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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Groups with just one character degree divisible by a given prime


Authors: I. M. Isaacs, Alexander Moretó, Gabriel Navarro and Pham Huu Tiep
Journal: Trans. Amer. Math. Soc. 361 (2009), 6521-6547
MSC (2000): Primary 20C15; Secondary 20C33
Published electronically: July 24, 2009
MathSciNet review: 2538603
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Abstract: The Ito-Michler theorem asserts that if no irreducible character of a finite group $ G$ has degree divisible by some given prime $ p$, then a Sylow $ p$-subgroup of $ G$ is both normal and abelian. In this paper we relax the hypothesis, and we assume that there is at exactly one multiple of $ p$ that occurs as the degree of an irreducible character of $ G$. We show that in this situation, a Sylow $ p$-subgroup of $ G$ is almost normal in $ G$, and it is almost abelian.


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Additional Information

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

Alexander Moretó
Affiliation: Departament d’Àlgebra, Universitat de València, 46100 Burjassot, València, Spain
Email: alexander.moreto@uv.es

Gabriel Navarro
Affiliation: Departament d’Àlgebra, Universitat de València, 46100 Burjassot, València, Spain
Email: gabriel@uv.es

Pham Huu Tiep
Affiliation: Department of Mathematics, University of Arizona, 617 N. Santa Rita Avenue, P.O. Box 210089, Tucson, Arizona 85721
Email: tiep@math.arizona.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-09-04718-7
PII: S 0002-9947(09)04718-7
Received by editor(s): October 2, 2007
Received by editor(s) in revised form: December 14, 2007
Published electronically: July 24, 2009
Additional Notes: This research was partially supported by the Spanish Ministerio de Educación y Ciencia, MTM2004-06067-C02-01 and MTM2004-04665, and the FEDER. The second author was also supported by the Programa Ramón y Cajal and the Generalitat Valenciana. The fourth author gratefully acknowledges the support of the NSF (grant DMS-0600967).
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.