# Directed power graph-equivalent groups

From Groupprops

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

## Contents

## Definition

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

## Finite version

If either of the groups or 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 | 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 |