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Proceedings of the American Mathematical Society

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Fixed point spaces, primitive character degrees and conjugacy class sizes

Authors: I. M. Isaacs, Thomas Michael Keller, U. Meierfrankenfeld and Alexander Moretó
Journal: Proc. Amer. Math. Soc. 134 (2006), 3123-3130
MSC (2000): Primary 20C99
Published electronically: May 12, 2006
MathSciNet review: 2231893
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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

Thomas Michael Keller
Affiliation: Department of Mathematics, Texas State University, San Marcos, Texas 78666
MR Author ID: 356408
ORCID: 0000-0003-3901-8585

U. Meierfrankenfeld
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824

Alexander Moretó
Affiliation: Departament d’Àlgebra, Universitat de València, 46100 Burjassot, València, Spain
ORCID: 0000-0002-6914-9650

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
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.