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.

Relation with other relations

Stronger relations

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
isomorphic groups 1-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