  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 $.
