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Algebra i Analiz, 2007, Volume 19, Issue 2, Pages 10–51 (Mi aa111)  

This article is cited in 15 scientific papers (total in 15 papers)

Research Papers

Overgroups of $\mathrm{EO}(n,R)$

N. A. Vavilov, V. A. Petrov

Saint-Petersburg State University

Abstract: Let $R$ be a commutative ring with 1, $n$ a natural number, and let $l=[n/2]$. Suppose that $2\in R^*$ and $l\ge 3$. We describe the subgroups of the general linear group $\operatorname{GL}(n,R)$ that contain the elementary orthogonal group $\operatorname{EO}(n,R)$. The main result of the paper says that, for every intermediate subgroup $H$, there exists a largest ideal $A\trianglelefteq R$ such that $\operatorname{EEO}(n,R,A)=\operatorname{EO}(n,R)E(n,R,A)\trianglelefteq H$. Another important result is an explicit calculation of the normalizer of the group $\operatorname{EEO}(n,R,A)$. If $R=K$ is a field, similar results were obtained earlier by Dye, King, Shang Zhi Li, and Bashkirov. For overgroups of the even split elementary orthogonal group $\operatorname{EO}(2l,R)$ and the elementary symplectic group $\operatorname{Ep}(2l,R)$, analogous results appeared in previous papers by the authors (Zapiski Nauchn. Semin. POMI, 2000, v. 272; Algebra i Analiz, 2003, v. 15, no. 3).

Keywords: General linear group, overgroup, split elementary orthogonal group.

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English version:
St. Petersburg Mathematical Journal, 2008, 19:2, 167–195

Bibliographic databases:

MSC: 20G35
Received: 20.11.2006

Citation: N. A. Vavilov, V. A. Petrov, “Overgroups of $\mathrm{EO}(n,R)$”, Algebra i Analiz, 19:2 (2007), 10–51; St. Petersburg Math. J., 19:2 (2008), 167–195

Citation in format AMSBIB
\by N.~A.~Vavilov, V.~A.~Petrov
\paper Overgroups of $\mathrm{EO}(n,R)$
\jour Algebra i Analiz
\yr 2007
\vol 19
\issue 2
\pages 10--51
\jour St. Petersburg Math. J.
\yr 2008
\vol 19
\issue 2
\pages 167--195

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    This publication is cited in the following articles:
    1. A. Yu. Luzgarev, “O nadgruppakh $\mathrm{E}(\mathrm{E}_6,R)$ i $\mathrm{E}(\mathrm{E}_7,R)$ v minimalnykh predstavleniyakh”, Voprosy teorii predstavlenii algebr i grupp. 11, Zap. nauchn. sem. POMI, 319, POMI, SPb., 2004, 216–243  mathnet  mathscinet  zmath; A. Yu. Luzgarev, “On overgroups of $\mathrm{E}(\mathrm{E}_6,R)$ and $\mathrm{E}(\mathrm{E}_7,R)$ in their minimal representations”, J. Math. Sci. (N. Y.), 134:6 (2006), 2558–2571  crossref
    2. N. A. Vavilov, A. Yu. Luzgarev, “Normalizator gruppy Shevalle tipa $\mathrm{E}_6$”, Algebra i analiz, 19:5 (2007), 37–64  mathnet  mathscinet  zmath; N. A. Vavilov, A. Yu. Luzgarev, “The normalizer of Chevalley groups of type $\mathrm{E}_6$”, St. Petersburg Math. J., 19:5 (2008), 699–718  crossref  isi
    3. N. A. Vavilov, “O podgruppakh simplekticheskoi gruppy, soderzhaschikh subsystem subgroup”, Voprosy teorii predstavlenii algebr i grupp. 16, Zap. nauchn. sem. POMI, 349, POMI, SPb., 2007, 5–29  mathnet  mathscinet  elib; N. A. Vavilov, “On subgroups of symplectic group containing a subsystem subgroup”, J. Math. Sci. (N. Y.), 151:3 (2008), 2937–2948  crossref  elib
    4. A. Yu. Luzgarëv, “Opisanie nadgrupp $\mathrm F_4$ v $\mathrm E_6$ nad kommutativnym koltsom”, Algebra i analiz, 20:6 (2008), 148–185  mathnet  mathscinet  zmath; A. Yu. Luzgarev, “Overgroups of $\mathrm{F}_4$ in $\mathrm{E}_6$ over commutative rings”, St. Petersburg Math. J., 20:6 (2009), 955–981  crossref  isi
    5. Vavilov N.A., Stepanov A.V., “Nadgruppy poluprostykh grupp”, Vestn. Samarskogo gos. un-ta. Estestvennonauchn. ser., 2008, no. 3, 51–95  mathscinet  zmath
    6. A. S. Ananievskiy, N. A. Vavilov, S. S. Sinchuk, “Overgroups of $E(m,R)\otimes E(n,R)$”, J. Math. Sci. (N. Y.), 161:4 (2009), 461–473  mathnet  crossref  elib
    7. Hazrat R., Vavilov N., “Bak's work on the K-theory of rings”, J. K-Theory, 4:1 (2009), 1–65  crossref  mathscinet  zmath  isi  elib
    8. N. A. Vavilov, V. G. Kazakevich, “More variations on decomposition of transvections”, J. Math. Sci. (N. Y.), 171:3 (2010), 322–330  mathnet  crossref
    9. 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
    10. Bakulin S.V., Vavilov N.A., “O podgruppakh, normalizuemykh $\mathrm{EO}(2L,R)$”, Vestn. Sankt-Peterburgskogo un-ta. Ser. 1. Matem. Mekh. Astronom., 2011, no. 4, 19–27  mathscinet  zmath  elib
    11. Stepanov A., “Subring subgroups in Chevalley groups with doubly laced root systems”, J. Algebra, 362 (2012), 12–29  crossref  mathscinet  zmath  isi  elib
    12. 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
    13. Hazrat R. Vavilov N. Zhang Z., “Relative Commutator Calculus in Chevalley Groups”, J. Algebra, 385 (2013), 262–293  crossref  mathscinet  zmath  isi  elib
    14. N. A. Vavilov, A. Yu. Luzgarev, “Normalizator gruppy Shevalle tipa $\mathrm E_7$”, Algebra i analiz, 27:6 (2015), 57–88  mathnet  mathscinet  elib; N. A. Vavilov, A. Yu. Luzgarev, “Normaliser of the Chevalley group of type $\mathrm E_7$”, St. Petersburg Math. J., 27:6 (2016), 899–921  crossref  isi
    15. Fasel J., “On the number of generators of ideals in polynomial rings”, Ann. Math., 184:1 (2016), 315–331  crossref  mathscinet  zmath  isi  elib  scopus
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