Non-Solvable Groups whose all Vanishing Class Sizes are Odd-Square-Free

Given a finite group G, a vanishing element is an element x 2 G for which there exists x 2 Irr(G) such that x (x) = 0. The conjugacy class of a vanishing element is called a vanishing class of G. Considering G as a finite non-solvable group with Sol(G) as its solvable radical, in this paper, we prove that if all vanishing class sizes of G are odd-square-free, then either G/Sol(G) is an almost simple group, or it has exactly two chief factors with the properties mentioned in Theorem 1.