Non-vanishing elements of finite groups

Abstract

Let G be a finite group, and let Irr ( G ) denote the set of irreducible complex characters of G. An element x of G is non-vanishing if, for every χ in Irr ( G ) , we have χ ( x ) ≠ 0 . We prove that, if x is a non-vanishing element of G and the order of x is coprime to 6, then x lies in the Fitting subgroup of G.