Zeros of Primitive Characters

Abstract

Let $G$ be a finite group. An irreducible character $\chi$ of $G$ is said to be primitive if $\chi \neq \vartheta^{G}$ for any character $\vartheta$ of a proper subgroup of $G$. In this paper, we consider about the zeros of primitive characters. Denote by ${\rm Irr_{pri} }(G)$ the set of all irreducible primitive characters of $G$. We proved  that  if $g\in G$ and the order of $gG'$ in the factor group $G/G'$ does not divide $|{\rm Irr_{pri}}(G)|$, then there exists $\varphi \in {\rm Irr_{pri}}(G)$ such that $\varphi(g)=0$.