FINITE GROUPS WITH FEW VANISHING ELEMENTS

Let G be a finite group, and Irr(G) the set of irreducible complex characters of G. We say that an element g is an element of G is a vanishing element of G if there exists chi in Irr(G) such that chi(g) = 0. Let Van(G) denote the set of vanishing elements of G, that is, Van(G) = {g is an element of G vertical bar chi(g) = 0 for some chi is an element of Irr(G)}. In this paper, we investigate the finite groups G with the following property: Van(G) contains at most four conjugacy classes of G.