\Bibitem{AnaVavSin09}
\by A.~S.~Ananievskiy, N.~A.~Vavilov, S.~S.~Sinchuk
\paper Overgroups of $E(m,R)\otimes E(n,R)$
\inbook Problems in the theory of representations of algebras and groups. Part~18
\serial Zap. Nauchn. Sem. POMI
\yr 2009
\vol 365
\pages 5--28
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl3463}
\elib{http://elibrary.ru/item.asp?id=15303447}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2009
\vol 161
\issue 4
\pages 461--473
\crossref{https://doi.org/10.1007/s10958-009-9576-y}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-70349587827}


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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. N. A. Vavilov, V. G. Kazakevich, “More variations on decomposition of transvections”, J. Math. Sci. (N. Y.), 171:3 (2010), 322–330  mathnet  crossref
    2. A. S. Ananevskii, N. A. Vavilov, S. S. Sinchuk, “O nadgruppakh $E(m,R)\otimes E(n,R)$. I. Urovni i normalizatory”, Algebra i analiz, 23:5 (2011), 55–98  mathnet  mathscinet  elib; A. S. Ananyevskiy, N. A. Vavilov, S. S. Sinchuk, “Overgroups of $E(m,R)\otimes E(n,R)$. I”, St. Petersburg Math. J., 23:5 (2012), 819–849  crossref  isi  elib
    3. Bakulin S.V., Vavilov N.A., “O podgruppakh, normalizuemykh $EO(2L,R)$*”, Vestnik Sankt-Peterburgskogo universiteta. Seriya 1: Matematika. Mekhanika. Astronomiya, 2011, no. 4, 19–27  mathscinet  zmath  elib
    4. N. A. Vavilov, A. V. Schegolev, “Nadgruppy subsystem subgroups v isklyuchitelnykh gruppakh: urovni”, Voprosy teorii predstavlenii algebr i grupp. 23, Zap. nauchn. sem. POMI, 400, POMI, SPb., 2012, 70–126  mathnet  mathscinet; N. A. Vavilov, A. V. Shchegolev, “Overgroups of subsystem subgroups in exceptional groups: levels”, J. Math. Sci. (N. Y.), 192:2 (2013), 164–195  crossref
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