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Nonabelian 1-cohomologies and conjugacy of finite subgroups in some extensions

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Abstract

We describe the conjugacy classes of finite subgroups in some split extensions using the notion of 1-cocycle and 1-coboundary with values in a noncommutative group. We prove that each finite subgroup in the automorphism group of a free Lie algebra of rank 3 is conjugated with a subgroup of the linear automorphism group provided that the group order does not divide the characteristic of the ground field.

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References

  1. Shestakov I. P. and Umirbaev U. U., “Poisson brackets and two-generated subalgebras of rings of polynomials,” J. Amer. Math. Soc., 17,No. 1, 181–196 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  2. Shestakov I. P. and Umirbaev U. U., “The tame and the wild automorphisms of polynomial rings in three variables,” J. Amer. Math. Soc., 17,No. 1, 197–227 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  3. Umirbaev U. U., “Defining relations of the tame automorphism group of polynomial algebras in three variables,” J. Reine Angew. Math. (Crelles Journal), No. 600, 203–235 (2006).

  4. Umirbaev U. U., “Defining relations for automorphism groups of free algebras,” J. Algebra, 314, No. 1, 209–225 (2007).

    Article  MATH  MathSciNet  Google Scholar 

  5. Cohn P. M., “The automorphism group of the free algebra of rank two,” Serdica Math. J., No. 28, 255–266 (2002).

  6. Shevelin M. A., “Periodic automorphisms of the free Lie algebra of rank 3,” Siberian Math. J., 52,No. 3, 544–553 (2011).

    Article  MATH  MathSciNet  Google Scholar 

  7. Serre J. P., Galois Cohomology, Springer-Verlag, Berlin (2002).

    MATH  Google Scholar 

  8. Magnus W., Karrass A., and Solitar D., Combinatorial Group Theory [Russian translation], Nauka, Moscow (1974).

    Google Scholar 

  9. Shirshov A. I., Selected Works. Rings and Algebras [in Russian], Nauka, Moscow (1984).

    Google Scholar 

  10. Witt E., “Die Unterringe der freien Lieschen Ringe,” Math. Z., Bd. 64, 195–216 (1956).

    Article  MATH  MathSciNet  Google Scholar 

  11. Cohn P. M., “Subalgebras of free associative algebras,” Proc. London Math. Soc., 56, 618–632 (1964).

    Article  Google Scholar 

  12. Andreadakis S., “On automorphisms of free groups and free nilpotent groups,” Proc. London Math. Soc., 15,No. 3, 239–268 (1965).

    Article  MATH  MathSciNet  Google Scholar 

  13. Kawazumi N., Cohomological Aspects of Magnus Expansions [Preprint], arXiv:math.GT/0505497.

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Authors and Affiliations

  1. Omsk State University, Omsk, Russia

    M. A. Shevelin

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  1. M. A. Shevelin
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Correspondence to M. A. Shevelin.

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Original Russian Text Copyright © 2012 Shevelin M.A.

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Omsk. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 53, No. 5, pp. 1166–1177, September–October, 2012.

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Shevelin, M.A. Nonabelian 1-cohomologies and conjugacy of finite subgroups in some extensions. Sib Math J 53, 934–942 (2012). https://doi.org/10.1134/S0037446612050187

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  • DOI: https://doi.org/10.1134/S0037446612050187

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