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 $n=3$, 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