Levi's characterization of 3-abelian groups

This article gives a proof/explanation of the equivalence of multiple definitions for the term 3-abelian group
View a complete list of pages giving proofs of equivalence of definitions

Statement

The following are equivalent for a group $G$:

1. $G$ is a 3-abelian group, i.e., the cube map of $G$ is an endomorphism of $G$.
2. $G$ is a 2-Engel group (also called a Levi group) and the derived subgroup of $G$ is either trivial or has exponent dividing three.

Related facts

Applications

• 3-abelian implies class three