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

Manfred Leitz. 

Proc. Amer. Math. Soc. 128 (2000), 3149-3152.

                                
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.