Dedekind group: Difference between revisions
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===Symbol-free definition=== | ===Symbol-free definition=== | ||
A [[group]] is termed a '''Dedekind | A [[group]] is termed a '''Dedekind group''' if it satisfies the following equivalent conditions: | ||
* Every [[subgroup]] is [[normal subgroup|normal]] | * Every [[subgroup]] is [[normal subgroup|normal]] | ||
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* The [[normal closure]] of any element is a [[cyclic group]] | * The [[normal closure]] of any element is a [[cyclic group]] | ||
A non-Abelian Dedekind group is termed a [[Hamiltonian group]]. | |||
===In terms of operators=== | ===In terms of operators=== | ||
Revision as of 08:37, 4 September 2007
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
Symbol-free definition
A group is termed a Dedekind group if it satisfies the following equivalent conditions:
- Every subgroup is normal
- Every cyclic subgroup is normal
- Every inner automorphism is a power automorphism
- The normal closure of any element is a cyclic group
A non-Abelian Dedekind group is termed a Hamiltonian group.
In terms of operators
- In terms of the Levi operator: The group property of being a Dedekind group (or Hamiltonian group) is obtained by applying the Levi operator to the group property of being cyclic.
- In terms of the Hamiltonian operator: The group property of being a Dedekind group (or Hamiltonian group) is obtained by applying the Hamiltonian operator to the subgroup property of being normal.
Relation with other properties
Stronger properties
Weaker properties
Metaproperties
Subgroups
This group property is subgroup-closed, viz., any subgroup of a group satisfying the property also satisfies the property
View a complete list of subgroup-closed group properties
Using the fact that normality satisfies the intermediate subgroup condition, viz any normal subgroup of the whole group is also normal in every intermediate subgroup, we conclude that any subgroup of a Dedekind group is again a Dedekind group.
Quotients
This group property is quotient-closed, viz., any quotient of a group satisfying the property also has the property
View a complete list of quotient-closed group properties
Using the fact that normality satisfies the image condition, viz the image of any normal subgroup via a quotient map is again a normal subgroup, we conclude that any quotient of a Dedekind group is again a Dedekind group.