Abstract:
Let $G$ be a finite group. A vanishing element of $G$ is $g\in G$ such that $\chi(g)=0$ for some $\chi\in\operatorname{Irr}(G)$ of the set of irreducible complex characters of $G$. Denote by $\operatorname{Vo}(G)$ the set of the orders of vanishing elements of $G$. A finite group $G$ is called a VCP-group if every element in $\operatorname{Vo}(G)$ is of prime power order. The main purpose of this paper is to investigate a new characterization related to $\operatorname{Vo}(G)$ for all finite nonabelian simple VCP-groups. We prove that if $G$ is a finite group and $M$ is a finite nonabelian simple VCP-group such that $\operatorname{Vo}(G)=\operatorname{Vo}(M)$ and $|G|=|M|$, then $G\cong M$.
Keywords:finite simple groups, zeros of characters.
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