A new characterization of some finite simple groups

Abstract

Let G be a finite group. A vanishing element of G is g ∈ G such that χ(g) = 0 for some χ ∈ Irr(G) of the set of irreducible complex characters of G. Denote by 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 Vo(G) is of prime power order. The main purpose of this paper is to investigate a new characterization related to 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 Vo(G) = Vo(M) and |G| = |M|, then G ≅ M.