On the Use of Gröbner Bases for Computing the Structure of Finite Abelian Groups

  • M. Borges-Quintana
  • M. A. Borges-Trenard
  • E. Martínez-Moro
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3718)


Some algorithmic properties are obtained related with the computation of the elementary divisors and a set of canonical generators of a finite abelian group, this properties are based on Gröbner bases techniques used as a theoretical framework. As an application a new algorithm for computing the structure of the abelian group is presented.


Abelian Group Great Common Divisor Elementary Divisor Residue Number System Canonical Structure 
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  1. 1.
    Borges Quintana, M.: On some Gröbner Bases Techniques and their Applications (Spanish). Phd Thesis. Universidad de Oriente, Santiago de Cuba, Cuba (2002)Google Scholar
  2. 2.
    Buchberger, B., Winkler, F.: Gröbner Bases and Applications. In: Proc. of the International Conference 33 Years of Gröbner Bases. London Mathematical Society Series, vol. 251. Cambridge University Press, Cambridge (1998)Google Scholar
  3. 3.
    Buchberger, B.: Introduction to Gröbner Bases. In: [2], pp. 3–31 (1998)Google Scholar
  4. 4.
    Buchmann, J., Jacobson Jr., M.J., Teske, E.: On Some Computational Problems in Finite Abelian Groups. Math. Comput. 66(220), 1663–1687 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Cohen, H.: A Course in Computational Algebraic Number Theory (3rd corrected printing), New York. Graduate Texts in Mathematics, vol. 138. Springer, Heidelberg (1996)Google Scholar
  6. 6.
    Cohen, H., Díaz y Díaz, F., Olivier, M.: Algorithmic Methods for Finitely Generated Abelian Groups. J. Symbolic Computation 31(1-2), 133–147 (2001)zbMATHCrossRefGoogle Scholar
  7. 7.
    Dumas, J.G., Saunders, B.D., Villard, G.: On Efficient Sparse Integer Matrix Smith Normal Form Computations. J. Symbolic Computation 32(1-2), 71–99 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Eberly, W., Giesbrecht, M.W., Villard, G.: Computing the determinant and Smith form of an integer matrix. In: The 41st Annual IEEE Symposium on Foundations of Computer Science, Redondo Beach, CA (2000)Google Scholar
  9. 9.
    Havas, G., Majewski, B.S.: Integer Matrix Diagonalization. J. Symbolic Computation 24(3-4), 399–408 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Iliopoulos, C.S.: Worst-Case Complexity Bounds on Algorithms for Computing the Canonical Structure of Finite Abelian Groups and the Hermite and Smith Normal Forms of an Integer Matrix. Siam J. Comput. 18/4, 658–669 (1989)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Lübeck, F.: On the Computation of Elementary Divisors of Integer Matrices. J. Symbolic Computation 33(1), 57–65 (2002)zbMATHCrossRefGoogle Scholar
  12. 12.
    Mora, T.: An Introduction to Commutative and Noncommutative Gröbner Bases. Theoretical Computer Science 134, 131–173 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Storjohann, A.: Near optimal algorithms for computing Smith normal forms of integer matrices. In: Lakshman, Y.N. (ed.) ISSAC 1996: Proceedings of the 1996 International Symposium on Symbolic and Algebraic Computation, Zurich, Switzerland, July 24-26, 1996, pp. 267–274. ACM Press, New York (1996)CrossRefGoogle Scholar
  14. 14.
    Storjohann, A.: Algorithms for Matrix Canonical Forms. Ph.D. Thesis, Institut für Wissenschaftliches Rechnen, ETH-Zentrum, Zürich, Switzerland (2000)Google Scholar
  15. 15.
    Terras, A.: Fourier analysis on finite groups and applications. LMS Student Texts, 43. Cambridge University Press, Cambridge (1999)Google Scholar
  16. 16.
    Teske, E.: A Space Efficient Algorithm for Group Structure Computation. Math. Comput. 67, 1637–1663 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Teske, E.: The Pohlig-Hellman Method Generalized for Group Structure Computation. J. Symbolic Computation 27(6), 521–534 (1999)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • M. Borges-Quintana
    • 1
  • M. A. Borges-Trenard
    • 1
  • E. Martínez-Moro
    • 2
  1. 1.Dpto. de MatemáticaFCMC, U. de OrienteSantiago de CubaCuba
  2. 2.Dpto. de Matemática AplicadaU. de ValladolidSpain

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