A Mathematica Notebook for Computing the Homology of Iterated Products of Groups

  • V. Álvarez
  • J. A. Armario
  • M. D. Frau
  • P. Real
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4151)


Let G be a group which admits the structure of an iterated product of central extensions and semidirect products of abelian groups G i (both finite and infinite). We describe a Mathematica 4.0 notebook for computing the homology of G, in terms of some homological models for the factor groups G i and the products involved. Computational results provided by our program have allowed the simplification of some of the formulae involved in the calculation of H n (G). Consequently the efficiency of the method has been improved as well. We include some executions and examples.


Abelian Group Homological Model Nilpotent Group Central Extension Semidirect Product 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
  2. 2.
    Álvarez, V., Armario, J.A., Frau, M.D., Real, P.: A genetic algorithm for cocyclic Hadamard matrices. In: Fossorier, M.P.C., Imai, H., Lin, S., Poli, A. (eds.) AAECC 2006. LNCS, vol. 3857, pp. 144–153. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  3. 3.
    Álvarez, V., Armario, J.A., Frau, M.D., Real, P.: (Co)homology of iterated products of semidirect products of abelian groups. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2006. LNCS, vol. 4194. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  4. 4.
    Álvarez, V., Armario, J.A., Frau, M.D., Real, P.: Comparison maps for relatively free resolutions. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2006. LNCS, vol. 4194, pp. 1–22. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  5. 5.
  6. 6.
    Dousson, X., Rubio, J., Sergeraert, F., Siret, Y.: The KENZO program. Institute Fourier, Grenoble (1998),
  7. 7.
    Dumas, J.G., Heckenbach, F., Saunders, D., Welker, V.: Computing simplicial homology based on efficient smith normal form algorithms. In: Algebra, Geometry and Software Systems, pp. 177–204. Springer, Heidelberg (2003)Google Scholar
  8. 8.
    Eilenberg, S., Mac Lane, S.: On the groups H(π, n) II. Annals of Math 66, 49–139 (1954)CrossRefMathSciNetGoogle Scholar
  9. 9.
    The GAP group, GAP- Group, Algorithms and programming, School of Mathematical and Computational Sciences, University of St. Andrews, Scotland (1998)Google Scholar
  10. 10.
    Grabmeier, J., Lambe, L.A.: Computing Resolutions Over Finite p-Groups. In: Betten, A., Kohnert, A., Lave, R., Wassermann, A. (eds.) Proceedings ALCOMA 1999. Lecture Notes in Computational Science and Engineering. Springer, Heidelberg (2000)Google Scholar
  11. 11.
    Ellis, G.: GAP package HAP, Homological Algebra Programming,
  12. 12.
    Huebschmann, J.: Cohomology of nilpotent groups of class 2. J. Algebra 126, 400–450 (1989)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Huebschmann, J.: Cohomology of metacyclic groups. Transactions of the American Mathematical Society 328(1), 1–72 (1991)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
  15. 15.
    Lambe, L.A.: Algorithms for the homology of nilpotent groups. In: Conf. on applications of computers to Geom. and Top., Lecture Notes in Pure and Applied Math., vol. 114, Marcel Dekker Inc., N.Y (1989)Google Scholar
  16. 16.
    Lambe, L.A.: Homological perturbation theory, Hochschild homology and formal groups. In: Proc. Conference on Deformation Theory and Quantization with Applications to Physics, Amherst, MA, June 1990. Cont. Math, vol. 134, pp. 183–218 (1992)Google Scholar
  17. 17.
    Lambe, L.A., Stasheff, J.D.: Applications of perturbation theory to iterated fibrations. Manuscripta Math. 58, 367–376 (1987)CrossRefMathSciNetGoogle Scholar
  18. 18.
    The MAGMA computational algebra system,
  19. 19.
    Real, P.: Homological Perturbation Theory and Associativity. Homology, Homotopy and Applications 2, 51–88 (2000)MATHMathSciNetGoogle Scholar
  20. 20.
    Rubio, J.: Integrating functional programming and symbolic computation. Mathematics and computers in simulation 44, 505–511 (1997)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Sergeraert, F.: The computability problem in Algebraic Topology. In: Advances in Math., vol. 1104, pp. 1–29 (1994)Google Scholar
  22. 22.
    Veblen, O.: Analisis situs, vol. 5. A.M.S. Publications, Providence, RI (1931)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • V. Álvarez
    • 1
  • J. A. Armario
    • 1
  • M. D. Frau
    • 1
  • P. Real
    • 1
  1. 1.Dpto. Matemática Aplicada IUniversidad de SevillaSevillaSpain

Personalised recommendations