Skip to main content
SpringerLink
Log in

The normal structure of the unipotent subgroup in Lie type groups and its extremal subgroups

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

We study the normal structure of the unipotent radical U of a Borel subgroup in a Lie type group over a field K. Thus, all maximal Abelian normal subgroups in U are described. This gives a new solution of C. Parker and P. Rowley’s problem about extremal subgroups in U and the description in finite groups U of the large normal (and, as proved, also normal large) Abelian subgroups.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. J. J. Barry, “Large Abelian subgroups of Chevalley groups,” J. Aust. Math. Soc. Ser. A, 27, No. 1, 59–87 (1979).

    Article  MathSciNet  MATH  Google Scholar 

  2. M. J. J. Barry and W. J. Wong, “Abelian 2-subgroups of finite symplectic groups in characteristic 2,” J. Aust. Math. Soc. Ser. A, 33, No. 3, 345–350 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  3. N. Bourbaki, Groupes et algebres de Lie (Chap. IV–VI ), Hermann, Paris (1968).

  4. R. Carter, Simple Groups of Lie Type, Wiley and Sons, New York (1972).

    MATH  Google Scholar 

  5. J. Gibbs, “Automorphisms of certain unipotent groups,” J. Algebra, 14, No. 2, 203–228 (1970).

    Article  MathSciNet  MATH  Google Scholar 

  6. C. K. Gupta, V. M. Levchuk, and Yu. Yu. Ushakov, “Hypercentral and monic automorphisms of classical algebras, rings and groups,” J. Sib. Fed. Univ., Math. Phys., 1, No. 4, 380–390 (2008).

    Google Scholar 

  7. A. S. Kondrat’ev, “Subgroups of finite Chevalley groups,” Usp. Mat. Nauk, 41, No. 1 (247), 57–96 (1986).

  8. F. Kuzucuoglu and V. M. Levchuk, “The automorphism group of certain radical rings,” J. Algebra, 243, 473–485 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  9. F. Kuzucuoglu and V. M. Levchuk, “Isomorphism of certain locally nilpotent finitary groups and associated rings,” Acta Appl. Math., 82, No. 2, 169–181 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  10. V. M. Levchuk, “Relation of the unitriangle group to certain rings,” Algebra Logic, 15, No. 5, 348–360 (1976).

    Article  Google Scholar 

  11. V. M. Levchuk, “Parabolic subgroups of some ABA-groups,” Mat. Zametki, 31, No. 4, 509–525 (1982).

    MathSciNet  MATH  Google Scholar 

  12. V. M. Levchuk, “Automorphisms of unipotent subgroups of Chevalley groups,” Algebra Logic, 29, 97–112, 211–224 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  13. V. M. Levchuk, “Chevalley groups and their unipotent subgroups,” in: Proc. of the Int. Conf. on Algebra Dedicated to the Memory of A. I. Mal’cev, held at Akademgorodok, Novosibirsk, USSR, Aug. 21–26, 1989, Part 1, Contemp. Math., Vol. 131, Amer. Math. Soc., Providence, RI (1992), pp. 227–242.

  14. V. M. Levchuk and G. S. Suleimanova, “Normal structure of the adjoint group in the radical rings R n (K, J),” Sib. Math. J., 43, No. 2, 334–349 (2002).

    Article  MathSciNet  Google Scholar 

  15. V. M. Levchuk and G. S. Suleimanova, “The normal structure of unipotent subgroups in Lie-type groups and related questions,” Dokl. Math., 77, No. 2, 284–287 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  16. V. M. Levchuk and G. S. Suleimanova, “Automorphisms and normal structure of unipotent subgroups of finitary Chevalley groups,” Tr. Inst. Mat. Mekh., 15, No. 2, 133–142 (2009).

    Google Scholar 

  17. V. M. Levchuk, G. S. Suleymanova, and T. Yu. Voitenko, “Some questions for the unipotent subgroup of the Chevalley group,” in: Int. Conf. “Algebra and Its Applications,” Krasnoyarsk, 2007, pp. 168–169.

