Group whose automorphism group is abelian: Difference between revisions
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===Symbol-free definition=== | ===Symbol-free definition=== | ||
A [[group]] is said to be ''' | 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=== | ===Definition with symbols=== | ||
A [[group]] <math>G</math> is said to be ''' | A [[group]] <math>G</math> is said to be '''aut-abelian''' if <math>Aut(G)</math> is an [[Abelian group]]. | ||
==Relation with other properties== | ==Relation with other properties== | ||
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===Stronger properties=== | ===Stronger properties=== | ||
* [[Cyclic group]] | * [[Weaker than::Cyclic group]]: {{proofofstrictimplicationat|[[cyclic implies aut-abelian]]|[[aut-abelian not implies cyclic]]}} | ||
===Weaker properties=== | ===Weaker properties=== | ||
* [[Metabelian group]] | * [[Stronger than::Group of nilpotence class two]] | ||
* [[Stronger than::Metabelian group]] | |||
==Facts== | ==Facts== | ||
Any aut-Abelian [[normal subgroup]] commutes with every element in the [[commutator subgroup]]. | Any aut-Abelian [[normal subgroup]] commutes with every element in the [[commutator subgroup]]. | ||
Revision as of 16:07, 14 February 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
Facts
Any aut-Abelian normal subgroup commutes with every element in the commutator subgroup.