Two remarks about non-vanishing elements in finite groups

Let G be a finite group. We prove that for x∈G we have χ(x)≠0 for all irreducible characters χ of G iff the class sum of x in the group algebra over C is a unit. From this we conclude that if G has a normal p-subgroup V and a Hall p′-subgroup, then G has non-vanishing elements different from 1. Hence we get another proof that a finite solvable group always has non-trivial non-vanishing elements. Moreover, we give an example for a finite solvable group G which has a non-vanishing involution not contained in an abelian normal subgroup of G.