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Archiv der Mathematik

, Volume 112, Issue 1, pp 5–11 | Cite as

Divisibility of degrees in McKay correspondences

  • Noelia RizoEmail author
Article
  • 43 Downloads

Abstract

If p is a prime and G is a finite solvable group, we prove that there is a McKay correspondence between the irreducible characters of G of degree not divisible by p and those of a p-Sylow normalizer of G which respects divisibility. This phenomenon does not happen outside solvable groups.

Keywords

The McKay conjecture Divisibility of degrees Solvable groups 

Mathematics Subject Classification

Primary 20D Secondary 20C15 

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Notes

Acknowledgements

I would like to thank E. Giannelli for very useful comments on this subject. Also, part of this work was done while the author stayed at the Mathematical Sciences Research Institute in Berkeley and I would like to thank MSRI for its hospitality.

References

  1. 1.
    Giannelli, E.: Characters of odd degree of symmetric groups. J. London Math. Soc. 96(1), 1–14 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Giannelli, E., Kleshchev, A., Navarro, G., Tiep, P.H.: Restriction of odd degree characters and natural correspondences. Int. Math. Res. Not. IMRN 20, 6089–6118 (2017)MathSciNetGoogle Scholar
  3. 3.
    Hartley, B., Turull, A.: On characters of coprime operator groups and the Glauberman character correspondence. J. Reine Angew. Math. 451, 175–219 (1994)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Isaacs, I.M.: Character Theory of Finite Groups. AMS-Chelsea, Providence (2006)zbMATHGoogle Scholar
  5. 5.
    Isaacs, I.M.: Finite Group Theory. Graduate Studies in Mathematics, vol. 92. AMS, Providence (2008)Google Scholar
  6. 6.
    Isaacs, I.M.: Characters of \(\pi \)-separable groups. J. Algebra 86(1), 98–128 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Isaacs, I.M., Navarro, G., Olsson, J.B., Tiep, P.H.: Character restrictions and multiplicities in symmetric groups. J. Algebra 478, 271–282 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Navarro, G.: Character Theory and the McKay Conjecture. Cambridge Studies in Advanced Mathematics, vol. 175. Cambrige (2018)Google Scholar
  9. 9.
    Navarro, G.: Number of Sylow subgroups in \(p\)-solvable groups. Proc. Am. Math. Soc. 131(10), 3019–3020 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Navarro, I.P.: A Divisibility problem in the McKay Conjecture, arXiv:1711.00642
  11. 11.
    Turull, A.: Above the Glauberman correspondence. Adv. Math. 217(5), 2170–2205 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Turull, A.: The number of Hall \(\pi \)-subgroups of a \(\pi \)-separable group. Proc. Am. Math. Soc. 132(9), 2563–2565 (2004)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Departament de MatemàtiquesUniversitat de ValènciaBurjassotSpain

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