Character degree graphs and normal subgroups

Isaacs, I.

doi:10.1090/S0002-9947-03-03462-7

Character degree graphs and normal subgroups
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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