Mathematische Zeitschrift

, Volume 268, Issue 1–2, pp 441–476 | Cite as

The structure of blocks with a Klein four defect group

  • David A. Craven
  • Charles W. Eaton
  • Radha Kessar
  • Markus LinckelmannEmail author


We prove Erdmann’s conjecture (J Algebra 76:505–518, 1982) stating that every block with a Klein four defect group has a simple module with trivial source, and deduce from this that Puig’s finiteness conjecture holds for source algebras of blocks with a Klein four defect group. The proof uses the classification of finite simple groups.


Normal Subgroup Conjugacy Class Simple Group Irreducible Character Defect Group 
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  1. 1.
    Berger, T.R.: Irreducible modules of solvable groups are algebraic. In: Proceedings of the Conference on Finite Groups, pp. 541–553. Univ. of Utah, Park City, UT, 1975 (1976)Google Scholar
  2. 2.
    Berger T.R.: Solvable groups and algebraic modules. J. Algebra 57, 387–406 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Boltje R., Külshammer B.: The ring of modules with endo-permutation source. Manuscripta Math. 120, 359–376 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Bonnafé, C.: Sur les caractères des groupes réductifs finis à centre non connexe: applications aux groupes spéciaux linéaires et unitaires. Astérisque, vol. vi, p. 165 (2006)Google Scholar
  5. 5.
    Borel A.: Linear Algebraic Groups. Graduate texts in mathematics, vol. 126. Springer, New York (1991)Google Scholar
  6. 6.
    Bourbaki N.: Lie Groups and Lie Algebras, Chap. 4–6. Springer, New York (2002)Google Scholar
  7. 7.
    Breuer, T.: Manual for the GAP Character Table Library, Version 1.1, Lehrstuhl D für Mathematik, Rheinisch Westfälische Technische Hochschule, Aachen, Germany (2004)Google Scholar
  8. 8.
    Cabanes M.: Extensions of p-groups and construction of characters. Comm. Algebra 15, 1297–1311 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Cabanes M., Enguehard M.: Unipotent blocks of finite reductive groups of a given type. Math. Z. 213, 479–490 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Cabanes M., Enguehard M.: Representation theory of finite reductive groups. New Mathematical Monographs. Cambridge University Press, Cambridge (2004)CrossRefGoogle Scholar
  11. 11.
    Carter R.: Finite Groups of Lie Type: Conjugacy Classes and Complex Characters. John Wiley & Sons Ltd, Chichester (1993)Google Scholar
  12. 12.
    Conway J.H., Curtis R.T., Norton S.P., Parker R.A., Wilson R.A.: Atlas of Finite Groups. Oxford University Press, Eynsham (1985)zbMATHGoogle Scholar
  13. 13.
    Curtis C.W.: The Steinberg character of a finite group with a (BN)-pair. J. Algebra 4, 433–441 (1966)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Deriziotis D.I.: The centralizers of semisimple elements of the Chevalley groups E 7 and E 8. Tokyo J. Math. 6, 191–216 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Digne F., Michel J.: Representations of finite groups of Lie type. London Mathematical Society Student Texts, vol. 21. Cambridge University Press, Cambridge (1991)Google Scholar
  16. 16.
    Erdmann K.: Blocks whose defect groups are Klein four groups: a correction. J. Algebra 76, 505–518 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Feit, W.: Irreducible modules of p-solvable groups. In: The Santa Cruz Conference on Finite Groups, pp. 405–411. Univ. California, Sant Cruz, CA, 1979 (1980)Google Scholar
  18. 18.
    Feit, W., Zuckerman, G.: Reality properties of conjugacy classes in spin groups and symplectic groups. In: Algebraists’ Homage: Papers in Ring Theory and Related Topics, pp. 239–253. New Haven, CT, 1981 (1982)Google Scholar
  19. 19.
    Fong P., Srinivasan B.: The blocks of finite general and unitary groups. Invent. Math. 69, 109–153 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    The GAP group, GAP–Groups, Algorithms, Programming.
  21. 21.
    Geck M.: Character values, Schur indices and character sheaves. Represent. Theory 7, 19–55 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Gorenstein D., Lyons R., Solomon R.: The classification of finite simple groups, Number 3. Mathematical Surveys and Monographs, vol. 40. American Mathematical Society, Providence (1998)Google Scholar
  23. 23.
