Groups whose set of vanishing elements is the union of at most three conjugacy classes

Abstract

Let $G$ be a finite group. We say that an element $g$ in $G$ is a vanishing element if there exists some irreducible character $\chi$ of $G$ such that $\chi(g)=0$. In this paper, we prove that if the set of vanishing elements of $G$ is the union of at most three conjugacy classes, then $G$ is solvable.