A new characterization of $E_8 (p)$ via its vanishing elements

Abstract

Let $G$ be a finite group, and $g \in G$. Then $g$ is said to be a vanishing element of $G$, if there exists an irreducible character $\chi$ of $G$ such that $\chi (g)=0$. Denote by ${\rm Vo} (G)$ the set of the orders of vanishing elements of $G$. We say a non-abelian group $G$ is V-recognizable, if any group $N$ with ${\rm Vo} (N) = {\rm Vo} (G)$ is isomorphic to $G$. In this paper, we investigate the V-recognizability of $E_8 (p)$, where $p$ is a prime number. As an application, among the 610 primes $p$ with $p<10000$ and $p \equiv 0,1,4\,(\!\!\!\mod 5)$, we obtain that the method is always valid for confirming the V-recognizability of $E_8 (p)$ for all such $p$ but $ 919,1289,1931,3911,4691,5381$ and $7589 $.