Skip to main content
Log in

Basic reductions in the description of normal subgroups

Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

Classification of subgroups in a Chevalley group G(Φ, R) over a commutative ring R, normalized by the elementary subgroup E(Φ, R), is well known. However, for exceptional groups, in the available literature neither the parabolic reduction nor the level reduction can be found. This is due to the fact that the Abe-Suzuki-Vaserstein proof relied on localization and reduction modulo the Jacobson radical. Recently, for the groups of types E 6, E 7, and F 4, the first-named author, M. Gavrilovich, and S. Nikolenko have proposed an even more straightforward geometric approach to the proof of structure theorems, similar to that used for exceptional cases. In the present paper, we give still simpler proofs of two key auxiliary results of the geometric approach. First, we carry through the parabolic reduction in full generality: for all parabolic subgroups of all Chevalley groups of rank ≥ 2. At that we succeeded in avoiding any reference to the structure of internal Chevalley modules, or explicit calculations of the centralizers of unipotent elements. Second, we prove the level reduction, also for the most general situation of double levels, which arise for multiply-laced root systems. Bibliography: 64 titles.

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

Access this article

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

Instant access to the full article PDF.

Institutional subscriptions

References

  1. E. Abe, “Automorphisms of Chevalley groups over commutative rings,” Algebra Analiz, 5, No. 2, 74–90 (1993).

    Google Scholar 

  2. H. Bass, Algebraic K-Theory [Russian translation], Mir, Moscow (1973).

    Google Scholar 

  3. A. Borel, “Properties and linear representations of Chevalley groups” [Russian translation],” in: Seminar on Algebraic Groups, Mir, Moscow (1973), pp. 9–59.

    Google Scholar 

  4. N. Bourbaki, Lie Groups and Algebras [Russian translation], Chaps IV–VI, Moscow (1972).

  5. N. A. Vavilov, “Computations in exceptional groups,” Vestn. Samara Univ., 33, No. 7, 11–24 (2007).

    Google Scholar 

  6. N. A. Vavilov and M. R. Gavrilovich, “An A2-proof of structure theorems for Chevalley groups of types E6 and E7,” Algebra Analiz, 16, No. 4, 54–87 (2004).

    MathSciNet  Google Scholar 

  7. N. A. Vavilov, M. R. Gavrilovich, and S. I. Nikolenko, “The structure of Chevalley groups: the proof from the Book,” Zap. Nauchn. Semin. POMI, 330, 36–76 (2006).

    MATH  Google Scholar 

  8. N. A. Vavilov and A. Yu. Luzgarev, “The normalizer of the Chevalley group of type E6,” Algebra Analiz, 19, No. 5, 35–62 (2007).

    MathSciNet  Google Scholar 

  9. N. A. Vavilov and S. I. Nikolenko, “An A2-proof of structure theorems for Chevalley groups of type F4,” Algebra Analiz, 20, No. 3 (2008).

  10. N. A. Vavilov, E. B. Plotkin, and A. V. Stepanov, “Computations in Chevalley groups over commutative rings,” Dokl. AN SSSR, 40, No. 1, 145–147 (1990).

    MathSciNet  Google Scholar 

  11. N. A. Vavilov and A. V. Stepanov, “The standard commutation formula,” Vestn. S.-Peterburg Gos. Univ., Ser. 1, No. 1, 9–14 (2008).

  12. I. Z. Golubchik, “Normal subgroups of linear and unitary groups over an associative ring,” in: Spaces over Algebras and Some Problems in the Theory of Nets, Ufa (1985), pp. 122–142.

  13. I. Z. Golubchik, “Lie type groups over PI-rings,” Fund. Prikl. Mat., 3, No. 2, 399–424 (1997).

    MATH  MathSciNet  Google Scholar 

  14. V. G. Kazakevich and A. K. Stavrova, “Subgroups normalized by the derived group of the Levi subgroup,” Zap. Nauchn. Semin. POMI, 319, 199–215 (2004).

    MATH  Google Scholar 

  15. R. Steinberg, Lectures on Chevalley Groups [Russian translation], Mir, Moscow (1975).

    MATH  Google Scholar 

  16. A. V. Stepanov, “On the normal structure of the general linear group over the ring of polynomials,” Zap. Nauchn. Semin. POMI, 236, 166–182 (1997).

    Google Scholar 

  17. A. A. Suslin, “On the structure of the special linear group over the ring of polynomials,” Izv. AN SSSR, Ser. Mat., 141, No. 2, 235–253 (1977).

