2-Engel group: Difference between revisions

Latest revision as of 00:33, 3 December 2024

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 closure of the 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 $[x,[y,z]]=[y,[z,x]]=[z,[x,y]]$ .

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]

VIEW: Definitions built on this | Facts about this: (facts closely related to 2-Engel group, all facts related to 2-Engel group) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
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
View a complete list of group properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

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 strictness (reverse implication failure)	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	2-Engel not implies class two for groups	\|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)
Bell group

Examples

Finite groups

Every finite abelian group, since abelian groups are 2-Engel.
The smallest non-abelian finite groups which are 2-Engel are both of the non-abelian groups of order 8: dihedral group:D8 and quaternion group. Note in particular that symmetric group:S3 is not 2-Engel.

@@ Line 6: / Line 6: @@
 | 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>.
+| 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 closure of the subgroup generated by <math>x</math> is abelian for all <math>x \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>
@@ Line 36: / Line 36: @@
 | [[Weaker than::Dedekind group]] || every subgroup is normal || || || {{intermediate notions short|Levi group|Dedekind group}}
 |-
-| [[Weaker than::group of nilpotency class two]] || [[nilpotency class]] at most two; or, quotient by [[center]] is an [[abelian group]] || || || {{intermediate notions short|Levi group|group of nilpotency class two}}
+| [[Weaker than::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]]|| {{intermediate notions short|Levi group|group of nilpotency class two}}
 |}
@@ Line 53: / Line 53: @@
 |-
 | [[nilpotent group]] (for [[finite group]]s) || || || ||
+|-
+| [[Stronger than::Bell group]] || || || ||
 |}
+==Examples==
+===Finite groups===
+* Every finite [[abelian group]], since abelian groups are 2-Engel.
+* The smallest non-abelian finite groups which are 2-Engel are both of the non-abelian groups of order 8: [[dihedral group:D8]] and [[quaternion group]]. Note in particular that [[symmetric group:S3]] is not 2-Engel.