Skip to main content
SpringerLink
Log in
Book cover

Algebraic Combinatorics and Applications pp 157–195Cite as

Computing Resolutions Over Finite p-Groups

  • Conference paper

Abstract

A uniform and constructive approach for the computation of resolutions and for (co)homology computations for any finite p-group is detailed. The resolutions we construct ([32]) are, as vector spaces, as small as the minimal resolution of IFp over the elementary abelian p-group of the same order as the group under study. Our implementations are based on the development of sophisticated algebraic data structures. Applications to calculating functional cocycles are given and the possibility of constructing interesting codes using such methods is presented.

Dedicated to Professor Adalbert Kerber at the occasion of his 60th birthday.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Donald W. Barnes and Larry A. Lambe. A fixed point approach to homological perturbation theory. Proc. Amer. Math. Soc., 112 (3): 881–892, 1991.

    Article  MathSciNet  MATH  Google Scholar 

  2. R. Brown. The twisted Eilenberg-Zilber theorem. In Simposio di Topologia (Messina, 1964), pages 33–37. Edizioni Oderisi, Gubbio, 1965.

    Google Scholar 

  3. Henri Cartan and Samuel Eilenberg. Homological algebra. Princeton University Press, Princeton, N. J., 1956.

    Google Scholar 

  4. Henri Cartan, J. C. Moore, R. Thom, and J-P. Serre. Algèbras d’Eilenberg- MacLane et homotopie. 2ieme ed., revue et corrig ee. Secretariat mathematique ( Hektograph ), Paris, 1956.

    Google Scholar 

  5. Torsten Ekedahl, Johannes Grabmeier, and Larry Lambe. A Generic Language for algebraic computations, 2000. In preparation.

    Google Scholar 

  6. G. Ellis and I. Kholodna. Second cohomology of finite groups with trivial coefficients. Homology, Homotopy & Appl, 1: 163–168, 1999.

    MathSciNet  MATH  Google Scholar 

  7. D. L. Flannery. Transgression and the calculation of cocyclic matrices. Aus- tralas. J. Combin., 11: 67–78, 1995.

    Google Scholar 

  8. D. L. Flannery. Calculation of cocyclic matrices. J. Pure Appl. Algebra, 112 (2): 181 — 190, 1996.

    Article  MathSciNet  MATH  Google Scholar 

  9. D. L. Flannery, K.J. Horadam, and W. de Launey. Cocyclic hadamard matrices and difference sets. Discrete Appl Math., 102: 47–61, 2000.

    Google Scholar 

  10. D. L. Flannery and E. A. O’Brien. Computing 2-cocycles for central extensions and relative difference sets. Comm. Algebra, 28 (4): 1939–1955, 2000.

    Article  MathSciNet  MATH  Google Scholar 

  11. R.L. Graham, D.E. Knuth, and O. Patashnik. Concrete Mathematics. Addison- Wesley, Reading, Massachusetts, 1989.

    Google Scholar 

  12. V. K. A. M. Gugenheim. On the chain-complex of a fibration. Illinois J. Math., 16: 398–414, 1972.

    MathSciNet  MATH  Google Scholar 

  13. V. K. A. M. Gugenheim and L. A. Lambe. Perturbation theory in differential homological algebra. I. Illinois J. Math., 33 (4): 566 - 582, 1989.

    Google Scholar 

  14. V. K. A. M. Gugenheim, L. A. Lambe, and J. D. Stasheff. Perturbation theory in differential homological algebra. II. Illinois J. Math., 35 (3): 357–373, 1991.

    Google Scholar 

  15. K. J. Horadam and W. de Launey. Cocyclic development of designs. J. Algebraic Combin., 2 (3): 267–290, 1993.

    Google Scholar 

  16. K. J. Horadam and W. de Launey. Erratum: “Cocylic development of designs”. J. Algebraic Combin., 3(1):129, 1994.

    Google Scholar 

  17. K. J. Horadam and W. de Launey. Generation of cocyclic Hadamard matrices. In Computational algebra and number theory (Sydney, 1992), volume 325 of Math. Appl, pages 279-290. Kluwer Acad. Publ., Dordrecht, 1995.

    Google Scholar 

  18. K. J. Horadam and A. A. I. Perera. Codes from cocycles. In Lecture Notes in Computer Science, volume 1255, pages 151 - 163. Springer-Verlag, Berlin- Heidelberg-New York, 1997.

    Google Scholar 

  19. Johannes Huebschmann. The homotopy type of FΨq. The complex and sym- plectic cases. In Applications of algebraic K-theory to algebraic geometry and number theory, Part I, II (Boulder, Colo., 1983), pages 487-518. Amer. Math. Soc., Providence, R.I., 1986.

