Group whose automorphism group is abelian: Difference between revisions
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* [[Stronger than::Group of nilpotency class two]]: {{proofofstrictimplicationat|[[aut-abelian implies class two]]|[[class two not implies aut-abelian]]}} | * [[Stronger than::Group of nilpotency class two]]: {{proofofstrictimplicationat|[[aut-abelian implies class two]]|[[class two not implies aut-abelian]]}} | ||
* [[Stronger than::Metabelian group]] | * [[Stronger than::Metabelian group]] | ||
===Related subgroup properties=== | |||
* [[Aut-abelian normal subgroup]] is a [[normal subgroup]] of a group that is aut-abelian as a group. | |||
==Facts== | ==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]]. | * [[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]]. | ||
Revision as of 13:24, 2 September 2009
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
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Definition
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 is said to be aut-abelian if is an Abelian group.
Relation with other properties
Stronger properties
- Cyclic group: For proof of the implication, refer cyclic implies aut-abelian and for proof of its strictness (i.e. the reverse implication being false) refer aut-abelian not implies cyclic.
Weaker properties
- Group of nilpotency class two: For proof of the implication, refer aut-abelian implies class two and for proof of its strictness (i.e. the reverse implication being false) refer class two not implies aut-abelian.
- Metabelian group
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.