Group whose automorphism group is abelian: Difference between revisions

Revision as of 23:52, 10 July 2011

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
For its use outside the wiki, please define the term when using it. If you are aware of an equivalent standard term, please leave a comment on the talk page
VIEW: Definitions built on this | Facts about this: (facts closely related to Group whose automorphism group is abelian, all facts related to Group whose automorphism group is abelian) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
Learn more about terminology local to the wiki | view a complete list of such terminology

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

Definition

QUICK PHRASES: abelian automorphism group, 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.

Definition with symbols

A group $G$ is said to be aut-abelian 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 aut-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 aut-abelian	aut-abelian not implies cyclic	\|FULL LIST, MORE INFO
locally cyclic group	every finitely generated subgroup is cyclic	locally cyclic implies aut-abelian	abelian and aut-abelian not implies locally cyclic	\|FULL LIST, MORE INFO

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
aut-nilpotent group	automorphism group is nilpotent	follows from abelian implies nilpotent	aut-nilpotent not implies aut-abelian	\|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

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.
Finite abelian and aut-abelian implies cyclic

@@ Line 27: / 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 aut-abelian]] || [[aut-abelian not implies cyclic]] || {{intermediate notions short|aut-abelian group|cyclic group}}
 |-
-| [[Weaker than::Locally cyclic group]] || every finitely generated subgroup is cyclic || [[locally cyclic implies aut-abelian]] || [[abelian and aut-abelian not implies locally cyclic]] || {{intermediate notions short|aut-abelian group|locally cyclic group}}
+| [[Weaker than::locally cyclic group]] || every finitely generated subgroup is cyclic || [[locally cyclic implies aut-abelian]] || [[abelian and aut-abelian not implies locally cyclic]] || {{intermediate notions short|aut-abelian group|locally cyclic group}}
 |}
@@ Line 37: / Line 37: @@
 ! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
 |-
-| [[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|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|aut-abelian group}}
 |-
-|[[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 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::Metabelian group]] || abelian normal subgroup with abelian quotient || (via class two) || (via class two) || {{intermediate notions short|metabelian group|aut-abelian group}}
+| [[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::Aut-nilpotent group]] || automorphism group is [[nilpotent group|nilpotent]] || follows from [[abelian implies nilpotent]] || [[aut-nilpotent not implies aut-abelian]] || {{intermediate notions short|aut-nilpotent group|aut-abelian group}}
+| [[Stronger than::aut-nilpotent group]] || automorphism group is [[nilpotent group|nilpotent]] || follows from [[abelian implies nilpotent]] || [[aut-nilpotent not implies aut-abelian]] || {{intermediate notions short|aut-nilpotent group|aut-abelian group}}
 |}