Amirkabir University of TechnologyAUT Journal of Mathematics and Computing2120210201Recognition by degree prime-power graph and order of some characteristically simple group1115412210.22060/ajmc.2020.18418.1033ENAfsaneBahriDepartment of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), Tehran, IranBehroozKhosraviDepartment of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic)MortezaBaniasad AzadDepartment of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic)Journal Article20200512In this paper, by the order of a group and triviality of $O_p(G)$ for some prime $p$, we give a new characterization for some characteristically simple groups. In fact, we prove that if pϵ{5, 17, 23, 37, 47, 73} and n≤p, where $n$ is a natural number, then $Gcong{{rm PSL}(2,p)}^{n}$ if and only if $ |G|=|{{rm PSL}(2,p)}|^{n}$ and $ O_p(G)=1$.}<br /><br />Recently in [Qin, Yan, Shum and Chen, Comm. Algebra, 2019], the degree prime-power graph of a finite group <br /> have been introduced and it is proved that the Mathieu groups are uniquely determined by their degree prime-power graphs and orders. As a consequence of our results, we show that ${rm PSL}(2,p)^n$, where pϵ{5,17,23,37,47,73} and n≤p are uniquely determined by their degree prime-power graphs and orders.https://ajmc.aut.ac.ir/article_4122_da8bcf27302b7048c4d44272e8b6fb02.pdf