  Many results have been established about determining whether or not an
element evaluates to zero on an irreducible character of a group. In this note
it is shown that if a group $G$ has a normal nilpotent subgroup $N$, and $P$ is
a Sylow $p$-subgroup of $G$, then no irreducible character of $G$ vanishes on
$N\cap Z(P)$.
