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 Zap. Nauchn. Sem. POMI, 2009, Volume 365, Pages 47–62 (Mi znsl3465)

Decomposition of transvections for automorphisms

N. A. Vavilov, V. G. Kazakevich

St.-Petersburg State University

Abstract: The method of decomposition of unipotents consists in writing elementary matrices as products of factors lying in proper parabolic subgroups, whose images under (abstract) inner automorphisms also fall into proper parabolic subgroups of certain types. For the general linear group, this method was first proposed by Stepanov in 1987 to simplify the proof of Suslin's normality theorem. Soon thereafter Vavilov and Plotkin generalised it to other classical groups and Chevalley groups. Subsequently, many further results of that type have been discovered. In the present paper, we show that a similar decomposition can be constructed for arbitrary standard automorphisms. This result emerged in the context of a simplified proof of the theorems due to Waterhouse, Golubchik, Mikhalev, Zelmanov, and Petechuk regarding the standard description of automorphisms of the general linear group, based exclusively on the use of unipotent elements. Bibl. – 27 titles.

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English version:
Journal of Mathematical Sciences (New York), 2009, 161:4, 483–491

Bibliographic databases:

UDC: 513.6

Citation: N. A. Vavilov, V. G. Kazakevich, “Decomposition of transvections for automorphisms”, Problems in the theory of representations of algebras and groups. Part 18, Zap. Nauchn. Sem. POMI, 365, POMI, St. Petersburg, 2009, 47–62; J. Math. Sci. (N. Y.), 161:4 (2009), 483–491

Citation in format AMSBIB
\Bibitem{VavKaz09} \by N.~A.~Vavilov, V.~G.~Kazakevich \paper Decomposition of transvections for automorphisms \inbook Problems in the theory of representations of algebras and groups. Part~18 \serial Zap. Nauchn. Sem. POMI \yr 2009 \vol 365 \pages 47--62 \publ POMI \publaddr St.~Petersburg \mathnet{http://mi.mathnet.ru/znsl3465} \zmath{https://zbmath.org/?q=an:05660144} \elib{http://elibrary.ru/item.asp?id=15311097} \transl \jour J. Math. Sci. (N. Y.) \yr 2009 \vol 161 \issue 4 \pages 483--491 \crossref{https://doi.org/10.1007/s10958-009-9578-9} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-70349598032} 

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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
2. N. A. Vavilov, “An $\mathrm A_3$-proof of the structure theorems for Chevalley groups of types $\mathrm E_6$ and $\mathrm E_7$. II. The main lemma”, St. Petersburg Math. J., 23:6 (2012), 921–942
3. N. A. Vavilov, A. V. Stepanov, “Linear groups over general rings. I. Generalities”, J. Math. Sci. (N. Y.), 188:5 (2013), 490–550
4. V. A. Petrov, “Decomposition of transvections: an algebro-geometric approach”, St. Petersburg Math. J., 28:1 (2017), 109–114
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