2-Engel group: Difference between revisions

Revision as of 07:57, 17 May 2012

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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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)

@@ Line 8: / Line 8: @@
 | 2 || normal closures abelian || the [[defining ingredient::normal closure]] of any [[cyclic group|cyclic]] subgroup (or the [[defining ingredient::normal subgroup generated by a subset|normal subgroup generated]] by any one-element subset) is [[defining ingredient::abelian group|abelian]] || the normal subgroup generated by <math>x</math> is abelian for all <math>x \in G</math>.
 |-
-| 3 || 2-Engel || the group is a <math>2</math>-[[defining ingredient::bounded Engel group|Engel group]]: the commutator between any element and its commutator with another element is the identity element. || the commutator <math>[x,[x,g]]</math> is the identity element for all <math>x,g \in G</math>.
+| 3 || union of abelian normal subgroups || the group is a ''union'' (as a set) of [[defining ingredient::abelian normal subgroup]]s || there is a collection of abelian normal subgroups <math>N_i, i \in I</math> of <math>G</math> such that <math>G = \bigcup_{i \in I} N_i</math>
 |-
-| 4 || union of abelian normal subgroups || the group is a ''union'' (as a set) of [[defining ingredient::abelian normal subgroup]]s || there is a collection of abelian normal subgroups <math>N_i, i \in I</math> of <math>G</math> such that <math>G = \bigcup_{i \in I} N_i</math>
+| 4 || 2-local class two || the 2-[[defining ingredient::local nilpotency class]] of the group is at most 2. || for any <math>x,g \in G</math>, the subgroup <math>\langle x,g \rangle</math> is a [[defining ingredient::group of nilpotency class two|group of class at most two]].
 |-
-| 5 || 2-local class two || the 2-[[defining ingredient::local nilpotency class]] of the group is at most 2. || for any <math>x,g \in G</math>, the subgroup <math>\langle x,g \rangle</math> is a [[defining ingredient::group of nilpotency class two|group of class at most two]].
+| 5 || 2-Engel || the group is a <math>2</math>-[[defining ingredient::bounded Engel group|Engel group]]: the commutator between any element and its commutator with another element is the identity element. || the commutator <math>[x,[x,g]]</math> is the identity element for all <math>x,g \in G</math>.
 |-
 |6 || cyclic property of triple commutators || triple commutators are preserved under cyclic permutation of the inputs. ||for all <math>x,y,z \in G</math>, we have <math>[x,[y,z]] = [y,[z,x]] = [z,[x,y]]</math>.