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 χ of G such that χ (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. © 2008 Elsevier Inc. All rights reserved.
Dolfi, S., Pacifici, E., Sanus, L., & Spiga, P. (2009). On the orders of zeros of irreducible characters. Journal of Algebra, 321(1), 345–352. https://doi.org/10.1016/j.jalgebra.2008.10.004