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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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 $ \vert G\vert$.


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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
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
Email: Alexander.Moreto@uv.es

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08383-3
PII: S 0002-9939(06)08383-3
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.