Directed power graph-equivalent groups

This article defines an equivalence relation over the collection of groups. View a complete list of equivalence relations on groups.

Definition

Two groups $G$ and $H$ are directed power graph-equivalent groups if the directed power graph of $G$ is isomorphic (as a directed graph) to the directed power graph of $H$.

Finite version

If either of the groups $G$ or $H$ is a finite group, so is the other group, and in this case they are both 1-isomorphic finite groups.

Facts

• Finite groups are 1-isomorphic iff their directed power graphs are isomorphic
• Directed power graph-equivalent not implies 1-isomorphic for infinite groups
• Undirected power graph determines directed power graph for finite group

Relation with other relations

Stronger relations

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
isomorphic groups				|FULL LIST, MORE INFO
1-isomorphic groups		1-isomorphic implies directed power graph-equivalent	directed power graph-equivalent not implies 1-isomorphic	|FULL LIST, MORE INFO

Weaker relations

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
undirected power graph-equivalent groups