Noetherian group: Difference between revisions
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A [[group]] is said to be '''slender''' or '''Noetherian''' or to satisfy the '''maximum condition on subgroups''' if it satisfies the following equivalent conditions: | A [[group]] is said to be '''slender''' or '''Noetherian''' or to satisfy the '''maximum condition on subgroups''' if it satisfies the following equivalent conditions: | ||
# Every [[subgroup]] is [[finitely generated group|finitely generated]] | |||
# Any [[ascending chain of subgroups]] stabilizes after a finite length | |||
# Any nonempty collection of subgroups has a maximal element: a member of that collection that is not contained in any other member of the collection. | |||
==Formalisms== | ==Formalisms== | ||
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* [[Stronger than::Finitely generated group]] | * [[Stronger than::Finitely generated group]] | ||
* [[Stronger than::Maximal-covering group]] | * [[Stronger than::Maximal-covering group]] | ||
* [[Stronger than::Group satisfying ascending chain condition on subnormal subgroups]] | |||
* [[Stronger than::Group satisfying subnormal join property]] | |||
* [[Stronger than::Group satisfying ascending chain condition on normal subgroups]] | * [[Stronger than::Group satisfying ascending chain condition on normal subgroups]] | ||
* [[Stronger than::Hopfian group]] | * [[Stronger than::Hopfian group]]: {{proofofstrictimplicationat|[[Slender implies Hopfian]]|[[Hopfian not implies slender]]}} | ||
* [[Stronger than::Direct product of finitely many indecomposable groups]] | * [[Stronger than::Direct product of finitely many indecomposable groups]] | ||
Revision as of 22:52, 20 October 2008
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
This is a variation of finiteness (groups)|Find other variations of finiteness (groups) |
Definition
Symbol-free definition
A group is said to be slender or Noetherian or to satisfy the maximum condition on subgroups if it satisfies the following equivalent conditions:
- Every subgroup is finitely generated
- Any ascending chain of subgroups stabilizes after a finite length
- Any nonempty collection of subgroups has a maximal element: a member of that collection that is not contained in any other member of the collection.
Formalisms
In terms of the hereditarily operator
This property is obtained by applying the hereditarily operator to the property: finitely generated group
View other properties obtained by applying the hereditarily operator
Relation with other properties
Stronger properties
Weaker properties
- Finitely generated group
- Maximal-covering group
- Group satisfying ascending chain condition on subnormal subgroups
- Group satisfying subnormal join property
- Group satisfying ascending chain condition on normal subgroups
- Hopfian group: For proof of the implication, refer Slender implies Hopfian and for proof of its strictness (i.e. the reverse implication being false) refer Hopfian not implies slender.
- Direct product of finitely many indecomposable groups