    MathSciNet  Google Scholar 

  18. E. Abe, “Chevalley groups over local rings,” Tôhoku Math. J. (2), 21, No. 3, 474–494 (1969).

    Article  MATH  Google Scholar 

  19. E. Abe, “Chevalley groups over commutative rings,” in: Proceedings of the Conference on Radical Theory (Sendai, 1988), Uchida Rokakuho, Tokyo (1989), pp. 1–23.

    Google Scholar 

  20. E. Abe, “Normal subgroups of Chevalley groups over commutative rings,” Contemp. Math., 83, 1–17 (1989).

    Google Scholar 

  21. E. Abe, “Chevalley groups over commutative rings: normal subgroups and automorphisms rings,” Contemp. Math., 184, 13–23 (1995).

    Google Scholar 

  22. E. Abe and J. Hurley, “Centers of Chevalley groups over commutative rings,” Comm. Algebra, 16, No. 1, 57–74 (1998).

    Article  MathSciNet  Google Scholar 

  23. E. Abe and K. Suzuki, “On normal subgroups of Chevalley groups over commutative rings,” Tôhoku Math. J., 28, No. 1, 185–198 (1976).

    Article  MATH  MathSciNet  Google Scholar 

  24. H. Azad, M. Barry, and G. M. Seitz, “On the structure of parabolic subgroups,” Comm. Algebra, 18, 551–562 (1990).

    Article  MATH  MathSciNet  Google Scholar 

  25. A. Bak, “The stable structure of quadratic modules,” Thesis, Columbia Univ. (1969).

  26. A. Bak, “Subgroups of the general linear group normalized by relative elementary groups,” Lecture Notes Math., 967, 1–22 (1982).

    Article  MathSciNet  Google Scholar 

  27. A. Bak, “Nonabelian K-theory: The nilpotent class of K1 and general stability,” K-Theory, 4, 363–397 (1991).

    Article  MATH  MathSciNet  Google Scholar 

  28. A. Bak, R. Hazrat, and N. Vavilov, “Localization-completion strikes again: relative K 1 is nilpotent by abelian,” K-Theory (2008).

  29. A. Bak and N. Vavilov, “Normality for elementary subgroups functors,” Math. Proc. Cambridge Phil. Soc., 118, No. 1, 35–47 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  30. A. Bak and N. Vavilov, “Structure of hyperbolic unitary groups. I. Elementary subgroups,” Algebra Colloquium, 7, No. 2, 159–196 (2000).

    MATH  MathSciNet  Google Scholar 

  31. A. Bak, R. Hazrat, and N. Vavilov, “Structure of hyperbolic unitary groups. II. Normal subgroups” Algebra Colloquium (to appear).

  32. H. Bass, “K-theory and stable algebra,” Inst. Hautes Études Sci. Publ. Math., No. 22, 5–60 (1964).

  33. H. Bass, “Unitary algebraic K-theory,” Lecture Notes Math., 343, 57–265 (1973).

    Article  MathSciNet  Google Scholar 

  34. R. Carter, Simple groups of Lie Type, John Wiley, London et al. (1972).

    MATH  Google Scholar 

  35. D. L. Costa and G. E. Keller, “The E(2, A) sections of SL(2, A),” Ann. Math. (2), 134, No. 1, 159–188 (1991).

    Article  MATH  MathSciNet  Google Scholar 

  36. D. L. Costa and G. E. Keller, “Radix redux: normal subgroups of symplectic groups,” J. Reine Angew. Math., 427, No. 1, 51–105 (1991).

    MathSciNet  Google Scholar 

  37. D. L. Costa and G. E. Keller, “On the normal subgroups of G2(A),” Trans. Amer. Math. Soc., 351, No. 12, 5051–5088 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  38. M. Demazure and A. Grothendieck, “Schémas en groupes. I, II, III,” Lecture Notes Math., 151, 1–564 (1971); 152, 1–654; 153, 1–529.

    Article  Google Scholar 

  39. A. J. Hahn and O. T. O’Meara, The Classical Groups and K-Theory, Springer-Verlag, Berlin et al. (1989).

    Google Scholar 

  40. R. Hazrat, “Dimension theory and non-stable K1 of quadratic module,” K-Theory, 27, 293–327 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  41. R. Hazrat and N. Vavilov, “K1 of Chevalley groups are nilpotent,” J. Pure Appl. Algebra, 179, 99–116 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  42. R. Hazrat and N. Vavilov, “Bak’s work on K-theory of rings” (to appear).

