Skip to main content
SpringerLink
Log in

Subgroups of Split Orthogonal Groups over a Commutative Ring

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

Abstract

The subgroups of the split orthogonal group Γ = GO(n,R) of degree n over a commutative ring R with 2 ∈R* that contain the elementary subgroup of a regularly embedded semisimple group F are described. It is shown that if the ranks of all irreducible components of F are at least 4, then the description of its overgroups is standard in the sense that for any intermediate subgroup H there exists a unique orthogonal net of ideals such that H lies between the corresponding net subgroup and its normalizer in Γ. A similar result for subgroups of the general linear group GL(n,R) with irreducible components of ranks at least 2 was obtained by Z. I. Borewicz and the present author. Examples which show that if F has irreducible components of ranks 2 or 3, then the standard description does not hold were constructed. The paper is based on the previous publications by the author, where similar results were obtained in some special cases, but the proof is based on a new computational trick. Bibliography: 76 titles.

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. E. Artin, Geometric Algebra [Russian translation], Nauka, Moscow (1969).

    Google Scholar 

  2. Z. I. Borevich, “On parabolic subgroups in linear groups over a semilocal ring,” Vestn. Leningr. Univ., 13, 16–24 (1976).

    Google Scholar 

  3. Z. I. Borevich, “A description of subgroups of the general linear group that contain the group of diagonal matrices,” Zap. Nauchn. Semin. LOMI, 64, 12–29 (1976).

    Google Scholar 

  4. Z. I. Borevich, “On subgroups of linear groups that are rich in transformations,” Zap. Nauchn. Semin. LOMI, 75, 22–31 (1978).

    Google Scholar 

  5. Z. I. Borevich and N. A. Vavilov, “The subgroups of the general linear group over a semilocal ring that contain the group of diagonal matrices,” Trudy Steklov Mat. Inst., 148, 43–57 (1978).

    Google Scholar 

  6. Z. I. Borevich and N. A. Vavilov, “Arrangement of subgroups that contain the group of block diagonal matrices in the general linear group over a ring,” Izv. Vuzov, Mat., 11, 11–15 (1982).

    Google Scholar 

  7. Z. I. Borevich and N. A. Vavilov, “On subgroups of the general linear group over a commutative ring,” Dokl. Akad. Nauk SSSR, 267,4, 777–778 (1982).

    Google Scholar 

  8. Z. I. Borevich and N. A. Vavilov, “On the definition of a net subgroup,” Zap. Nauchn. Semin. LOMI, 132, 26–33 (1984).

    Google Scholar 

  9. Z. I. Borevich and N. A. Vavilov, “Arrangement of subgroups in the general linear group over a commutative ring,” Trudy Steklov Mat. Inst, 165, 24–42 (1984).

    Google Scholar 

  10. Z. I. Borevich N. A. Vavilov, and V. Narkevich, “On subgroups of the general linear group over a Dedekind ring,” Zap. Nauchn. Semin. LOMI, 94, 13–20 (1970).

    Google Scholar 

  11. N. Bourbaki, Lie Groupes and Algebras [Russian translation], Chaps. IV-VI, Mir, Moscow (1972); Chaps. VII, VIII (1978).

    Google Scholar 

  12. N. A. Vavilov, “On parabolic subgroups of Chevalley groups over a semilocal ring,” Zap. Nauchn. Semin. LOMI, 75, 43–58 (1978).

    Google Scholar 

  13. N. A. Vavilov, “On the subgroups of the general linear group over a semilocal ring that contain the group of diagonal matrices,” Vestn. Leningr. Univ., 1, 10–15 (1981).

    Google Scholar 

  14. N. A. Vavilov, “Parabolic subgroups of Chevalley groups over a commutative ring,” Zap. Nauchn. Semin. LOMI, 116, 20–43 (1982).

    Google Scholar 

  15. N. A. Vavilov, “The subgroups of the general linear group over a semilocal ring that contain the group of block diagonal matrices,” Vestn. Leningr. Univ., 1, 16–21 (1983).

    Google Scholar 

  16. N. A. Vavilov, “On the subgroups of the general linear group over a Dedekind ring of arithmetic type,” Izv. Vuzov, Mat., 17, 14–20 (1987).

    Google Scholar 

  17. N. A. Vavilov, “Subgroups of split classical groups,” Doctoral Thesis, Leningrad (1987).

  18. N. A. Vavilov, “The structure of split classical groups over a commutative ring,” Dokl. Akad. Nauk SSSR, 299,6, 1300–1303 (1988).

    Google Scholar 

  19. N. A. Vavilov, “On subgroups of split orthogonal groups. I, II,” Sib. Mat. Zhur., 29,3, 12–25 (1988); Zap. Nauchn. Semin. LOMI, 265, 42–63 (1999).

    Google Scholar 

  20. N. A. Vavilov, “On subgroups of split orthogonal groups over a ring,” Sib. Mat. Zhur., 29,4, 31–43 (1988).

    Google Scholar 

  21. N. A. Vavilov, “Semisimple root elements and triples of unipotent root subgroups in Chevalley groups,” in: Proceedings of the 10th All-Union Symposium on Group Theory. Problems in Algebra, Minsk (1989), pp. 162–173.

