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 $x,g\in G$.

Formalisms

In terms of the Levi operator

This property is obtained by applying the Levi operator to the property: Abelian group
View other properties obtained by applying the Levi operator

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
abelian group				|FULL LIST, MORE INFO
Dedekind group	every subgroup is normal			|FULL LIST, MORE INFO
group of nilpotency class two	nilpotency class at most two; or, quotient by center is an abelian group			|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
group generated by abelian normal subgroups	generated by abelian normal subgroups			|FULL LIST, MORE INFO
bounded Engel group	$k$-Engel group for some finite $k$			|FULL LIST, MORE INFO
Engel group	For any two elements $x$ and $y$, the iterated commutator of $x$ with $y$ eventually becomes trivial			|FULL LIST, MORE INFO
nilpotent group (for finite groups)