  43. Li Fuan and Liu Mulan, “Generalized sandwich theorem,” K-Theory, 1, 171–184 (1987).

    Article  MathSciNet  Google Scholar 

  44. H. Matsumoto, “Sur les sous-groupes arithmétiques des groupes semi-simples déployés,” Ann. Sci. École Norm. Sup. (4), 2, 1–62 (1969).

    MATH  Google Scholar 

  45. V. M. Petechuk, “Stability structure of linear group over rings,” Mat. Studii, 16, No. 1, 13–24 (2001).

    MATH  MathSciNet  Google Scholar 

  46. G. E. Roehrle, “On the structure of parabolic subgroups in algebraic groups,” J. Algebra, 157, No. 1, 80–115 (1993).

    Article  MATH  MathSciNet  Google Scholar 

  47. A. Stavrova, “Normal structure of maximal parabolic subgroups in Chevalley groups over rings,” Algebra Colloquium (to appear).

  48. M. R. Stein, “Generators, relations, and coverings of Chevalley groups over commutative rings,” Amer. J. Math., 93, No. 4, 965–1004 (1971).

    Article  MATH  MathSciNet  Google Scholar 

  49. A. Stepanov and N. Vavilov, “Decomposition of transvections: a theme with variations,” K-Theory, 19, 109–153 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  50. K. Suzuki, “On normal subgroups of twisted Chevalley groups over local rings,” Sci. Rep. Tokyo Kyoiku Daigaku, 13, 237–249 (1997).

    Google Scholar 

  51. K. Suzuki, “Normality of the elementary subgroups of twisted Chevalley groups over commutative rings,” J. Algebra, 175, No. 3, 526–536 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  52. G. Taddei, “Normalité des groupes élémentaire dans les groupes de Chevalley sur un anneau,” Contemp. Math., 55, 693–710 (1986).

    MathSciNet  Google Scholar 

  53. L. N. Vaserstein, “On the normal subgroups of the GLn of a ring,” Lecture Notes Math., 854, 454–465 (1981).

    MathSciNet  Google Scholar 

  54. L. N. Vaserstein, “The subnormal structure of general linear groups,” Math. Proc. Cambridge Phil. Soc., 99, 425–431 (1986).

    Article  MATH  MathSciNet  Google Scholar 

  55. L. N. Vaserstein, “On normal subgroups of Chevalley groups over commutative rings,” Tôhoku Math. J. (2), 36, No. 5, 219–230 (1986).

    Article  MathSciNet  Google Scholar 

  56. L. N. Vaserstein, “The subnormal structure of general linear groups over rings,” Math. Proc. Cambridge Phil. Soc., 108, No. 2, 219–229 (1990).

    Article  MATH  MathSciNet  Google Scholar 

  57. L. N. Vaserstein and You Hong, “Normal subgroups of classical groups over rings,” J. Pure Appl. Algebra, 105, No. 1, 93–106 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  58. N. Vavilov, “A note on the subnormal structure of general linear groups,” Math. Proc. Cambridge Phil. Soc., 107, No. 2, 193–196 (1990).

    Article  MATH  MathSciNet  Google Scholar 

  59. N. Vavilov, “Structure of Chevalley groups over commutative rings,” in: Proceedings of the Conference on nonassociative Algebras and Related Topics (Hiroshima, 1990), World Sci. Publ., London et al. (1991), pp. 219–335.

    Google Scholar 

  60. N. Vavilov, “A third look at weight diagrams,” Rendiconti del Rend. Sem. Mat. Univ. Padova, 204, No. 1, 201–250 (2000).

    MathSciNet  Google Scholar 

  61. N. Vavilov, “An A3-proof of structure theorems for Chevalley groups of types E6 and E7,” Intern. J. Algebra Comput., 17, No. 5–6, 1283–11298 (2007).

    Article  MATH  MathSciNet  Google Scholar 

  62. N. Vavilov, “Structure of exceptional groups over rings,” in: Proceedings of the 3rd International Congress on Algebra and Combinatorics (Beijing and Xian, 2007) (to appear).

  63. N. A. Vavilov and E. B. Plotkin, “Chevalley groups over commutative rings. I. Elementary calculations,” Acta Appl. Math., 45, 73–115 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  64. Zhang, Zuhong, “Stable sandwich classification theorem for classical-like groups,” Math. Proc. Cambridge Phil. Soc., 143, No. 3, 607–619 (2007).

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

__________

Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 349, 2007, pp. 30–52.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Vavilov, N.A., Stavrova, A.K. Basic reductions in the description of normal subgroups. J Math Sci 151, 2949–2960 (2008). https://doi.org/10.1007/s10958-008-9019-1

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-008-9019-1

Keywords

Navigation