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Zap. Nauchn. Sem. POMI, 2001, Volume 281, Pages 35–59 (Mi znsl1488)  

This article is cited in 8 scientific papers (total in 8 papers)

Subgroups of split orthogonal groups over a commutative ring

N. A. Vavilov

Saint-Petersburg State University

Abstract: We describe subgroups of the split orthogonal group $\Gamma=\mathrm{GO}(n,R)$ of degree $n$ over a commutative ring $R$ such that $2\in R^*$, which contain the elementary subgroup of a regularly embedded semi-simple group $F$. We show that if the ranks of all irreducible components of $F$ are at least 4, then the description of its overgroups is standard in the sense that for any intermediate subgroup $H$ there exists a unique orthogonal net of ideals such that $H$ lies between the corresponding net subgroup and its normalaser in $\Gamma$. A similar result for subgroups of the general linear group $\mathrm{GL}(n,R)$ with irreducible components of ranks at least 2 was obtained by Z. I. Borevich and the present author. We construct examples which show that if $F$ has irreducible components of ranks 2 or 3, then the standard description does not hold. The paper is based on the previous publications by the author, where similar results were obtained in some special cases, but the proof is based on a new computational trick.

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English version:
Journal of Mathematical Sciences (New York), 2004, 120:4, 1501–1512

Bibliographic databases:

UDC: 513.6
Received: 21.05.2001

Citation: N. A. Vavilov, “Subgroups of split orthogonal groups over a commutative ring”, Problems in the theory of representations of algebras and groups. Part 8, Zap. Nauchn. Sem. POMI, 281, POMI, St. Petersburg, 2001, 35–59; J. Math. Sci. (N. Y.), 120:4 (2004), 1501–1512

Citation in format AMSBIB
\by N.~A.~Vavilov
\paper Subgroups of split orthogonal groups over a~commutative ring
\inbook Problems in the theory of representations of algebras and groups. Part~8
\serial Zap. Nauchn. Sem. POMI
\yr 2001
\vol 281
\pages 35--59
\publ POMI
\publaddr St.~Petersburg
\jour J. Math. Sci. (N. Y.)
\yr 2004
\vol 120
\issue 4
\pages 1501--1512

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    This publication is cited in the following articles:
    1. N. A. Vavilov, V. A. Petrov, “On supergroups of $\mathrm{Ep}(2l,R)$”, St. Petersburg Math. J., 15:4 (2004), 515–543  mathnet  crossref  mathscinet  zmath  scopus
    2. N. A. Vavilov, M. R. Gavrilovich, “$\mathrm A_2$-proof of structure theorem for Chevaller groups of type $\mathrm E_6$ and $\mathrm E_7$”, St. Petersburg Math. J., 16:4 (2005), 649–672  mathnet  crossref  mathscinet  zmath  scopus
    3. N. A. Vavilov, V. A. Petrov, “Overgroups of $\mathrm{EO}(n,R)$”, St. Petersburg Math. J., 19:2 (2008), 167–195  mathnet  crossref  mathscinet  zmath  isi  elib  scopus
    4. N. A. Vavilov, “On subgroups of symplectic group containing a subsystem subgroup”, J. Math. Sci. (N. Y.), 151:3 (2008), 2937–2948  mathnet  crossref  mathscinet  zmath  elib  elib  scopus
    5. N. A. Vavilov, I. M. Pevzner, “Triples of long root subgroups”, J. Math. Sci. (N. Y.), 147:5 (2007), 7005–7020  mathnet  crossref  mathscinet  elib  elib  scopus
    6. 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  mathnet  crossref  mathscinet  isi  elib  elib  scopus
    7. N. A. Vavilov, A. V. Shchegolev, “Overgroups of subsystem subgroups in exceptional groups: levels”, J. Math. Sci. (N. Y.), 192:2 (2013), 164–195  mathnet  crossref  mathscinet  zmath  scopus
    8. 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  mathnet  crossref  mathscinet  zmath  scopus
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