Character degree graphs and normal subgroups
HTML articles powered by AMS MathViewer
- by I. M. Isaacs PDF
- Trans. Amer. Math. Soc. 356 (2004), 1155-1183 Request permission
Abstract:
We consider the degrees of those irreducible characters of a group $G$ whose kernels do not contain a given normal subgroup $N$. We show that if $N \subseteq G’$ and $N$ is not perfect, then the common-divisor graph on this set of integers has at most two connected components. Also, if $N$ is solvable, we obtain bounds on the diameters of the components of this graph and, in the disconnected case, we study the structure of $N$ and of $G$.References
- S. Garrison, On groups with a small number of character degrees. Ph.D. Thesis, University of Wisconsin, Madison, 1973.
- B. Huppert, Endliche Gruppen. I, Die Grundlehren der mathematischen Wissenschaften, Band 134, Springer-Verlag, Berlin-New York, 1967 (German). MR 0224703, DOI 10.1007/978-3-642-64981-3
- I. Martin Isaacs, Character theory of finite groups, Pure and Applied Mathematics, No. 69, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976. MR 0460423
- I. M. Isaacs and Greg Knutson, Irreducible character degrees and normal subgroups, J. Algebra 199 (1998), no. 1, 302–326. MR 1489366, DOI 10.1006/jabr.1997.7191
- Mark L. Lewis, Solvable groups whose degree graphs have two connected components, J. Group Theory 4 (2001), no. 3, 255–275. MR 1839998, DOI 10.1515/jgth.2001.023
- Mark L. Lewis, Bounding Fitting heights of character degree graphs, J. Algebra 242 (2001), no. 2, 810–818. MR 1848974, DOI 10.1006/jabr.2001.8831
- Olaf Manz, Degree problems. II. $\pi$-separable character degrees, Comm. Algebra 13 (1985), no. 11, 2421–2431. MR 807482, DOI 10.1080/00927878508823281
- Olaf Manz and Thomas R. Wolf, Representations of solvable groups, London Mathematical Society Lecture Note Series, vol. 185, Cambridge University Press, Cambridge, 1993. MR 1261638, DOI 10.1017/CBO9780511525971
- J. McVey, Bounding graph diameters of solvable groups, J. of Algebra (Submitted).
- P. P. Pálfy, On the character degree graph of solvable groups. II. Disconnected graphs, Studia Sci. Math. Hungar. 38 (2001), 339–355. MR 1877790, DOI 10.1556/SScMath.38.2001.1-4.25
- Thomas Yuster, Orbit sizes under automorphism actions in finite groups, J. Algebra 82 (1983), no. 2, 342–352. MR 704759, DOI 10.1016/0021-8693(83)90155-2
- Jiping Zhang, A note on character degrees of finite solvable groups, Comm. Algebra 28 (2000), no. 9, 4249–4258. MR 1772504, DOI 10.1080/00927870008827087
Additional Information
- I. M. Isaacs
- Affiliation: Department of Mathematics, University of Wisconsin, 480 Lincoln Dr., Madison Wisconsin 53706
- Email: isaacs@math.wisc.edu
- Received by editor(s): November 6, 2002
- Published electronically: October 6, 2003
- Additional Notes: This research was partially supported by a grant from the U. S. National Security Agency
- © Copyright 2003 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 356 (2004), 1155-1183
- MSC (2000): Primary 20C15
- DOI: https://doi.org/10.1090/S0002-9947-03-03462-7
- MathSciNet review: 2021616