Undirected power graph determines directed power graph for finite group

Statement
For a finite group, the isomorphism class of its undirected power graph determines the isomorphism class of its directed power graph. In particular, since the directed power graph determines a finite group up to 1-isomorphism, the undirected power graph also determines the group up to 1-isomorphism. In other words, if two finite groups have isomorphic undirected power graphs, they are defining ingredient::1-isomorphic groups.

Related facts

 * Undirected power graph need not determine directed power graph for infinite group