Characterization of a Finite Nonabelian Simple Group by Its Order and Vanishing Prime Graph

Source abstract

Let G be a finite group. A vanishing element of G is an element g ∈ G such that χ ( g ) = 0 for some irreducible complex character χ of G . Denote by V o ( G ) the set of orders of vanishing elements of G . The vanishing prime graph Γ( G ) of G is defined as follows. The vertex set of Γ( G ) is the set { p | p is a prime divisor of some element of V o ( G )}. Two distinct vertices p and q of Γ( G ) are adjacent if and only if there exists an element a of V o ( G ) such that p q | a . For vanishing prime graphs, there is an interesting problem as follows: Which groups G are uniquely determined by their order | G | and vanishing prime graph Γ( G )? In this paper, we establish a number of results related to vanishing prime graphs and this problem.