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Finite groups whose irreducible characters of principal blocks have prime power degrees

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Abstract

In this note, we characterize finite nonsolvable groups whose principal p-blocks consist of ordinary irreducible characters of prime power degrees for every prime p. In addition, we give an upper bound of the nilpotent length for the solvable situation.

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References

  1. Bessenrodt, C., Zhang, J.: Block separations and inclusions. Adv. Math. 218, 485–495 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bessenrodt, C., Zhang, J.: Character separation and principal covering. J. Algebra 327, 170–185 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Dagger, S.W.: On the blocks of the Chevalley groups. J. Lond. Math. Soc. s2–3(1), 21–29 (1971). doi:10.1112/jlms/s2-3.1.21

  4. Humphreys, J.E.: Defect groups for finite groups of Lie type. Math. Z. 119, 149–152 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  5. Isaacs, I.M.: Character Theory of Finite Groups. Academic Press, New York (1994)

    MATH  Google Scholar 

  6. Isaacs, I.M., Smith, S.D.: A note on groups of \(p\)-length \(1\). J. Algebra 38, 531–535 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  7. James, G.: The Representation Theory of the Symmetric Groups. Lecture Notes in Math, vol. 682. Springer, New York (1978)

  8. Manz, O.: Endliche auflösbare Gruppen, deren sämtliche Charaktergrade Primzahlpotenzen sind. (German) [Finite solvable groups all of whose character degrees are prime powers]. J. Algebra 94, 211–255 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  9. Nagao, H., Tsushima, Y.: Representations of Finite Groups. Academic Press, London (1989)

    MATH  Google Scholar 

  10. Navarro, G.: Characters and Blocks of Finite Groups. London Mathematical Society Lecture Note Series, vol. 250. Cambridge University Press, Cambridge (1998)

  11. Navarro, G., Turull, A., Wolf, T.R.: Block separation in solvable groups. Arch. Math. (Basel) 85, 293–296 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  12. The GAP Group, GAP-Groups, algorithms, and programming, Version 4.7.5 (2014) http://www.gap-system.org

  13. Willems, W.: Blocks of defect zero and degree problems. In: Proceedings of Symposia in Pure Mathematics, vol. 47, Part I, pp. 481–484. American Mathematical Society, Providence (1987)

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Acknowledgments

The author would like to thank Professor Wolfgang Willems for some helpful conversations and Professor Jiping Zhang for his persistent encouragement.

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Authors and Affiliations

  1. College of Mathematics and Information Science, Jiangxi Normal University, Nanchang, 330022, China

    Yanjun Liu

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  1. Yanjun Liu
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Correspondence to Yanjun Liu.

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Communicated by J. S. Wilson.

Y. Liu was supported by the National Natural Science Foundation of China (11201194) and (11471054). In addition, he was supported by Jiangxi Province Science Foundation for Youths (20142BAB211011).

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Liu, Y. Finite groups whose irreducible characters of principal blocks have prime power degrees. Monatsh Math 181, 117–122 (2016). https://doi.org/10.1007/s00605-015-0836-2

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  • DOI: https://doi.org/10.1007/s00605-015-0836-2

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