    Gow R.: Real representations of the finite orthogonal and symplectic groups of odd characteristic. J. Algebra 96, 249–274 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Hiss G.: Regular and semisimple blocks of finite reductive groups. J. Lond. Math. Soc. (2) 41, 63–68 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Hiss G., Kessar R.: Scopes reduction and Morita equivalence classes of blocks in finite classical groups. J. Algebra 230, 378–423 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Hiss G., Kessar R.: Scopes reduction and Morita equivalence classes of blocks in finite classical groups II. J. Algebra 283, 522–563 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Hiss G., Shamash J.: 2-blocks and 2-modular characters of the Chevalley groups G 2(q). Math. Comput. 59, 645–672 (1992)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Iwahori, N.: Centralizers of involutions in finite Chevalley groups. In: Seminar on Algebraic Groups and Related Finite Groups, pp. 267–295. The Institute for Advanced Study, Princeton, NJ, 1968/69 (1970)Google Scholar
  29. 29.
    Kessar R.: Blocks and source algebras for the double covers of the symmetric and alternating groups. J. Algebra 186, 872–933 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Kessar R.: Equivalences for blocks of the Weyl groups. Proc. Am. Math. Soc. 128, 337–346 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Kessar R.: Source algebra equivalences for blocks of finite general linear groups over a fixed field. Manuscripta Math. 104, 145–162 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Kessar R.: Scopes reduction for blocks of finite alternating groups. Q. J. Math. 53, 443–454 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Kessar R.: The Solomon system \({\mathcal{F}_{\rm Sol}(3)}\) does not occur as fusion system of a 2-block. J. Algebra 296, 409–425 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Kessar R.: On duality inducing automorphisms and sources of simple modules in classical groups. J. Group Theory 12, 331–349 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Külshammer B.: Morita equivalent blocks in Clifford theory of finite groups. Astérisque 181–182, 209–215 (1990)Google Scholar
  36. 36.
    Landrock P.: The non-principal 2-blocks of sporadic simple groups. Comm. Algebra 6, 1865–1891 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Linckelmann M.: The source algebras of blocks with a Klein four defect group. J. Algebra 167, 821–854 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Linckelmann M.: The isomorphism problem for cyclic blocks and their source algebras. Invent. Math. 12, 265–283 (1996)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Narkiewicz W.: Elementary and Analytic Theory of Algebraic Numbers, 3rd edn. Springer, Berlin (2004)zbMATHGoogle Scholar
  40. 40.
    Navarro G., Tiep P.H., Turull A.: Brauer characters with cyclotomic field of values. J. Pure Appl. Algebra 212, 628–635 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Puig L.: On Joanna Scopes’ criterion of equivalence for blocks of symmetric groups. Algebra Colloq. 1, 25–55 (1994)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Riese, U.: 2-blocks of defect 1 in finite simple groups, Darstellungstheorietage Jena 1996, Sitzungsber. Math.-Naturwiss. Kl., 7, pp. 197–204. Akad. Gemein. Wiss. Erfurt (1996)Google Scholar
  43. 43.
    Salminen A.: On the sources of simple modules in nilpotent blocks. J. Algebra 319, 4559–4574 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Thévenaz J.: G-Algebras and Modular Representation. Theory Oxford Mathematical Monographs. Clarendon Press, Oxford (1995)Google Scholar
  45. 45.
    Thévenaz J.: Endo-permutation Modules: A Guided Tour. Group Representation Theory, pp. 115–147. EPFL press, Lausanne (2007)Google Scholar
  46. 46.
    Tiep P.H., Zalesski A.E.: Real conjugacy classes in algebraic groups and finite groups of Lie type. J. Group Theory 8, 291–315 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Vinroot C.R.: A factorization in GSp(V). Linear Multilinear Algebra 52, 385–403 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Ward H.: On Ree’s series of simple groups. Trans. Am. Math. Soc. 121, 62–89 (1966)zbMATHGoogle Scholar
  49. 49.
    Wonenburger M.J.: Transformations which are products of two involutions. J. Math. Mech. 16, 327–338 (1966)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • David A. Craven
    • 1
  • Charles W. Eaton
    • 2
  • Radha Kessar
    • 3
  • Markus Linckelmann
    • 3
    Email author
  1. 1.Mathematical InstituteUniversity of OxfordOxfordUK
  2. 2.School of MathematicsThe University of ManchesterManchesterUK
  3. 3.Institute of MathematicsKing’s College, University of AberdeenAberdeenScotland, UK

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