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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
Published electronically: August 31, 2006
MathSciNet review: 2262862
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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.


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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: http://dx.doi.org/10.1090/S0002-9939-06-08651-5
PII: S 0002-9939(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.