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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Fixed point spaces, primitive character degrees and conjugacy class sizes
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by I. M. Isaacs, Thomas Michael Keller, U. Meierfrankenfeld and Alexander Moretó PDF
Proc. Amer. Math. Soc. 134 (2006), 3123-3130 Request permission

Abstract:

Let $G$ be a finite group that acts on a nonzero finite dimensional vector space $V$ over an arbitrary field. Assume that $V$ is completely reducible as a $G$-module, and that $G$ fixes no nonzero vector of $V$. We show that some element $g\in G$ has a small fixed-point space in $V$. Specifically, we prove that we can choose $g$ so that $\dim \mathbf {C}_V(g)\le (1/p)\dim V$, where $p$ is the smallest prime divisor of $|G|$.
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Additional Information
  • I. M. Isaacs
  • Affiliation: Department of Mathematics, University of Wisconsin, Madison, 480 Lincoln Drive, Madison, Wisconsin 53706
  • Email: isaacs@math.wisc.edu
  • Thomas Michael Keller
  • Affiliation: Department of Mathematics, Texas State University, San Marcos, Texas 78666
  • MR Author ID: 356408
  • ORCID: 0000-0003-3901-8585
  • Email: tk04@txstate.edu
  • U. Meierfrankenfeld
  • Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824
  • Email: meier@math.msu.edu
  • Alexander Moretó
  • Affiliation: Departament d’Àlgebra, Universitat de València, 46100 Burjassot, València, Spain
  • ORCID: 0000-0002-6914-9650
  • Email: Alexander.Moreto@uv.es
  • Received by editor(s): June 2, 2005
  • Published electronically: May 12, 2006
  • Additional Notes: The fourth author was partially supported by the Spanish Ministerio de Educación y Ciencia, grants MTM2004-04665 and MTM2004-06067-C02-01, the FEDER and the Programa Ramón y Cajal
  • Communicated by: Jonathan I. Hall
  • © Copyright 2006 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 134 (2006), 3123-3130
  • MSC (2000): Primary 20C99
  • DOI: https://doi.org/10.1090/S0002-9939-06-08383-3
  • MathSciNet review: 2231893