2-Engel group

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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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Definition

Symbol-free definition

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

Any two conjugate elements of the group commute.
The normal closure of any cyclic subgroup is Abelian.
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:

$x$ commutes with $g x g^{- 1}$ for all $x, g \in G$ .
The normal subgroup generated by $x$ is Abelian for all $x \in G$ .
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

Weaker properties