    Google Scholar 

  20. Johannes Huebschmann. Perturbation theory and free resolutions for nilpotent groups of class 2. J. Algebra, 126 (2): 348 - 399, 1989.

    Article  MathSciNet  MATH  Google Scholar 

  21. Johannes Huebschmann and Tornike Kadeishvili. Small models for chain algebras. Math. Z., 207 (2): 245 - 280, 1991.

    Google Scholar 

  22. Bertram Huppert and Norman Blackburn. Finite groups. II. Springer-Verlag, Berlin-New York, 1982. Grundlehren der Mathematischen Wissenschaften, Band 242.

    Google Scholar 

  23. N. Jacobson. Restricted Lie algebras of characteristic p. Trans. Amer. Math. Soc50: 15–25, 1941.

    Google Scholar 

  24. Richard D. Jenks and Robert S. Sutor. Axiom. The scientific computation system. Springer-Verlag, Berlin, Heidelberg, New York, 1992.

    Google Scholar 

  25. S. A. Jennings. The structure of the group ring of a p-group over a modular field. Trans. Amer. Math. Soc., 50: 175 – 185, 1941.

    MathSciNet  Google Scholar 

  26. Leif Johansson and Larry Lambe. Transferring algebra structures up to homology equivalence. Math. Scand., 88 (2), 2001.

    Google Scholar 

  27. Leif Johansson, Larry Lambe, and Emil Sköldberg. On constructing resolutions over the polynomial algebra, 2000. Preprint.

    Google Scholar 

  28. Larry Lambe. Next generation computer algebra systems AXIOM and the scratchpad concept: applications to research in algebra. In Analysis, algebra, and computers in mathematical research (Lulea, 1992), volume 156 of Lecture Notes in Pure and Appl. Math., pages 201 - 222. Dekker, New York, 1994.

    Google Scholar 

  29. Larry Lambe. The 1996 Adams Lectures at Manchester University: New computational methods in algebra and topology, May 20 1996.

    Google Scholar 

  30. Larry Lambe and Jim Stasheff. Applications of perturbation theory to iterated fibrations. Manuscripta Math., 58 (3): 363 - 376, 1987.

    Article  MathSciNet  MATH  Google Scholar 

  31. Larry A. Lambe. Resolutions via homological perturbation. J. Pure Appl. Algebra, 12: 71–87, 1991.

    MATH  Google Scholar 

  32. Larry A. Lambe. Homological perturbation theory, Hochschild homology, and formal groups. In Deformation theory and quantum groups with applications to mathematical physics (Amherst, MA, 1990), volume 134 of Contemp. Math., pages 183 - 218. Amer. Math. Soc., Providence, RI, 1992.

    Google Scholar 

  33. Larry A. Lambe. Resolutions which split off of the bar construction. J. Pure Appl. Algebra, 84 (3): 311 - 329, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  34. Saunders Mac Lane. Homology. Classics in Mathematics. Springer-Verlag, Berlin, 1995. Reprint of the 1975 edition.

    Google Scholar 

  35. J. Peter May. The cohomology of restricted Lie algebras and of Hopf algebras. Bull. Amer. Math. Soc., 71: 372 - 377, 1965.

    Article  MathSciNet  MATH  Google Scholar 

  36. D. Quillen. On the associated graded ring of a group ring. J. Algebra, 10:411- 418, 1968.

    Google Scholar 

  37. Jean-Pierre Serre. Lie algebras and Lie groups. 1964 lectures, given at Harvard University, 2nd ed., volume 1500 of Lecture Notes in Mathematics. Springer- Verlag, Berlin-Heidelberg-New York, 1992.

    Google Scholar 

  38. C.T.C. Wall. Resolutions for extensions of groups. Proc. Phil. Soc., 57: 251 - 255, 1961.

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. IBM Deutschland Informationssysteme GmbH, Postfach 103068, D-69020, Heidelberg, Germany

    Johannes Grabmeier

  2. Centre for Innovative Computation University of Wales, Bangor, Gwynedd, LL57 1UT, UK

    Larry A. Lambe

Authors
  1. Johannes Grabmeier
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Larry A. Lambe
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Lehrstuhl II für Mathematik, Universität Bayreuth, 95440, Bayreuth, Germany

    Anton Betten , Axel Kohnert , Reinhard Laue  & Alfred Wassermann , ,  & 

Rights and permissions

Reprints and permissions

Copyright information

© 2001 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Grabmeier, J., Lambe, L.A. (2001). Computing Resolutions Over Finite p-Groups. In: Betten, A., Kohnert, A., Laue, R., Wassermann, A. (eds) Algebraic Combinatorics and Applications. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-59448-9_12

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-59448-9_12

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-41110-9

  • Online ISBN: 978-3-642-59448-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation