N. A. Vavilov, A. V. Stepanov, “Linear groups over general rings. I. Generalities”, Problems in the theory of representations of algebras and groups. Part 22, Zap. Nauchn. Sem. POMI, 394, POMI, St. Petersburg, 2011, 33–139; J. Math. Sci. (N. Y.), 188:5 (2013), 490

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Zapiski Nauchnykh Seminarov POMI, 2011, Volume 394, Pages 33–139 (Mi znsl4630)

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

Linear groups over general rings. I. Generalities

N. A. Vavilov^a, A. V. Stepanov^ba

^a Saint-Petersburg State University, Saint-Petersburg, Russia
^b Abdus Salam School of Mathematical Sciences, Lahore, Pakistan

Full-text PDF (1150 kB) Citations (15)

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Abstract: This paper is the first part of a systematic survey on the structure of classical groups over general rings. We intend to cover various proofs of the main structure theorems, commutator formulae, finiteness and stability conditions, stability and pre-stability theorems, nilpotency of

K_{1}

$\mathrm K_1$ , centrality of

K_{2}

$\mathrm K_2$ , automorphism and homomorphisms, etc. This first part covers background material such as one-sided inverses, elementary transformations, definitions of obvious subgroups, Bruhat and Gauss decompositions, relative subgroups, finitary phenomens, and transvections.

Key words and phrases: linear groups, general linear group, associative rings, one-sided inverses, weakly finite rings, IBN rings, elementary transvections, linear transvections, congruence subgroups, elementary subgroups, Bruhat decomposition, Gauss decomposition, parabolic subgroups, group of finitary matrices, Whitehead type lemmas.

Received: 17.08.2011

English version:
Journal of Mathematical Sciences (New York), 2013, Volume 188, Issue 5, Pages 490–550
DOI: https://doi.org/10.1007/s10958-013-1146-7

Bibliographic databases:

Document Type: Article

UDC: 512.5

Language: Russian

Citation: N. A. Vavilov, A. V. Stepanov, “Linear groups over general rings. I. Generalities”, Problems in the theory of representations of algebras and groups. Part 22, Zap. Nauchn. Sem. POMI, 394, POMI, St. Petersburg, 2011, 33–139; J. Math. Sci. (N. Y.), 188:5 (2013), 490–550

Citation in format AMSBIB

\Bibitem{VavSte11}

\by N.~A.~Vavilov, A.~V.~Stepanov

\paper Linear groups over general rings.~I. Generalities

\inbook Problems in the theory of representations of algebras and groups. Part~22

\serial Zap. Nauchn. Sem. POMI

\yr 2011

\vol 394

\pages 33--139

\publ POMI

\publaddr St.~Petersburg

\mathnet{http://mi.mathnet.ru/znsl4630}

\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2870172}

\transl

\jour J. Math. Sci. (N. Y.)

\yr 2013

\vol 188

\issue 5

\pages 490--550

\crossref{https://doi.org/10.1007/s10958-013-1146-7}

\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84884413693}

Linking options:

https://www.mathnet.ru/eng/znsl4630

https://www.mathnet.ru/eng/znsl/v394/p33

This publication is cited in the following 15 articles:

George Ioniţă, Frank Kutzschebauch, “Holomorphic factorization of vector bundle automorphisms”, Bulletin des Sciences Mathématiques, 199 (2025), 103565
N. A. Vavilov, “St. Petersburg School of Linear Groups: II. Early Works by Suslin”, Vestnik St.Petersb. Univ.Math., 57:1 (2024), 30
N. A. Vavilov, “Bounded generation of relative subgroups in Chevalley groups”, Algebra i teoriya chisel. 6, Zap. nauchn. sem. POMI, 523, POMI, SPb., 2023, 7–18
N. A. Vavilov, “Saint Petersburg School of the Theory of Linear Groups. I. Prehistory”, Vestnik St.Petersb. Univ.Math., 56:3 (2023), 273
N. A. Vavilov, “Sankt-Peterburgskaya shkola teorii lineinykh grupp. I. Predystoriya”, Vestnik Sankt-Peterburgskogo universiteta. Matematika. Mekhanika. Astronomiya, 10:3 (2023), 381–405
M. A. Buryakov, N. A. Vavilov, “Relative decomposition of transvections: explicit bounds”, Voprosy teorii predstavlenii algebr i grupp. 38, Zap. nauchn. sem. POMI, 513, POMI, SPb., 2022, 9–21
N. Vavilov, Z. Zhang, “Commutators of Relative and Unrelative Elementary Groups, Revisited”, J Math Sci, 251:3 (2020), 339
S. V. Sidorov, “On the Similarity of Certain Integer Matrices with Single Eigenvalue over the Ring of Integers”, Math. Notes, 105:5 (2019), 756–762
N. Vavilov, “Commutators of congruence subgroups in the arithmetic case”, Algebra i teoriya chisel. 2, Zap. nauchn. sem. POMI, 479, POMI, SPb., 2019, 5–22
N. Vavilov, Z. Zhang, “Commutators of relative and unrelative elementary groups, revisited”, Teoriya predstavlenii, dinamicheskie sistemy, kombinatornye metody. XXXI, Zap. nauchn. sem. POMI, 485, POMI, SPb., 2019, 58–71
J. Math. Sci. (N. Y.), 243:4 (2019), 515–526
J. Math. Sci. (N. Y.), 243:4 (2019), 527–534
E. Yu. Voronetsky, “Normalizers of elementary overgroups of $\mathrm{Ep}(2,A)$”, J. Math. Sci. (N. Y.), 232:5 (2018), 610–621
R. Hazrat, N. Vavilov, Z. Zhang, “The commutators of classical groups”, J. Math. Sci. (N. Y.), 222:4 (2017), 466–515
J. Math. Sci. (N. Y.), 200:6 (2014), 742–768

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

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