power graphs based on the order of their groups

The power graph p(g) of a group g is a graph with vertex setg, where two vertices u and v are adjacent if and only if u ̸= v and um = vor vm = u for some positive integer m. the present paper aims to classifypower graphs based on group orders, which can be a new look at the powergraphs classication. we raise and study the following question: for whichnatural numbers n every two groups of order n with isomorphic power graphsare isomorphic? we denote the set of all such numbers by  s and consider theelements of  s. moreover, we show that if two nite groups have isomorphicpower graphs and one of them is nilpotent or has a normal hall subgroup,the same is true with the other one.