Silvio Dolfi, Emanuele Pacifici, Lucia Sanus, Pablo Spiga
August 31, 2009
Let G be a finite group, and Irr( G ) the set of irreducible complex characters of G . We say that an element g ∈ G is a vanishing element of G if there exists χ in Irr( G ) such that χ ( g ) = 0. In this paper, we consider the set of orders of the vanishing elements of a group G , and we define the prime graph on it, which we denote by Γ( G ). Focusing on the class of solvable groups, we prove that Γ( G ) has at most two connected components, and we characterize the case when it is disconnected. Moreover, we show that the diameter of Γ( G ) is at most 4. Examples are given to round out our understanding of this matter. Among other things, we prove that the bound on the diameter is best possible, and we construct an infinite family of examples showing that there is no universal upper bound on the size of an independent set of Γ( G ).