Undirected power graph determines directed power graph for finite group

This article gives a proof/explanation of the equivalence of multiple definitions for the term 1-isomorphic finite groups
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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 1-isomorphic groups.

Related facts

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

References

• The power graph of a finite group, II by Peter J. Cameron, ,  : ^{official online copy} The article has not yet been printed.

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