  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 one is the power of the other. 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, we determine prove that all such n are cube-free and
are not multiples of 16. Moreover, we show that if two finite 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.
