Algebras and Representation Theory

, Volume 3, Issue 4, pp 303–335

Torsion Endo-Trivial Modules

  • Jon F. Carlson
  • Jacques Thévenaz


We prove that the group T(G) of endo-trivial modules for a noncyclic finite p-group G is detected on restriction to the family of subgroups which are either elementary Abelian of rank 2 or (almost) extraspecial. This result is closely related to the problem of finding the torsion subgroup of T(G). We give the complete structure of T(G) when G is dihedral, semi-dihedral, or quaternion.

endo-permutation module Dade group group algebras endo-trivial modules modular group representations 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [Al1]
    Alperin, J. L.: construction of endo-permutation modules, Preprint, 1999.Google Scholar
  2. [Al2]
    Alperin, J. L.: Lifting endo-trivial modules, Preprint, 1999.Google Scholar
  3. [Be]
    Benson, D. J.: Representations and Cohomology I, II, Cambridge Univ. Press, 1991.Google Scholar
  4. [BeCa]
    Benson, D. J. and Carlson, J. F.: The cohomology of extraspecial groups, Bull. London Math. Soc. 24 (1992), 209–235.Google Scholar
  5. [BoCa]
    Bosma, W. and Cannon, J.: Handbook of Magma Functions, Magma Computer Algebra, Sydney, 1999.Google Scholar
  6. [Bo]
    Bouc, S.: Tensor induction of relative syzygies, J. Reine angew. Math. (Crelle) 523 (2000), 113–171.Google Scholar
  7. [BoTh]
    Bouc, S. and Thévenaz, J.: The group of endo-permutation modules, Invent. Math. 139 (2000), 275–349.Google Scholar
  8. [Ca1]
    Carlson, J. F.: Endo-trivial modules over (p, p)-groups, Illinois J. Math. 24 (1980), 287–295.Google Scholar
  9. [Ca2]
    Carlson, J. F.: Induction from elementary Abelian subgroups, Quarterly J. Math. 51 (2000), 169–181.Google Scholar
  10. [Ca3]
    Carlson, J. F.: A characterization of endo-trivial modules over p-groups, Manuscripta Math. 97 (1998), 303–307.Google Scholar
  11. [CaRo]
    Carlson, J. F. and Rouquier, R.: Self-equivalences of stable module categories, Math. Z. 233 (2000), 165–178.Google Scholar
  12. [Ch]
    Chouinard, L.: Projectivity and relative projectivity over group rings, J. Pure Appl. Algebra 7 (1976), 278–302.Google Scholar
  13. [Da1]
    Dade, E. C.: Une extension de la théorie de Hall et Higman, J. Algebra 20 (1972), 570–609.Google Scholar
  14. [Da2]
    Dade, E. C.: Endo-permutation modules over p-groups, I, II, Ann. Math. 107 (1978), 459– 494, 108 (1978), 317–346.Google Scholar
  15. [Ev]
    Evens, L.: The Cohomology of Groups, Oxford University Press, 1991.Google Scholar
  16. [Pu1]
    Puig, L.: Affirmative answer to a question of Feit, J. Algebra 131 (1990), 513–526.Google Scholar
  17. [Pu2]
    Puig, L.: Une correspondance de modules pour les blocs à groupes de défaut abéliens, Geom. Dedicata 37 (1991), 9–43.Google Scholar
  18. [Se]
    Serre, J. P.: Corps locaux, Hermann, Paris, 1968.Google Scholar
  19. [Th1]
    Thévenaz, J.: G-algebras and Modular Representation Theory, Oxford University Press, 1995.Google Scholar
  20. [Th2]
    Thévenaz, J.: Representations of finite groups in characteristic p r, J. Algebra 72 (1981), 478–500.Google Scholar

Copyright information

© Kluwer Academic Publishers 2000

Authors and Affiliations

  • Jon F. Carlson
    • 1
  • Jacques Thévenaz
    • 2
  1. 1.Department of MathematicsUniversity of GeorgiaAthensU.S.A.
  2. 2.Institut de MathématiquesUniversité de LausanneLausanneSwitzerland. e-mail

Personalised recommendations