# 3-abelian group

From Groupprops

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism

View a complete list of group propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

## Contents

## Definition

A group is termed **3-abelian** if it is n-abelian for , i.e., the cube map is an endomorphism of the group.

## Facts

- Levi's characterization of 3-abelian groups
- Cube map is endomorphism implies class three
- Cube map is surjective endomorphism implies abelian
- Cube map is endomorphism iff abelian (if order is not a multiple of 3)

For more on power maps being endomorphisms, see n-abelian group.

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

abelian group | ||||

group of exponent three |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

2-Engel group | Follows from Levi's characterization of 3-abelian groups |