Non-vanishing elements in finite groups

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 $N\cap Z(P)$.

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