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Character degree graphs that are complete graphs


Authors: Mariagrazia Bianchi, David Chillag, Mark L. Lewis and Emanuele Pacifici
Journal: Proc. Amer. Math. Soc. 135 (2007), 671-676
MSC (2000): Primary 20C15; Secondary 05C25
DOI: https://doi.org/10.1090/S0002-9939-06-08651-5
Published electronically: August 31, 2006
MathSciNet review: 2262862
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ G$ be a finite group, and write $ \operatorname{cd}(G)$ for the set of degrees of irreducible characters of $ G$. We define $ \Gamma(G)$ to be the graph whose vertex set is $ \operatorname{cd}(G)-\{1\}$, and there is an edge between $ a$ and $ b$ if $ (a,b)>1$. We prove that if $ \Gamma(G)$ is a complete graph, then $ G$ is a solvable group.


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

  • 1. J. H. CONWAY, R. T. CURTIS, S. P. NORTON, R. A. PARKER, AND R. A. WILSON, ``Atlas of Finite Groups,'' Oxford University Press, London, 1984. MR 0827219 (88g:20025)
  • 2. Y. BERKOVICH, Finite groups with small sums of degrees of some non-linear irreducible characters, J. Algebra 171 (1995), 426-443. MR 1315905 (96c:20015)
  • 3. R. W. CARTER ``Finite Groups of Lie Type,'' Wiley, New York, 1985. MR 0794307 (87d:20060)
  • 4. B. HUPPERT, ``Character Theory of Finite Groups,'' Walter DeGruyter, Berlin, 1998. MR 1645304 (99j:20011)
  • 5. I. M. ISAACS, ``Character Theory of Finite Groups,'' Academic Press, San Diego, 1976. MR 0460423 (57:417)
  • 6. I. M. ISAACS, Character correspondences in solvable groups, Adv. in Math. 43 (1982), 284-306. MR 0648802 (83i:20007)
  • 7. M. L. LEWIS, A. MORETÓ, AND T. R. WOLF, Non-divisibility among character degrees, J. Group Theory 8 (2005), 561-588. MR 2165291
  • 8. M. L. LEWIS, An overview of graphs associated with character degrees and conjugacy class sizes in finite groups, to appear in Rocky Mountain J. Math.
  • 9. J. K. MCVEY, Bounding graph diameters of nonsolvable groups, J. Algebra 282 (2004), 260-277. MR 2097583 (2005h:20011)
  • 10. P. SCHMID, Rational matrix groups of a special type, Linear Algebra Appl. 71 (1985), 289-293. MR 0813053 (87e:20018)
  • 11. P. SCHMID, Extending the Steinberg representation, J. Algebra 150 (1992), 254-256. MR 1174899 (93k:20016)
  • 12. A. TURULL, Generic fixed point free action of arbitrary finite groups, Math. Z. 187 (1984), 491-503. MR 0760049 (86a:20020)
  • 13. D. L. WHITE, Degree graphs of simple groups of exceptional Lie type, Comm. Algebra 32 (2004), 3641-3649. MR 2097483 (2005h:20022)
  • 14. D. L. WHITE, Degree graphs of simple linear and unitary groups, to appear in Comm. Algebra.
  • 15. D. L. WHITE, Degree graphs of simple orthogonal and symplectic groups, to appear in J. Algebra.

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Additional Information

Mariagrazia Bianchi
Affiliation: Dipartimento di Matematica “F. Enriques”, Università Degli Studi Di Milano, Via C. Saldini 50, 20133 Milano, Italy
Email: Mariagrazia.Bianchi@mat.unimi.it

David Chillag
Affiliation: Department of Mathematics, Technion, Israel Institute of Technology, Haifa 32000, Israel
Email: chillag@techunix.technion.ac.il

Mark L. Lewis
Affiliation: Department of Mathematical Sciences, Kent State University, Kent, Ohio 44242
Email: lewis@math.kent.edu

Emanuele Pacifici
Affiliation: Dipartimento di Matematica “F. Enriques”, Università Degli Studi Di Milano, Via C. Saldini 50, 20133 Milano, Italy
Email: Emanuele.Pacifici@mat.unimi.it

DOI: https://doi.org/10.1090/S0002-9939-06-08651-5
Received by editor(s): October 4, 2005
Published electronically: August 31, 2006
Communicated by: Jonathan I. Hall
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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