Artinian group: Difference between revisions
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===Stronger properties=== | ===Stronger properties=== | ||
* [[Finite group]] | * [[Weaker than::Finite group]] | ||
===Weaker properties=== | ===Weaker properties=== | ||
* [[Minimax group]] | * [[Stronger than::Periodic group]]: {{proofofstrictimplicationat|[[Artinian implies periodic]]|[[Periodic not implies Artinian]]}} | ||
* [[Stronger than::Co-Hopfian group]]: {{proofofstrictimplicationat|[[Artinian implies co-Hopfian]]|[[Co-Hopfian not implies Artinian]]}} | |||
* [[Stronger than::Minimax group]] | |||
Latest revision as of 23:15, 1 March 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
This is a variation of finiteness (groups)|Find other variations of finiteness (groups) |
Definition
A group is said to be Artinian or to satisfy the minimum condition on subgroups if every descending chain of subgroups stabilizes after a finite stage.
Relation with other properties
Stronger properties
Weaker properties
- Periodic group: For proof of the implication, refer Artinian implies periodic and for proof of its strictness (i.e. the reverse implication being false) refer Periodic not implies Artinian.
- Co-Hopfian group: For proof of the implication, refer Artinian implies co-Hopfian and for proof of its strictness (i.e. the reverse implication being false) refer Co-Hopfian not implies Artinian.
- Minimax group