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 \emph{vanishing element} of $G$ if there exists an irreducible character $\chi$ of $G$ such that $\chi(g)=0$. 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.