2-Engel group

Definition

No.	Shorthand	A group is termed a Levi group or a 2-Engel group if ...	A group $G$ is termed a Levi group or 2-Engel group if ...
1	conjugates commute	any two conjugate elements of the group commute.	$x$ commutes with $gxg^{-1}$ for all $x,g\in G$
2	normal closures abelian	the normal closure of any cyclic subgroup (or the normal subgroup generated by any one-element subset) is abelian	the normal closure of the subgroup generated by $x$ is abelian for all $x\in G$.
3	union of abelian normal subgroups	the group is a union (as a set) of abelian normal subgroups	there is a collection of abelian normal subgroups $N_{i},i\in I$ of $G$ such that $G=\bigcup _{i\in I}N_{i}$
4	2-local class two	the 2-local nilpotency class of the group is at most 2.	for any $x,g\in G$, the subgroup $\langle x,g\rangle$ is a group of class at most two.
5	2-Engel	the group is a $2$-Engel group: the commutator between any element and its commutator with another element is the identity element.	the commutator $[x,[x,g]]$ is the identity element for all $x,g\in G$.
6	cyclic property of triple commutators	triple commutators are preserved under cyclic permutation of the inputs.	for all $x,y,z\in G$, we have $[x,[y,z]]=[y,[z,x]]=[z,[x,y]]$.

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This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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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		2-Engel not implies class two for groups	|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
group in which order of commutator divides order of element	For any two elements $x$ and $y$, if the order of $x$ is finite, the order of $[x,y]$ divides the order of $x$			|FULL LIST, MORE INFO
nilpotent group (for finite groups)
Bell group

Examples

Finite groups

• Every finite abelian group, since abelian groups are 2-Engel.
• The smallest non-abelian finite groups which are 2-Engel are both of the non-abelian groups of order 8: dihedral group:D8 and quaternion group. Note in particular that symmetric group:S3 is not 2-Engel.