On Common Zeros of Characters of Finite Groups

Let G be a finite group, and let Irr(G) denote the set of the irreducible complex characters of G. An element g∈G is called a vanishing element of G if there exists χ∈Irr(G) such that χ(g)=0 (i.e., g is a zero of χ) and, in this case, the conjugacy class gG of g in G is called a vanishing conjugacy class. In this paper we consider several problems concerning vanishing elements and vanishing conjugacy classes; in particular, we consider the problem of determining the least number of conjugacy classes of a finite group G such that every non-linear χ∈Irr(G) vanishes on one of them. We also consider the related problem of determining the minimum number of non-linear irreducible characters of a group such that two of them have a common zero.