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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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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
Trans. Amer. Math. Soc. 361 (2009), 6521-6547 Request permission

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
  • 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