3-abelian group
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 properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions
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 |