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

 

Elementary proof of Brauer's and Nesbitt's theorem on zeros of characters of finite groups


Author: Manfred Leitz
Journal: Proc. Amer. Math. Soc. 128 (2000), 3149-3152
MSC (2000): Primary 20C15
Published electronically: March 3, 2000
MathSciNet review: 1676316
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Abstract:

The following has been proven by Brauer and Nesbitt. Let $G$ be a finite group, and let $p$ be a prime. Assume $\chi$ is an irreducible complex character of $G$ such that the order of a $p$-Sylow subgroup of $G$ divides the degree of $\chi$. Then $\chi$ vanishes on all those elements of $G$ whose order is divisible by $p$. The two only known proofs of this theorem use profound methods of representation theory, namely the theory of modular representations or Brauer's characterization of generalized characters. The purpose of this paper is to present a more elementary proof.


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Additional Information

Manfred Leitz
Affiliation: Fachbereich Informatik und Mathematik, Fachhochschule Regensburg, Postfach 120327, 93025 Regensburg, Germany
Email: manfred.leitz@mathematik.fh-regensburg.de

DOI: http://dx.doi.org/10.1090/S0002-9939-00-05422-8
PII: S 0002-9939(00)05422-8
Received by editor(s): December 5, 1998
Published electronically: March 3, 2000
Communicated by: Ronald M. Solomon
Article copyright: © Copyright 2000 American Mathematical Society