- Volume 52 Issue 2
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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
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.
연구 과제 주관 기관 : NSF of China
- A. Espuelas, Large character degrees of groups of odd order, Illinois J. Math. 35 (1991), no. 3, 499-505.
- The GAP Group, GAP-Groups, algorithms, and programming, version 4.5, http://www.gap-system.org, 2012.
- 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
- L. G. He, Notes on non-vanishing elements of finite solvable groups, Bull. Malays. Math. Sci. Soc. (2) 35 (2012), no. 1, 163-169.
- I. M. Isaacs, Character Theory of Finite Groups, Academic Press, New York, 1976.
- 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
- 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
- O. Manz, Degree problems II: -separable character degrees, Comm. Algebra 13 (1985), no. 11, 2421-2431. https://doi.org/10.1080/00927878508823281
- 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
- O. Manz and T. R. Wolf, Representations of Solvable Groups, Cambridge Univ. Press, Cambridge, 1993.
- 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
- P. P. Palfy, On the character degree graph of solvable groups. II: disconnected graphs, Studia Sci. Math. Hungar. 28 (2001), 339-355.
- 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