Generators of Automorphism Groups of Modules

  • H. Ishibashi
Part of the NATO ASI Series book series (ASIC, volume 333)


This note consists of two parts. In the first part we treat the length problem for unitary groups and symplectic groups over a quasisemilocal semihereditary ring. We treat orthogonal groups over valuation rings and quasisemilocal semihereditary domains. Further, we find some minimal or small sets of generators for these groups over finite fields. Throughout part one we assume that the rings have 2 as a unit and the modules on which these groups act are nonsingular.

In the second part, we treat the symplectic groups Sp(V) on a free module V of rank n over a local ring R with an alternating bilinear map \(f:V\times Vn \to R\) Define \(M=\left\{{x\epsilon V|f(x,V)=R}\right\}\) and for \(\theta \neq S\subseteq M\) let TR(S) denote the subgroup generated by all transvections with axis s⊥ for s in S. Then T R(M) = Sp(V) if f is nonsingular, and we find necessary and sufficient conditions for S to satisfy TR(S) = T R(M). Further, we determine the smallest number of elements of a subset S’ of S such that T R(S′) = TR(M). Throughout part two, we do not assume that R has 2 as a unit, neither do we assume that V is nonsingular.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Artin, E. ”Geometric Algebra”, Interscience, New York, 1957.zbMATHGoogle Scholar
  2. 2.
    Bachmann, F. ”Aufbau der Geometrie aus dem Spiegelungsbegriff”, 2nd ed. Springer, New York, 1973.zbMATHCrossRefGoogle Scholar
  3. 3.
    Brown, R. and Humphries, S. P. ”Orbits under symplectic transvecti-ons I”, Proceedings of London Math. Soc. 52 (1986), 517–531.MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Brown, R. and Humphries, S. P. ”Orbits under symplectic transvections II”, Proceedings of London Math. Soc. 52 (1986), 532–555.MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chang, C. ”The structures of symplectic groups over semilocal domains”, J. Algebra 35 (1975), 457–476.MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Dieudonné, J. ”La géométrie des groupes classiques”, Springer, Berlin, 1955.zbMATHGoogle Scholar
  7. 7.
    Dieudonné, J. ”Sur les générateurs des groupes classiques”, Summa Bras. Math. 3 (1955), 149–178.Google Scholar
  8. 8.
    Djoković, D. Ž. and Malzan, J. ”Products of reflections in the general linear group over a division ring”, Linear Algebra Appl. 28 (1979), 53–62.MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Dye, R. H. ”Symmetric groups as maximal subgroups of orthogonal and symplectic groups over the field of two elements”, J. London Math. Soc. 20 (1979), 227–237.MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Ellers, E. W. ”Decomposition of equiaffinities into reflections”, Geom. Dedi-cata 6 (1977), 297–304.MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Ellers, E. W. ”Products of axial affinities and products of central collinea-tions”, The Geometric Vein, Springer, New York (1982), 465–470.Google Scholar
  12. 12.
    Ellers, E. W. and Ishibashi, H. ”Factorizations of Transformations over a Valuation Ring”, Linear Algebra and Its Applications 85 (1987), 17–27.MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Götzky, M. ”Über die Erzeugenden der engeren unitären Gruppen”, Arch. Math. XIX (1968), 383–389.CrossRefGoogle Scholar
  14. 14.
    Hahn, A. J. and O’Meara, O. T. ”The Classical Groups and K-theory”, Springer-Verlag, 1989.Google Scholar
  15. 15.
    Hall, J. I. ”Symplectic geometry and mapping class groups”, Geometrical Combinatorics (ed. Holroyd, F. D. and Wilson, R. J.), Research Notes in Mathematics 114, Pitman, London (1985), 21–33.Google Scholar
  16. 16.
    Ishibashi, H. ”On Some System of Generators of the Orthogonal Groups”, Science Reports of The Tokyo Kyoiku Daigaku, Sect. A, 11 (1972), No. 287, 96–105.MathSciNetzbMATHGoogle Scholar
  17. 17.
    Ishibashi, H. ”Generators of O n(V) over a Quasi-Semilocal Semihereditary Domain”, Communications in Algebra 7 (1979), 1043–1064.MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Ishibashi, H. ”Decomposition of Isometries of U n(V) over Finite Fields into Simple Isometries”, Czechoslovak Mathematical Journal 31 (1981), 301–305.MathSciNetGoogle Scholar
  19. 19.
    Ishibashi, H. ”Generators of Sp n(V) over a Quasi-Semilocal Semihereditary Ring”, Journal of Pure and Applied Algebra 22 (1981), 121–129.MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Ishibashi, H. ”Generators of Orthogonal Groups over Valuation Rings”, Canadian Journal of Mathematics 33 (1981), 116–128.MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Ishibashi, H. ”Generators of U n(V) over a Quasi-Semilocal Semihereditary Ring”, Canadian Journal of Mathematics 33 (1981), 1232–1244.MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Ishibashi, H. ”Small Systems of Generators of Isotropic Unitary Groups over Finite Fields of Characteristic not Two”, Journal of Algebra 93 (1985), 324–331.MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Ishibashi, H. ”Simple Coefficient Extensions of Symplectic Groups”, Communications in Algebra 9 (1981), 47–65.MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Ishibashi, H. ”Generation of symplectic groups by transvections over local rings with at least 3 residue classes”, Journal of Algebra 112 (1988), 151–158.MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Ishibashi, H. ”Necessary and sufficient condition for certain subsets of transvections to generate a subgroup of the symplectic group over local rings”, Bulletin of the London Math. Soc. 21 (1989), 551–556.MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Kantor, W. ”Classical Groups from a Non-classical Viewpoint”, Lecture Notes, Mathematical Institute, Oxford, 1978.Google Scholar
  27. 27.
    Klingenberg, W. ”Orthogonale Gruppen über lokalen Ringen”, Amer. J. Math. 83 (1961), 281–320.MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    McDonald, B. R. ”Geometric Algebra over Local Ringss”, Marcel Dekker, New York, 1976.Google Scholar
  29. 29.
    McDonald, B. R. ”Linear Algebraover Commutative Rings”, Marcel Dekker, New York, 1984.Google Scholar
  30. 30.
    McLaughlin, J. ”Some subgroups of SL n(F 2)”, Illinois J. Math. 13 (1969), 108–115.MathSciNetzbMATHGoogle Scholar
  31. 31.
    Orlik, P. and Solomon, L. ”A character formula for the unitary group over a finite field”, J. Algebra 84 (1983), 136–141.MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Scherk, P. ”On the decomposition of orthogonalities into symmetries”, Proc. Amer. Math. Soc. 1 (1950), 481–491.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1991

Authors and Affiliations

  • H. Ishibashi
    • 1
  1. 1.Department of MathematicsJosai UniversitySakado, SaitamaJapan

Personalised recommendations