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

This article is cited in 16 scientific papers (total in 16 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, “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  mathnet  crossref  mathscinet  zmath
    2. 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  mathnet  crossref  mathscinet  zmath  isi
    3. 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  elib  elib
    4. A. Yu. Luzgarev, “Overgroups of $\mathrm{F}_4$ in $\mathrm{E}_6$ over commutative rings”, St. Petersburg Math. J., 20:6 (2009), 955–981  mathnet  crossref  mathscinet  zmath  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. 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
    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. Shchegolev, “Overgroups of subsystem subgroups in exceptional groups: levels”, J. Math. Sci. (N. Y.), 192:2 (2013), 164–195  mathnet  crossref  mathscinet
    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, “Normaliser of the Chevalley group of type $\mathrm E_7$”, St. Petersburg Math. J., 27:6 (2016), 899–921  mathnet  crossref  mathscinet  isi  elib
    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
    16. R. A. Lubkov, I. I. Nekrasov, “Yavnye uravneniya na vneshnii kvadrat polnoi lineinoi gruppy”, Voprosy teorii predstavlenii algebr i grupp. 33, Zap. nauchn. sem. POMI, 470, POMI, SPb., 2018, 120–137  mathnet
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