A new characterization of some families of finite simple groups

Foroudi Ghasemabadi, M.; Iranmanesh, Ali; Ahanjideh, M.

Rendiconti del Seminario Matematico della Università di Padova, Tome 137 (2017), pp. 57-74.

Let $G$ be a finite group. A vanishing element of $G$ is an element $g\in G$ such that $\chi \left(g\right)=0$ for some irreducible complex character $\chi$ of $G$. Denote by $\mathrm{Vo}\left(G\right)$ the set of the orders of vanishing elements of $G$. In this paper, we prove that if $G$ is a finite group such that $\mathrm{Vo}\left(G\right)=\mathrm{Vo}\left(M\right)$ and $\left|G\right|=\left|M\right|$, then $G\cong M$, where $M$ is a sporadic simple group, an alternating group, a projective special linear group $L_2\left(p\right)$, 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.