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 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 3-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)