The power graph of a torsion-free group determines the directed power graph

The directed power graph G?(G) of a group G is the simple digraph with vertex set such that x?y if y x. G, denoted by G(G), underlying graph. In this paper, for groups and H, following proved. If has no quasicyclic subgroup Cp? which trivial intersection every cyclic K K??Cp?, then G(G)?G(H) implies G?(G)?G?(H). particular, any two torsion-free having isomorphic graphs have graphs.