대한수학회보 (Bulletin of the Korean Mathematical Society)

  • 제52권2호
  • /
  • Pages.363-366
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  • 2015
  • /
  • 1015-8634(pISSN)
  • /
  • 2234-3016(eISSN)
대한수학회 (Korean Mathematical Society)
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TWO APPLICATIONS OF LEWIS' THEOREM ON CHARACTER DEGREE GRAPHS OF SOLVABLE GROUPS

  • He, Liguo (Department of Mathematics Shenyang University of Technology) ;
  • Zhao, Yuanhe (Department of Mathematics Shenyang University of Technology) ;
  • Bi, Jianxing (Department of Mathematics Shenyang University of Technology)
  • 투고 : 2013.03.31
  • 발행 : 2015.03.31
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초록

In this note, we prove Gluck's conjecture and Isaacs-Navarro-Wolf Conjecture are true for the solvable groups with disconnected graphs by using Lewis' group structure theorem with respect to the disconnected character degree graphs.

키워드

참고문헌

  1. A. Espuelas, Large character degrees of groups of odd order, Illinois J. Math. 35 (1991), no. 3, 499-505.
  2. The GAP Group, GAP-Groups, algorithms, and programming, version 4.5, http://www.gap-system.org, 2012.
  3. D. Gluck, The largest irreducible character degree of a finite group, Canad. J. Math. 37 (1985), no. 2, 442-451. https://doi.org/10.4153/CJM-1985-026-8
  4. L. G. He, Notes on non-vanishing elements of finite solvable groups, Bull. Malays. Math. Sci. Soc. (2) 35 (2012), no. 1, 163-169.
  5. I. M. Isaacs, Character Theory of Finite Groups, Academic Press, New York, 1976.
  6. I. M. Isaacs, G. Navarro, and T. R. Wolf, Finite group elements where no irreducible character vanishes, J. Algebra 222 (1999), no. 2, 413-423. https://doi.org/10.1006/jabr.1999.8007
  7. M. 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
  8. O. Manz, Degree problems II: -separable character degrees, Comm. Algebra 13 (1985), no. 11, 2421-2431. https://doi.org/10.1080/00927878508823281
  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. O. Manz and T. R. Wolf, Representations of Solvable Groups, Cambridge Univ. Press, Cambridge, 1993.
  11. A. Moreto and T. R. Wolf, Orbit sizes, character degrees and Sylow subgroups, Adv. Math. 184 (2004), no. 1, 18-36. https://doi.org/10.1016/S0001-8708(03)00093-8
  12. P. P. Palfy, On the character degree graph of solvable groups. II: disconnected graphs, Studia Sci. Math. Hungar. 28 (2001), 339-355.
  13. J. P. Zhang, A note on character degrees of finite solvable groups, Comm. Algebra 28 (2000), no. 9, 4249-4258. https://doi.org/10.1080/00927870008827087