The directed power graph (G) over right arrow (G) of a group G is the simple digraph with vertex set G in which x -> y if y is a power of x, and the power graph is the underlying simple graph G(G). In this paper, three versions of the definition of the power graph are discussed, and it is proved that the power graph by any of the three versions of the definition determines the other two up to isomorphism. It is also proved that if G is a torsion-free group of nilpotency class 2 and if H is a group such that G(H) congruent to to G(G), then G and H have isomorphic directed power graphs, which was an open problem proposed by Cameron, Guerra and Jurina 9.
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