  Let $N$ be a normal subgroup of a finite group $G$. In this paper, we
consider the elements $g$ of $N$ such that $\chi(g)\neq 0$ for all irreducible
characters $\chi$ of $G$. Such an element is said to be non-vanishing in $G$.
Let $p$ be a prime. If all $p$-elements of $N$ satisfy the previous property,
then we prove that $N$ has a normal Sylow $p$-subgroup. As a consequence, we
also study certain arithmetical properties of the $G$-conjugacy class sizes of
the elements of $N$ which are zeros of some irreducible character of $G$. In
particular, if $N=G$, then new contributions are obtained.
