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Acta Mathematica Sinica, English Series

, Volume 31, Issue 11, pp 1683–1702 | Cite as

Nonsolvable D 2-groups

  • Yang LiuEmail author
  • Zi Qun Lu
Article

Abstract

Let G be a finite group. Let Irr1(G) be the set of nonlinear irreducible characters of G and cd1(G) the set of degrees of the characters in Irr1(G). A group G is said to be a D 2-group if |cd1(G)| = |Irr1(G)| − 2. The main purpose of this paper is to classify nonsolvable D 2-groups.

Keywords

Character degree degree multiplicity nonsolvable group 

MR(2010) Subject Classification

20C15 20C33 

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References

  1. [1]
    Atlas of online. http://web.mat.bham.ac.uk/atlas/v2.0/Google Scholar
  2. [2]
    Berkovich, Y.: Finite solvable groups in which only two nonlinear irreducible characters have equal degrees. J. Algebra, 184, 584–603 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Berkovich, Y., Chillag, D., Herzog, M.: Finite groups in which the degrees of the nonlinear irreducible characters are distinct. Proc. Amer. Math. Soc., 115(4), 955–959 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Berkovich, Y., Isaacs, I. M., Kazarin, L.: Groups with distinct monolithic character degrees. J. Algebra, 216, 448–480 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    Berkovich, Y., Kazarin, L.: Finite nonsolvable groups in which only two nonlinear irreducible characters have equal degrees. J. Algebra, 184, 538–560 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Bianchi, M., Chillag, D., Lewis, M. L., et al.: Character degree graphs that are complete graphs. Proc. Amer. Math. Soc, 135(3), 671–676 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Carter, R. W.: Finite Groups of Lie Type: Conjugacy Classes and Complex Characters, Wiley, New York, 1985zbMATHGoogle Scholar
  8. [8]
    Connor, T., Leemans, D.: http://homepages.ulb.ac.be/ tconnor/atlaslat/Google Scholar
  9. [9]
    Conway, J. H., Curtis, R. T., Norton, S. P., et al.: Atlas of Finite Groups, Oxford University Press, Oxford, 1984Google Scholar
  10. [10]
    Craven, D.: Symmetric group character degrees and hook numbers. Proc. Lond. Math. Soc., 96(1), 26–50 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    Curtis, C. W., Reiner, I.: Representation Theory of Finite Groups and Associated Algebras, Interscience Publishers, New York, 1962Google Scholar
  12. [12]
    Dornhoff, L.: Group Representation Theory, Part A: Ordinary Representation Theory, Marcel Dekker, New York, 1971Google Scholar
  13. [13]
    GAP: http://www.gap-system.org/Google Scholar
  14. [14]
    Isaacs, I. M.: Character Theory of Finite Groups, Academic Press, New York, 1976zbMATHGoogle Scholar
  15. [15]
    Jansen, C., Lux, K., Parker, R., et al.: An Atlas of Brauer Characters, Clarendon Press, Oxford, 1995zbMATHGoogle Scholar
  16. [16]
    James, G., Kerber, A.: The Representation Theory of the Symmetric Group, Cambridge Univ. Press, Cambridge, 2009zbMATHGoogle Scholar
  17. [17]
    Liu, Y. J., Song, X. L., Xiong, H.: Almost simple DD-groups. Beijing Daxue Xuebao Ziran Kexue Ban, 49(5), 741–753 (2013)MathSciNetzbMATHGoogle Scholar
  18. [18]
    Malle, G.: Extensions of unipotent characters and the inductive Mckay condition. J. Algebra, 320, 2963–2980 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    Manz, O., Wolf, T.: Representations of Solvable Groups, Cambridge university press, Cambridge, 1993CrossRefzbMATHGoogle Scholar
  20. [20]
    Moretó, A.: Complex group algebras of finite groups: Brauer’s problem 1. Adv. Math., 208(1), 236–248 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    Nagao, H., Tsushima, Y.: Representations of Finite Groups, Academic Press, New York, 1988Google Scholar
  22. [22]
    Pfeiffer, G.: http://schmidt.nuigalway.ie/subgroups/alt.htmlGoogle Scholar
  23. [23]
    Qian, G. H., Wang, Y. M., Wei, H. Q.: Finite solvable groups with at most two nonlinear irreducible characters of each degree. J. Algebra, 320(8), 3172–3186 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    Schmid, P.: Extending the Steinberg representation. J. Algebra, 150, 254–256 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    Simpson, W. A., Frame, J. S.: The character tables of SL(3, q), SU(3, q), PSL(3, q), PSU(3, q 2). Canad. J. Math, 25, 486–494 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    Tong-viet, H. P.: Finite nonsolvable groups with many distinct character degrees. Pacific J. Math., 268(2), 477–492 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Beijing International Center for Mathematical ResearchPeking UniversityBeijingP. R. China
  2. 2.Department of MathematicsTsinghua UniversityBeijingP. R. China

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