On the orders of zeros of irreducible characters

Let G be a finite group and p a prime number. We say that an element g in G is a vanishing element of G if there exists an irreducible character X of G such that X(g) = 0. The main result of this paper shows that, if G does not have any vanishing element of p-power order, then G has a normal Sylow p-subgroup. Also, we prove that this result is a generalization of some classical theorems in Character Theory of finite groups. (c) 2008 Elsevier Inc. All rights reserved.