Skip to main content
SpringerLink
Log in

Subgroups of the General Linear Group That Contain Elementary Subgroup Over a Rank 2 Commutative Ring Extension

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

Let R = \( \prod \limits_{i\in I}{F}_i \) be the direct product of fields, and let\( S=R\left[\sqrt{d}\right]=\prod \limits_{i\in I}{F}_i\left[\sqrt{d_i}\right] \) be a rank 2 extension of R. The subgroups of the general linear group GL(2n,R), n ≥ 3, that contain the elementary group E (n, S) are described. It is shown that for every such a subgroup H there exists a unique ideal A ⊴ R such that E (n, S)E(2n,R,A) ≤ H ≤ NGL(2n,R) (E (n, S)E(2n,R,A)).

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. Z. I. Borevich and N. A. Vavilov, “Subgroups of the general linear group over a commutative ring,” Dokl. Akad. Nauk, 267, No. 4, 777–778 (1982).

    MathSciNet  Google Scholar 

  2. L. N. Vaserstein, “Normal subgroups of the general linear groups over von Neumann regular rings,” J. Pure Appl. Algebra, 52, 187–195 (1988).

    Article  MathSciNet  Google Scholar 

  3. L. N. Vaserstein, “On normal subgroups of GLn over a ring,” Lect. Notes Math., 854, 456–465 (1981).

    Article  Google Scholar 

  4. S. Li, “Overgroups in GL (nr, F) of certain subgroups of SL (n,K),” J. Algebra, 125, 115–135 (1989).

    Article  MathSciNet  Google Scholar 

  5. A. A. Suslin, “The structure of the special linear group over rings of polynomials,” Izv. Akad. Nauk Ser. Mat., 41, No. 2, 235–252 (1977).

    MathSciNet  MATH  Google Scholar 

  6. L. N. Vaserstein and A. A. Suslin, “Serre’s problem on projective modules over polynomial rings, and algebraic K-theory,” Izv. Akad. Nauk Ser. Mat., 40, No. 5, 993–1054 (1976).

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  8. N. A. Vavilov and V. A. Petrov, “On overgroups of EO(2l,R),” Zap. Nauchn. Semin. POMI, 272, 68–85 (2000).

    MATH  Google Scholar 

  9. N. A. Vavilov and V. A. Petrov, “On overgroups of Ep(2l,R),” Algebra Analiz, 15, No. 3, 72–114 (2003).

    MathSciNet  Google Scholar 

  10. N. A. Vavilov and V. A. Petrov, “On overgroups of EO (n,R),” Algebra Analiz, 19, No. 2, 10–51 (2007).

    MathSciNet  Google Scholar 

  11. J. Tits, “Systèmes générateurs de groupes de congruence,” C. R. Acad. Sci. Paris Ser A-B, 283, A693-A695 (1976).

    MATH  Google Scholar 

  12. N. H. T. Nhat and T. N. Hoi, “The normalizer of the elementary linear group of a module arising under extension of the base ring,” Zap. Nauchn. Semin. POMI, (2017), this volume.

Download references

Author information

Authors and Affiliations

  1. University of Science, VNU-HCM, Ho Chi Minh City, Vietnam

    T. N. Hoi & N. H. T. Nhat

Authors
  1. T. N. Hoi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. N. H. T. Nhat
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to T. N. Hoi.

Additional information

Published in Zapiski Nauchnykh Seminarov POMI, Vol. 455, 2017, pp. 209–225.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hoi, T.N., Nhat, N.H.T. Subgroups of the General Linear Group That Contain Elementary Subgroup Over a Rank 2 Commutative Ring Extension. J Math Sci 234, 256–267 (2018). https://doi.org/10.1007/s10958-018-4001-z

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-018-4001-z

Search

Navigation