Finite groups with the same power graph

Abstract

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 and or for some positive integer m. In this paper, we raise and study the following question: For which natural numbers n every two groups of order n with isomorphic power graphs are isomorphic? In particular, it is proved that all such odd number n are cube-free and also they are not multiples of 16 in general. Moreover, we show that if two finite groups have isomorphic power graphs and one of them is nilpotent, the same is true for the other one.