Zeros of Primitive Characters (Journal of Mathematical Study, 2022)
Let $G$ be a finite group. An irreducible character $χ$ of $G$ is said to be primitive if $χ eq ϑ^{G}$ for any character $ϑ$ 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 $φ \in {\rm Irr_{pri}}(G)$ such that $φ(g)=0$.
Keywords: Finite group; primitive character; vanishing element.