Amirkabir University of Technology AUT Journal of Mathematics and Computing 2783-2449 Articles in Press 2025 07 01 On the simple $K_5$-groups 5781 10.22060/ajmc.2025.23312.1249 EN Behnam Ebrahimzadeh Department of Mathematics, Persian Gulf university, Boshehr, Iran 0000-0001-5696-275X Journal Article 2024 06 29 After classification  of finite simple groups, the  researchers dissucced about  groups characterization by property. Properties, such as element order, the set of elements with the same order, graphs,etc. In other words  if $G$ be a finite group  and  $M$ be a property  then we say the group $G$ is characterized by property $M$ if by isomorphic $G$ be a only group by property $M$.  One of the methods, is group characterization by largest element order. In other wrds, we say the group $G$ is characterized by largest element order $k(G)$  and order of $G$ if there exists the group $H$, so that  if $k(G)=k(H)$ and $|G|=|H|$, then $G\cong H$.  In this paper, we prove that the simple $K_5$-groups $PSL(6,2)$ and $PSU(6,2)$ can be uniquely determined by their order and the largest order of elements.