2-Engel group

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Definition

Symbol-free definition

A group is termed a Levi group or a 2-Engel group if it satisfies the following equivalent conditions:

1. Any two conjugate elements of the group commute.
2. The normal closure of any cyclic subgroup is Abelian.
3. The group is a $2$-Engel group: the commutator between any element and its commutator with another element is the identity element.

Definition with symbols

A group $G$ is termed a Levi-group or a 2-Engel group if it satisfies the following equivalent conditions:

1. $x$ commutes with $gx{g}^{-1}$ for all $x,g\in G$.
2. The normal subgroup generated by $x$ is Abelian for all $x\in G$.
3. The commutator $[x,[x,g]]$ is the identity element for all $x,g\in G$.

Relation with other properties

Stronger properties

• Abelian group
• Dedekind group
• Group of nilpotence class two

Weaker properties

• Group generated by Abelian normal subgroups
• Bounded Engel group
• Engel group
• Nilpotent group (for finite groups)