Recognition by the set of orders of vanishing elements and order of PSL(3, p)

Abstract

Let G be a finite group. We say that an element g of G is a vanishing element if there exists an irreducible complex character χ of G such that χ ( g ) = 0. Denote by V o ( G ) the set of order of vanishing elements of G , and we prove that G ∼ = PSL (3 ,p ) if and only if V o ( G ) = V o ( PSL (3 ,p )) and | G | = | PSL (3 ,p ) | , where p is a prime number.