DOI QR코드

DOI QR Code

TWO APPLICATIONS OF LEWIS' THEOREM ON CHARACTER DEGREE GRAPHS OF SOLVABLE GROUPS

  • He, Liguo ;
  • Zhao, Yuanhe ;
  • Bi, Jianxing
  • Received : 2013.03.31
  • Published : 2015.03.31

Abstract

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.

Keywords

solvable group;character;fitting subgroup;non-vanishing element

References

  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.
  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

Acknowledgement

Supported by : NSF of China