Characterization of the Chevalley group $G_{2}(5)$ by the set of numbers of the same order elements

Abstract

Let $G$ be a group and $omega(G)={o(g)|gin G}$ be the set of element orders of $G$. Let $kinomega(G)$ and $s_{k}=|{gin G|o(g)=k}|$. Let $nse(G)={s_{k}|kinomega(G)}.$ In this paper, we prove that if $G$ is a group and $G_{2}(5)$ is the Chevalley simple group of type $G_{2}$ over $GF(5)$ such that $nse(G)=nse(G_{2}(5))$, then $Gcong G_{2}(5)$.