Notes on Non-Vanishing Elements of Finite Solvable Groups

Let G be a finite solvable group. The element gϵG is said to be a non-vanishing element of G if χ(g)≠0 for all χ ϵ Irr (G). It is conjectured that all of non-vanishing elements of G lie in its Fitting subgroup F(G). In this note, we prove that this conjecture is true for nilpotent-by-supersolvable groups. Write V(G) to denote the subgroup generated by all non-vanishing elements of G, and Fn(G) the nth term of the ascending Fitting series. It is proved that V(Fn(G))≤Fn−1(G) whenever G is solvable. If this conjecture were not true, then it is proved that the minimal counterexample is a solvable primitive permutation group and the more detailed information is presented. Some other related results are proved.