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Journal of Mathematical Sciences

, Volume 83, Issue 5, pp 617–625 | Cite as

The subgroups of the special linear group over a skew field that contain the group of diagonal matrices

  • Bui Xuan Hai
Article
  • 19 Downloads

Abstract

For any (noncommutative) skew field T, the lattice of subgroups of the special linear group Λ=SL(n,T) that contain the subgroup Δ=SD(n,T) of diagonal matrices (with Dieudonné determinants equal to 1) is studied. It is established that for any subgroup H, Δ≤H≤Λ, there exists a uniquely determined unital net σ such that Λ(σ)≤H≤N(σ), where Λ(σ) is the net subgroup associated with the net σ and N(σ) is its normalizer in Λ. Bibliography: 11 titles.

Keywords

Parabolic Subgroup Diagonal Matrice Standard Description Special Linear Group Reducible Subgroup 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    E. Artin,Geometric Algebra, New York-London (1957).Google Scholar
  2. 2.
    Z. I. Borevich, “Description of subgroups of the general linear group that contain the group of diagonal matrices,”Zap. Nauchn. Semin. LOMI,64, 12–29 (1976).MathSciNetMATHGoogle Scholar
  3. 3.
    Z. I. Borevich, “On parabolic subgroups of linear groups over a semilocal ring,”Vestn. LGU, No. 13, 16–24 (1976).MathSciNetMATHGoogle Scholar
  4. 4.
    Bui Xuan Hai, “On the distribution of subgroups in the special linear group over a skew field with infinite center,”Zap. Nauchn. Semin. LOMI (1989).Google Scholar
  5. 5.
    N. A. Vavilov, “On subgroups of the special linear group that contain the group of diagonal matrices,”Vestn. LGU, Mat., No. 22, 3–7 (1985).MATHGoogle Scholar
  6. 6.
    N. A. Vavilov, “Linear groups generated by homoparametric groups of homogeneous transformations,”Usp. Mat. Nauk,44, 189–190 (1989).Google Scholar
  7. 7.
    D. A. Suprunenko, “Matrix groups,”Translation of Mathematical Monographs,45 (1976).Google Scholar
  8. 8.
    N. Bourbaki,Groupes et Algèbres de Lie, Chap. IV–VI.Google Scholar
  9. 9.
    G. M. Seitz, “Subgroups of finite groups of Lie type,”J Algebra,61, No. 1, 16–27 (1979).MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    N. A. Vavilov and E. V. Dybkova, “Subgroups of the general symplectic group containing the group of diagonal matrices,”Zap. Nauchn. Semin. LOMI,103, 31–47 (1980).MathSciNetMATHGoogle Scholar
  11. 11.
    V. A. Koibaev, “Examples of nonmonomial linear groups without transvections,”Zap. Nauchn. Semin. LOMI,71, 153–154 (1977).MathSciNetGoogle Scholar

Copyright information

© Plenum Publishing Corporation 1997

Authors and Affiliations

  • Bui Xuan Hai

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