Finite groups are 1-isomorphic iff their directed power graphs are isomorphic

This article gives a proof/explanation of the equivalence of multiple definitions for the term 1-isomorphic finite groups
View a complete list of pages giving proofs of equivalence of definitions

Statement

Suppose $G$ and $H$ are Finite group (?)s. Then, $G$ and $H$ are 1-isomorphic groups (?) (in particular, 1-isomorphic finite groups (?)if and only if their directed power graphs are isomorphic as directed graphs.

Related facts

• Undirected power graph determines directed power graph for finite group
• Undirected power graph need not determine directed power graph for infinite group
• Directed power graph-equivalent not implies 1-isomorphic for infinite groups