A new characterization of some families of finite simple groups

Abstract

Let $G$ be a finite group. A vanishing element of $G$ is an element $g\in G$ such that $\chi (g)=0$ for some irreducible complex character $\chi$ of $G$. Denote by $\mathrm{Vo}(G)$ the set of the orders of vanishing elements of $G$. In this paper, we prove that if $G$ is a finite group such that $\mathrm{Vo}(G)=\mathrm{Vo}(M)$ and $|G|=|M|$, then $G\cong M$, where $M$ is a sporadic simple group, an alternating group, a projective special linear group $L_2(p)$, where $p$ is an odd prime or a finite simple $K_n$-group, where $n\in \{3,4\}$. These results confirm the conjecture posed in [17] for the simple groups under study.