Group whose automorphism group is abelian: Difference between revisions

Latest revision as of 19:18, 20 June 2013

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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

QUICK PHRASES: abelian automorphism group, any two automorphisms commute

Symbol-free definition

A group is said to be a group whose automorphism group is abelian or a group with abelian automorphism group if its automorphism group is an abelian group or equivalently, if any two automorphisms of the group commute.

Definition with symbols

A group $G$ is said to be a group whose automorphism group is abelian or a group with abelian automorphism group if $A u t (G)$ is an abelian group.

Formalisms

In terms of the supergroup property collapse operator

This group property can be defined in terms of the collapse of two subgroup properties in the following sense. Whenever the given group is embedded as a subgroup satisfying the first subgroup property (normal subgroup), in some bigger group, it also satisfies the second subgroup property (normal subgroup contained in centralizer of commutator subgroup), and vice versa.
View other group properties obtained in this way

A group $H$ is a group whose automorphism group is abelian if and only if for every group $G$ containing $H$ as a normal subgroup, $H$ is also contained in the centralizer of derived subgroup of $G$ .

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
cyclic group	generated by one element	cyclic implies abelian automorphism group	follows from abelian automorphism group not implies abelian	\|FULL LIST, MORE INFO
locally cyclic group	every finitely generated subgroup is cyclic	locally cyclic implies abelian automorphism group	abelian and abelian automorphism group not implies locally cyclic	\|FULL LIST, MORE INFO
group whose automorphism group is cyclic	automorphism group is a cyclic group	(follows from cyclic implies abelian)	follows from any example of a cyclic group whose automorphism group is not cyclic, e.g., cyclic group:Z8.

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
group whose inner automorphism group is central in automorphism group	inner automorphism group is in center of automorphism group			\|FULL LIST, MORE INFO
group of nilpotency class two	inner automorphism group is abelian	aut-abelian implies class two	class two not implies aut-abelian	\|FULL LIST, MORE INFO
metabelian group	abelian normal subgroup with abelian quotient	(via class two)	(via class two)	\|FULL LIST, MORE INFO
group whose automorphism group is nilpotent	automorphism group is nilpotent	follows from abelian implies nilpotent	nilpotent automorphism group not implies abelian automorphism group	\|FULL LIST, MORE INFO

Related subgroup properties

Aut-abelian normal subgroup is a normal subgroup of a group that is aut-abelian as a group.

Facts

Derived subgroup centralizes normal subgroup whose automorphism group is abelian: Any normal subgroup whose automorphism group is abelian commutes with every element in the derived subgroup. Hence, it is contained in the centralizer of commutator subgroup.
Finite abelian and abelian automorphism group implies cyclic

@@ Line 3: / Line 3: @@
 ==Definition==
+{{quick phrase|[[quick phrase::abelian automorphism group]], [[quick phrase::any two automorphisms commute]]}}
 ===Symbol-free definition===
-A [[group]] is said to be '''aut-abelian''' if its [[automorphism group]] is an [[Abelian group]] or equivalently, if any two automorphisms of the group commute.
+A [[group]] is said to be a '''group whose automorphism group is abelian''' or a '''group with abelian automorphism group''' if its [[defining ingredient::automorphism group]] is an [[defining ingredient::abelian group]] or equivalently, if any two automorphisms of the group commute.
 ===Definition with symbols===
-A [[group]] <math>G</math> is said to be '''aut-abelian''' if <math>Aut(G)</math> is an [[Abelian group]].
+A [[group]] <math>G</math> is said to be a '''group whose automorphism group is abelian''' or a '''group with abelian automorphism group''' if <math>\operatorname{Aut}(G)</math> is an [[abelian group]].
+==Formalisms==
+{{supergroup property collapse|normal subgroup|normal subgroup contained in centralizer of commutator subgroup}}
+A group <math>H</math> is a group whose automorphism group is abelian if and only if for every group <math>G</math> containing <math>H</math> as a [[normal subgroup]], <math>H</math> is also contained in the [[centralizer of derived subgroup]] of <math>G</math>.
 ==Relation with other properties==
@@ Line 19: / Line 27: @@
 ! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
 |-
-| [[Weaker than::Cyclic group]] || generated by one element || [[cyclic implies aut-abelian]] || [[aut-abelian not implies cyclic]] || {{intermediate notions short|aut-abelian group|cyclic group}}
+| [[Weaker than::cyclic group]] || generated by one element || [[cyclic implies abelian automorphism group]] || follows from [[abelian automorphism group not implies abelian]] || {{intermediate notions short|group whose automorphism group is abelian|cyclic group}}
+|-
+| [[Weaker than::locally cyclic group]] || every finitely generated subgroup is cyclic || [[locally cyclic implies abelian automorphism group]] || [[abelian and abelian automorphism group not implies locally cyclic]] || {{intermediate notions short|group whose automorphism group is abelian|locally cyclic group}}
+|-
+| [[Weaker than::group whose automorphism group is cyclic]] || automorphism group is a [[cyclic group]] || (follows from [[cyclic implies abelian]]) || follows from any example of a cyclic group whose automorphism group is not cyclic, e.g., [[cyclic group:Z8]]. ||
 |}
@@ Line 27: / Line 39: @@
 ! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
 |-
-|[[Stronger than::Group of nilpotency class two]] || inner automorphism group is abelian || [[aut-abelian implies class two]] || [[class two not implies aut-abelian]]  || {{intermediate notions short|group of nilpotency class two|aut-abelian group}}
+| [[Stronger than::group whose inner automorphism group is central in automorphism group]] || inner automorphism group is in [[center]] of automorphism group || || || {{intermediate notions short|group whose inner automorphism group is central in automorphism group|group whose automorphism group is abelian}}
+|-
+|[[Stronger than::group of nilpotency class two]] || inner automorphism group is abelian || [[aut-abelian implies class two]] || [[class two not implies aut-abelian]]  || {{intermediate notions short|group of nilpotency class two|group whose automorphism group is abelian}}
+|-
+| [[Stronger than::metabelian group]] || abelian normal subgroup with abelian quotient || (via class two) || (via class two) || {{intermediate notions short|metabelian group|group whose automorphism group is abelian}}
 |-
-| [[Stronger than::Metabelian group]] || abelian normal subgroup with abelian quotient || (via class two) || (via class two) || {{intermediate notions short|metabelian group|aut-abelian group}}
+| [[Stronger than::group whose automorphism group is nilpotent]] || automorphism group is [[nilpotent group|nilpotent]] || follows from [[abelian implies nilpotent]] || [[nilpotent automorphism group not implies abelian automorphism group]] || {{intermediate notions short|group whose automorphism group is nilpotent|group whose automorphism group is abelian}}
 |}
@@ Line 38: / Line 54: @@
 ==Facts==
-* [[Commutator subgroup centralizes aut-abelian normal subgroup]]: Any [[aut-abelian normal subgroup]] commutes with every element in the [[commutator subgroup]]. Hence, it is contained in the [[centralizer of commutator subgroup]].
+* [[Derived subgroup centralizes normal subgroup whose automorphism group is abelian]]: Any [[normal subgroup whose automorphism group is abelian]] commutes with every element in the [[derived subgroup]]. Hence, it is contained in the [[centralizer of commutator subgroup]].
-* [[Finite abelian and aut-abelian implies cyclic]]
+* [[Finite abelian and abelian automorphism group implies cyclic]]