Power Graphs Based on the Order of Their Groups

The power graph P(G) of a group G is a graph with vertex set G, where two vertices u and v are adjacent if and only if u ̸= v and um = v or vm = u for some positive integer m. The present paper aims to classify power graphs based on group orders, which can be a new look at the power graphs classi cation. We raise and study the following question: For which natural numbers n every two groups of order n with isomorphic power graphs are isomorphic? We denote the set of all such numbers by S and consider the elements of S. Moreover, we show that if two nite groups have isomorphic power graphs and one of them is nilpotent or has a normal Hall subgroup, the same is true with the other one.