Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Character degree graphs and normal subgroups


Author: I. M. Isaacs
Journal: Trans. Amer. Math. Soc. 356 (2004), 1155-1183
MSC (2000): Primary 20C15
DOI: https://doi.org/10.1090/S0002-9947-03-03462-7
Published electronically: October 6, 2003
MathSciNet review: 2021616
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. S. Garrison, On groups with a small number of character degrees. Ph.D. Thesis, University of Wisconsin, Madison, 1973.
  • 2. B. Huppert, Endliche Gruppen I, Springer-Verlag, Berlin-New York, 1967. MR 37:302
  • 3. I. M. Isaacs, Character Theory of Finite Groups, Dover, New York, 1994. MR 57:417 (review of 1976 original edition)
  • 4. I. M. Isaacs and G. Knutson, Irreducible character degrees and normal subgroups, J. of Algebra 199 (1998) 302-326. MR 98m:20013
  • 5. M. L. Lewis, Solvable groups whose degree graphs have two connected components, J. of Group Theory 4 (2001) 255-275. MR 2002g:20015
  • 6. M. L. Lewis, Bounding Fitting heights of character degree graphs, J. of Algebra 242 (2001) 810-818. MR 2003d:20010
  • 7. O. Manz, Degree problems II. $\pi $-separable character degrees, Comm. in Algebra 13 (1985) 2421-2431. MR 86m:20009
  • 8. O. Manz and T. R. Wolf, Representations of Solvable Groups, Cambridge University Press, Cambridge, 1993. MR 95c:20013
  • 9. J. McVey, Bounding graph diameters of solvable groups, J. of Algebra (Submitted).
  • 10. P. P. Pálfy, On the character degree graph of solvable groups II. Disconnected graphs, Studia Sci. Math. Hungarica 38 (2001) 339-355. MR 2002m:20012
  • 11. T. Yuster, Orbit sizes under automorphism actions in finite groups. J. of Algebra 82 (1983) 342-352. MR 84m:20013
  • 12. J. Zhang, A note on character degrees of finite solvable groups, Comm. in Algegra 28 (2000) 4249-4248. MR 2001f:20020

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 20C15

Retrieve articles in all journals with MSC (2000): 20C15


Additional Information

I. M. Isaacs
Affiliation: Department of Mathematics, University of Wisconsin, 480 Lincoln Dr., Madison Wisconsin 53706
Email: isaacs@math.wisc.edu

DOI: https://doi.org/10.1090/S0002-9947-03-03462-7
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
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society