Groups with just one character degree divisible by a given prime
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- by I. M. Isaacs, Alexander Moretó, Gabriel Navarro and Pham Huu Tiep PDF
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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.References
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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
- ORCID: 0000-0002-6914-9650
- Email: alexander.moreto@uv.es
- Gabriel Navarro
- Affiliation: Departament d’Àlgebra, Universitat de València, 46100 Burjassot, València, Spain
- MR Author ID: 129760
- 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
- MR Author ID: 230310
- Email: tiep@math.arizona.edu
- 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).
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 361 (2009), 6521-6547
- MSC (2000): Primary 20C15; Secondary 20C33
- DOI: https://doi.org/10.1090/S0002-9947-09-04718-7
- MathSciNet review: 2538603