  22. N. A. Vavilov, “On subgroups of split classical groups,” Trudy Steklov Mat. Inst., 183, 29–41 (1990).

    Google Scholar 

  23. N. A. Vavilov, “Subgroups of Chevalley groups that contain a maximal torus,” Trudy Leningr. Mat. Obshch., 1, 64–109 (1990).

    Google Scholar 

  24. N. A. Vavilov, “On subgroups of the spinor group that contain a split maximal torus,” Zap. Nauchn. Semin. LOMI, 191, 49–75 (1991).

    Google Scholar 

  25. N. A. Vavilov and E. V. Dybkova, “On subgroups of full symplectic group that contain the group of diagonal matrices. I, II,” Zap. Nauchn. Semin. LOMI, 103, 31–47 (1980); 132, 44–56 (1983).

    Google Scholar 

  26. N. A. Vavilov and E. B. Plotkin, “Net subgroups of Chevalley groups,” Zap. Nauchn. Semin. LOMI, 94, 40–49 (1979).

    Google Scholar 

  27. N. A. Vavilov, E. B. Plotkin, and A. V. Stepanov, “Computations in Chevalley groups over commutative rings,” Dokl. Akad. Nauk SSSR, 304,41, 788–791 (1989).

    Google Scholar 

  28. N. A. Vavilov and A. V. Stepanov, “On subgroups of the general linear group over a ring satisfying stability conditions,” Izv. Vuzov, Mat., 10, 19–25 (1989).

    Google Scholar 

  29. I. Z. Golubchik, “On normal subgroups of the orthogonal group over an associative ring with involution,” Usp. Mat. Nauk, 30,6, 165 (1975).

    Google Scholar 

  30. I. Z. Golubchik, “On subgroups of the general linear group GLn(R) over an associative ring R,” Usp. Mat. Nauk, 39,1, 125–126 (1984).

    Google Scholar 

  31. I. Z. Golubchik, “On normal subgroups of linear and unitary groups over associative rings,” in: Spaces over Algebras and Some Problems in Net Theory, Ufa (1985), pp. 122–142.

  32. V. E. Dybkova, “On some congruence subgroups of the symplectic group,” Zap. Nauchn. Semin. POMI, 64, 80–91 (1976).

    Google Scholar 

  33. E. B. Dynkin, “Semisimple subalgebras of semisimple Lie algebras,” Matem. Sb., 30,2, 349–462 (1952).

    Google Scholar 

  34. J. Dieudonné, Geometry of Classical Groups [Russian translation], Mir, Moscow (1974).

    Google Scholar 

  35. A. E. Zalessky, “Linear groups,” Usp. Mat. Nauk, 36,5, 56–107 (1981).

    Google Scholar 

  36. A. E. Zalessky, “Linear groups,” in: Progress in Science. Algebra, Topology, Geometry, 21, Moscow (1985), pp. 135–182.

    Google Scholar 

  37. V. A. Koibaev, “On subgroups of the general linear group that contain the group of elementary, block diagonal matrices,” Vestn. Leningr. Univ., 15, 169–177 (1982).

    Google Scholar 

  38. A. S. Kondrat'ev, “Subgroups of finite Chevalley groups,” Usp. Mat. Nauk, 41,1, 57–96 (1986).

    Google Scholar 

  39. V. I. Kopeiko, “Stabilization of symplectic groups over a ring of polynomials,” Mat. Sb., 106,1, 94–107 (1987).

    Google Scholar 

  40. A. V. Stepanov, “Stabilization conditions in the theory of linear groups,” Ph. D. Thesis, Leningrad (1987).

  41. A. A. Suslin, “On the structure of the special linear group over a ring of polynomials,” Izv. Akad. Nauk SSSR, Ser. Mat., 41,2, 235–253 (1977).

    Google Scholar 

  42. A. A. Suslin, “On a theorem of Cohn,” Zap. Nauchn. Semin. LOMI, 64, 127–130 (1976).

    Google Scholar 

  43. A. A. Suslin and A. I. Kopeiko, “Quadratic modules and orthogonal groups over rings of polynomials,” Zap. Nauchn. Semin. POMI, 71, 216–250 (1977).

    Google Scholar 

  44. M. Aschbacher, “On the maximal subgroups of the finite classical groups,” Invent. Math., 76, 469–514 (1984).

    Google Scholar 

  45. H. Bass, “Clifford algebras and spinor norms over a commutative ring,” Amer. J. Math., 96, 156–206 (1974).

    Google Scholar 

  46. R. W. Carter, Simple Groups of Lie Yype, Wiley, London (1972).

    Google Scholar 

  47. R. W. Carter, Finite Groups of Lie Type: Conjugacy Classes and Complex Characters, Wiley, London (1985).

    Google Scholar 

  48. A. Hahn and O. T. O'Meara, The Classical Groups and K-Theory, Springer, Berlin (1989).

    Google Scholar 

  49. J.-Y. Hée, “Groupes de Chevalley et groupes classiques,” Publ. Math. Univ. Paris VII, 17, 1–54 (1984).

    Google Scholar 

  50. W. Kantor, “Subgroups of classical groups generated by long root elements,” Trans. Amer. Math. Soc., 248, 347–379 (1979).