  18. A. I. Mal’cev, “Commutative subalgebras of semisimple Lie algebras,” Izv. Akad. Nauk SSSR, Ser. Mat., 9, No. 4, 291–300 (1945).

    MathSciNet  Google Scholar 

  19. L. A. Martynova, Normal Structure and Automorphisms of Lie-Type Unipotent Subgroups [in Russian], Candidate’s Dissertation, Moscow Univ. (1994).

  20. C. Parker and P. Rowley, “Extremal subgroups in Chevalley groups,” J. London Math. Soc., 55, No. 2, 387–399 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  21. C. Parker and P. Rowley, “Extremal subgroups in twisted Lie type groups,” J. Reine Angew. Math., 498, 135–152 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  22. C. Parker and P. Rowley, Symplectic Amalgams, Springer Monographs Math., London, Springer (2002).

    Book  MATH  Google Scholar 

  23. C. Parker and P. Rowley, “Unique node extremal subgroups in Chevalley groups,” Commun. Algebra, 31, No. 7, 3471–3486 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  24. J.-P. Serre, Algebres de Lie Semi-Simple Complexes, New York, Benjamin (1966).

    Google Scholar 

  25. E. I. Spitznagel, “Terminality of the maximal unipotent groups of Chevalley groups,” Math. Z., 103, No. 2, 112–116 (1968).

    Article  MathSciNet  MATH  Google Scholar 

  26. E. I. Spitznagel, “Structure and terminality of the maximal unipotent groups of Steinberg groups,” J. Math., 13, No. 2, 400–405 (1969).

    MathSciNet  MATH  Google Scholar 

  27. R. Steinberg, Lectures on Chevalley Groups, Yale Univ. (1967).

  28. G. S. Suleimanova, Structure Descriptions and Relations of Nilpotent Matrix Groups and Associated Rings, Candidate’s Dissertation, Krasnoyarsk Univ. (2002).

  29. G. S. Suleimanova, “The normal structure and Abelian normal subgroups in the unipotent subgroup of the symplectic group,” Vladikavkaz. Mat. Zh., 10, No. 1, 79–83 (2008).

    MathSciNet  Google Scholar 

  30. G. S. Suleimanova, “On a conjugacy in a Chevalley group of large Abelian subgroups of the unipotent subgroup,” J. Math. Sci., 169, No. 5, 696–794 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  31. G. S. Suleimanova and V. V. Yakoby, “The normal structure of the unipotent subgroup of a Chevalley group of type E6, E7, E8,” J. Sib. Fed. Univ., Math. Phys., 1, No. 2, 152–157 (2008).

    Google Scholar 

  32. E. P. Vdovin, “Large Abelian unipotent subgroups of finite Chevalley groups,” Algebra Logic, 40, No. 5, 292–305 (2001).

    Article  MathSciNet  Google Scholar 

  33. A. J. Weir, “Sylow p-subgroups of the general linear group over finite fields of characteristic p,” Proc. Amer. Math. Soc., 6, No. 3, 454–464 (1955).

    MathSciNet  MATH  Google Scholar 

  34. W. J. Wong, “Abelian unipotent subgroups of finite orthogonal groups,” J. Aust. Math. Soc. Ser. A, 32, No. 2, 223–245 (1982).

    Article  MATH  Google Scholar 

  35. W. J. Wong, “Abelian unipotent subgroups of finite unitary and symplectic groups,” J. Aust. Math. Soc. Ser. A, 33, No. 2, 331–344 (1982).

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Siberian Federal University, Krasnoyarsk, Russia

    V. M. Levchuk

  2. Khakas Technical Institute of Siberian Federal University, Abakan, Russia

    G. S. Suleimanova

Authors
  1. V. M. Levchuk
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. G. S. Suleimanova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. M. Levchuk.

Additional information

Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 17, No. 1, pp. 155–168, 2011/12.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Levchuk, V.M., Suleimanova, G.S. The normal structure of the unipotent subgroup in Lie type groups and its extremal subgroups. J Math Sci 185, 448–457 (2012). https://doi.org/10.1007/s10958-012-0927-8

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-012-0927-8

Keywords

Search

Navigation