Amirkabir University of TechnologyAUT Journal of Mathematics and Computing2783-24491120200201The validity of a Thompsonâ€™s problem for $\rm{PSL(4,7)}$8994374610.22060/ajmc.2019.16174.1022ENBehroozKhosraviDepartment of Mathematics and Computer Science, Amirkabir University of Technology, 424, Hafez Ave., Tehran 15914, IranCyrusKalantarpourDepartment of Mathematics and Computer Science, Amirkabir University of Technology, 424, Hafez Ave., Tehran 15914, IranJournal Article20190421Let $\pi_e(G)$ be the set of elements orders of $G$. Also let $s_n$ be the number of elements of order $n$ in $G$ and ${\rm nse}(G)=\{s_n| n\in\pi_e(G)\}$. In this paper we prove that if $G$ is a group such that ${\rm nse}(G)= {\rm nse}(\rm PSL(4,7))$, $19\big\vert|G|$ and $19^2\nmid|G|$, then $G\cong{\rm PSL(4,7)}$. As a consequence of this result it follows that Thompson's problem is satisfied for the simple group $\rm{PSL(4,7)}$.https://ajmc.aut.ac.ir/article_3746_c15795a34e3e77400bdbc8563d65144d.pdf