A new characterization of some families of finite simple groups

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 {\rm 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 {\rm Vo}(G)={\rm 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.