2-Engel group: Difference between revisions

Revision as of 18:33, 10 December 2010

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 [[defining ingredient::normal subgroup generated by a subset\|normal subgroup generated) by any one-element subset) is abelian	the normal subgroup generated by $x$ is abelian for all $x\in G$ .
3	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 $x,g\in G$ .
4	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 $G=\bigcup _{i\in I}N_{i}$

This article defines a term that has been used or referenced in a journal article or standard publication, but may not be generally accepted by the mathematical community as a standard term.[SHOW MORE]

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View other semistandard definitions in group theory

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	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 1: / Line 1: @@
-{{semistddef}}
-{{group property}}
 ==Definition==
-===Symbol-free definition===
+{| class="sortable" border="1"
+! No. !! Shorthand !! A group is termed a Levi group or a 2-Engel group if ... !! A group <math>G</math> is termed a Levi group or 2-Engel group if ...
+|-
+| 1 || conjugates commute || any two [[conjugate elements]] of the group commute. || <math>x</math> commutes with <math>gxg^{-1}</math> for all <math>x,g \in G</math>
+|-
+| 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>.
+|-
+| 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>
+|}
-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.
+{{semistddef}}
-# The [[normal closure]] of any cyclic subgroup is [[Abelian group|Abelian]].
+{{group property}}
-# The group is a <math>2</math>-Engel group: the commutator between any element and its commutator with another element is the identity element.
-===Definition with symbols===
-A [[group]] <math>G</math> is termed a Levi-group or a 2-Engel group if it satisfies the following equivalent conditions:
-# <math>x</math> commutes with <math>gxg^{-1}</math> for all <math>x,g \in G</math>.
-# The normal subgroup generated by <math>x</math> is Abelian for all <math>x \in G</math>.
-# The commutator <math>[x,[x,g]]</math> is the identity element for all <math>x,g \in G</math>.
 ==Formalisms==