Amirkabir University of Technology AUT Journal of Mathematics and Computing 2783-2449 1 1 2020 02 01 The validity of a Thompson’s problem for $\rm{PSL(4,7)}$ 89 94 3746 10.22060/ajmc.2019.16174.1022 EN Behrooz Khosravi Department of Mathematics and Computer Science, Amirkabir University of Technology, 424, Hafez Ave., Tehran 15914, Iran Cyrus Kalantarpour Department of Mathematics and Computer Science, Amirkabir University of Technology, 424, Hafez Ave., Tehran 15914, Iran Journal Article 2019 04 21 Let $\pi_e(G)$ 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 $\rm{PSL(4,7)}$. Thompson’s problem Characterization Number of elements of the same order Projective special linear group Hall subgroup NSE Sporadic groups Python https://ajmc.aut.ac.ir/article_3746_e8542a04d734d0cae36d648b3f519e5c.pdf