    Google Scholar 

  51. O. H. King, “On some maximal subgroups of the classical groups,” J. Algebra, 68, 109–120 (1981).

    Google Scholar 

  52. O. H. King, “Maximal subgroups of the special classical groups associated with nonisotropic subspaces of a vector space,” J. Algebra, 73, 350–375 (1981).

    Google Scholar 

  53. O. H. King, “Imprimitive maximal subgroups of the orthogonal, special orthogonal, unitary, and special unitary groups,” Math. Z., 182, 193–203 (1983).

    Google Scholar 

  54. O. H. King, “Imprimitive maximal subgroups of the symplectic, orthogonal, and unitary groups,” Geom. Dedic., 15, 339–353 (1984).

    Google Scholar 

  55. O. H. King, “On subgroups of the special linear group containing the special orthogonal group,” J. Algebra, 96, 178–193 (1985).

    Google Scholar 

  56. P. Kleidman and M. Liebeck, “A survey of the maximal subgroups of the finite simple groups,” Geom. Dedic., 25, 375–389 (1988).

    Google Scholar 

  57. P. Kleidman and M. Liebeck, The Subgroup Structure of the Finite Classical Groups, Cambridge Univ. Press (1990).

  58. Li Shang Zhi, “Maximal subgroups containing root subgroups in finite classical groups,” Kexue Tongbao, 29, 14–18 (1984).

    Google Scholar 

  59. Li Shang Zhi, “Maximal subgroups of PΩ(n, F, Q) with root subgroups,” Scientia Sinica, Ser. A, 28, 826–838 (1990).

    Google Scholar 

  60. Li Shang Zhi, “Maximal subgroups in classical groups over arbitrary fields,” Proc. Symp. Pure Math., 47, Part 2, 487–493 (1987).

    Google Scholar 

  61. B. R. McDonald, Geometric Algebra over Local Rings, Marcel Dekker, New York-Basel (1976).

    Google Scholar 

  62. B. R. McDonald, Linear Algebra over Commutative Rings, Marcel Dekker, New York-Basel (1984).

    Google Scholar 

  63. B. S. Stark, “Some subgroups of Ω(V) generated by groups of root type 1,” Illinois J. Math., 17, 584–607 (1973).

    Google Scholar 

  64. B. S. Stark, “Some subgroups of Ω(V) generated by groups of root type,” J. Algebra, 29, 33–41 (1974).

    Google Scholar 

  65. B. S. Stark, “Irreducible subgroups of orthogonal groups generated by groups of root type 1,” Pacif. J. Math., 53, 611–625 (1974).

    Google Scholar 

  66. K. Suzuki, “On parabolic subgroups of Chevalley groups over local rings,” Tôhoku Math. J., 28, 57–66 (1976).

    Google Scholar 

  67. K. Suzuki, “On parabolic subgroups of Chevalley groups over commutative rings,” Sci. Repts Tokyo Daigaku, A13,Nos. 366–382, 225–232 (1977).

    Google Scholar 

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

    Google Scholar 

  69. N. A. Vavilov, “On subgroups of split orthogonal groups in even dimension,” Bull. Acad. Polon. Sci., Ser. Sci. Math., 29,Nos. 9–10, 425–429 (1981).

    Google Scholar 

  70. A. Bak and N. A. Vavilov, “Structure of hyperbolic unitary groups. I. Elementary subgroups,” Algebra Colloquium, 7, 159–196 (2000).

    Google Scholar 

  71. E. V. Dybkova, “On net subgroups of hyperbolic unitary groups,” Algebra Analiz, 9,4, 79-(1997).

    Google Scholar 

  72. E. V. Dybkova, “Overdiagonal subgroups of hyperbolic unitary groups over a good form ring over a field,” Zap. Nauchn. Semin. POMI, 236, 87–96 (1996).

    Google Scholar 

  73. E. V. Dybkova, “Form nets and lattices of overdiagonal subgroups in the symplectic group over a field of characteristic 2,” Algebra Analiz, 10,4, 113–129 (1998).

    Google Scholar 

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

    Google Scholar 

  75. N. A. Vavilov, “Intermediate subgroups in Chevalley groups,” in: Proceeding of the Conference on Groups of Lie Type and Their Geometries, Como-1993 (1995), pp. 233–280.

  76. N. A. Vavilov and E. B. Plotkin “Chevalley groups over commutative rings. I. Elementary calculations,” Acta Appl. Math., 45, 73–113 (1996).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. St.Petersburg State University, Russia

    N. A. Vavilov

Authors
  1. N. A. Vavilov
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Vavilov, N.A. Subgroups of Split Orthogonal Groups over a Commutative Ring. Journal of Mathematical Sciences 120, 1501–1512 (2004). https://doi.org/10.1023/B:JOTH.0000017881.22871.49

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/B:JOTH.0000017881.22871.49

Keywords

Search

Navigation