Bulletin of the Korean Mathematical Society (대한수학회보)

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CONJECTURES THAT SOLVABLE GROUPS WHOSE CHARACTER GRAPHS HAVING DIAMETER 3 SATISFY

  • Meng, Qingyun (College of Sciences Henan University of Technology)
  • Received : 2018.04.12
  • Accepted : 2018.07.05
  • Published : 2019.03.31
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Abstract

In this note, we prove that Gluck's conjecture, Isaacs-Navarro-Wolf Conjecture and Taketa's inequality are true for solvable groups whose character graphs having diameter 3.

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References

  1. C. Casolo, S. Dolfi, E. Pacifici, and L. Sanus, Groups whose character degree graph has diameter three, Israel J. Math. 215 (2016), no. 2, 523-558. https://doi.org/10.1007/s11856-016-1387-5
  2. D. Gluck, The largest irreducible character degree of a finite group, Canad. J. Math. 37 (1985), no. 3, 442-451. https://doi.org/10.4153/CJM-1985-026-8
  3. L. He, Y. Zhao, and J. Bi, Two applications of Lewis' theorem on character degree graphs of solvable groups, Bull. Korean Math. Soc. 52 (2015), no. 2, 363-366. https://doi.org/10.4134/BKMS.2015.52.2.363
  4. I. M. Isaacs, Character Theory of Finite Groups, Academic Press, New York, 1976.
  5. M. L. Lewis, Derived lengths and character degrees, Proc. Amer. Math. Soc. 126 (1998), no. 7, 1915-1921. https://doi.org/10.1090/S0002-9939-98-04391-3
  6. M. L. Lewis, Solvable groups whose degree graphs have two connected components, J. Group Theory 4 (2001), no. 3, 255-275. https://doi.org/10.1515/jgth.2001.023
  7. M. L. Lewis, A solvable group whose character degree graph has diameter 3, Proc. Amer. Math. Soc. 130 (2002), no. 3, 625-630. https://doi.org/10.1090/S0002-9939-01-06091-9
  8. M. L. Lewis and C. B. Sass, The Taketa problem and character degree graphs with diameter three, Algebr. Represent. Theory 18 (2015), no. 5, 1395-1399. https://doi.org/10.1007/s10468-015-9546-7
  9. O. Manz, R. Staszewski, and W. Willems, On the number of components of a graph related to character degrees, Proc. Amer. Math. Soc. 103 (1988), no. 1, 31-37. https://doi.org/10.1090/S0002-9939-1988-0938639-1
  10. P. Palfy, On the character degree graph of solvable groups. I. Three primes, Period. Math. Hungar. 36 (1998), no. 1, 61-65. https://doi.org/10.1023/A:1004659919371
  11. C. B. Sass, Character degree graphs of solvable groups with diameter three, J. Group Theory 19 (2016), no. 6, 1097-1127. https://doi.org/10.1515/jgth-2016-0029