2-Engel group: Difference between revisions

Revision as of 18:28, 10 December 2010

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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

Property	Meaning	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	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)

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 | [[Stronger than::Engel group]] || For any two elements <math>x</math> and <math>y</math>, the iterated commutator of <math>x</math> with <math>y</math> eventually becomes trivial || || || {{intermediate notions short|Engel group|Levi group}}
+|-
+| [[Stronger than::group in which order of commutator divides order of element]] || For any two elements <math>x</math> and <math>y</math>, if the order of <math>x</math> is finite, the order of <math>[x,y]</math> divides the order of <math>x</math> || || || {{intermediate notions short|group in which order of commutator divides order of element|Levi group}}
 |-
 | [[nilpotent group]] (for [[finite group]]s) || || || ||
 |}