Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-06-08383-3
Published electronically: May 12, 2006
MathSciNet review: 2231893
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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$.


References [Enhancements On Off] (What's this?)

  • 1. J. H. CONWAY, R. T. CURTIS, S. P. NORTON, R. A. PARKER, AND R. A. WILSON, Atlas of Finite Groups, Oxford University Press, London, 1984. MR 0827219 (88g:20025)
  • 2. S. DOLFI, Prime factors of conjugacy class lengths and irreducible character degrees in finite soluble groups, J. Algebra 174 (1995), 749-752. MR 1337168 (96c:20011)
  • 3. R. M. GURALNICK AND W. M. KANTOR, Probabilistic generation of finite simple groups. J. Algebra 234 (2000), no. 2, 743-792.MR 1800754 (2002f:20038)
  • 4. I. M. ISAACS, Primitive characters, normal subgroups, and $ M$-groups, Math. Z. 177 (1981), 267-287. MR 0612879 (82f:20026)
  • 5. P. M. NEUMANN, A study of some finite permutation groups, Ph.D. Thesis, Oxford, 1966.
  • 6. P. M. NEUMANN AND M. R. VAUGHAN-LEE, An essay on BFC-groups, Proc. London Math. Soc. 35 (1977), 213-237. MR 0463311 (57:3264)
  • 7. D. SEGAL AND A. SHALEV, On groups with bounded conjugacy classes, Quart. J. Math. Oxford 50 (1999), 505-516. MR 1726791 (2000j:20053)
  • 8. T. R. WOLF, Sylow $ p$-subgroups of $ p$-solvable subgroups of $ \operatorname{GL}(n,p)$, Arch. Math. 43 (1984), 1-10.MR 0758331 (86g:20064)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 20C99

Retrieve articles in all journals with MSC (2000): 20C99


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: https://doi.org/10.1090/S0002-9939-06-08383-3
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.

American Mathematical Society