Characterization of the Chevalley group G_{۲}(۵) by the set of numbers of the same order elements

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_{۲}(۵) is the Chevalley simple group of type G_{۲} over GF(۵) such that nse(G)=nse(G_{۲}(۵)), then G\cong G_{۲}(۵).

Keywords:

Element order , Thompson's problem , Number of the same order elements