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 $\omega(G)=\{o(g)|g\in G\}$ be the set of element orders of $G$. Let $k\in\omega(G)$ and $s_{k}=|\{g\in G|o(g)=k\}|$. Let $nse(G)=\{s_{k}|k\in\omega(G)\}.$ 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 $G\cong G_{2}(5)$.

Keywords

• element order
• thompson's problem
• number of the same order elements