Noetherian group: Difference between revisions
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| [[Weaker than::Polycyclic group]] || [[solvable group]] || {{intermediate notions short|slender group|polycyclic group}} || {{intermediate notions short|polycyclic group|solvable group}} | | [[Weaker than::Polycyclic group]] || [[solvable group]] || {{intermediate notions short|slender group|polycyclic group}} || {{intermediate notions short|polycyclic group|solvable group}} | ||
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===Related properties=== | |||
{| class="sortable" border="1" | |||
! Property !! Meaning !! Proof of one non-implication !! Proof of other non-implication !! Notions stronger than both !! Notions weaker than both | |||
|- | |||
| [[Artinian group]] || any descending chain of subgroups stabilizes after finitely many steps || [[slender not implies Artinian]] || [[Artinian not implies slender]] || {{stronger than both short|slender group|Artinian group}} || {{weaker than both short|slender group|Artinian group}} | |||
|- | |||
| [[Finitely presentable group]] || has a finite [[presentation of a group|presentation]] || [[slender not implies finitely presentable]] || [[finitely presentable not implies slender]] || {{stronger than both short|slender group|finitely presentable group}} || {{weaker than both short|slender group|finitely presentable group}} | |||
|} | |} | ||
Revision as of 17:11, 25 May 2010
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
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| Finite group | slender group|finite group} | |||
| Finitely generated abelian group | finitely generated and abelian | |FULL LIST, MORE INFO | ||
| Finitely generated nilpotent group | finitely generated and nilpotent | |FULL LIST, MORE INFO | ||
| Supersolvable group | has a normal series where all successive quotients are cyclic | |FULL LIST, MORE INFO | ||
| Polycyclic group | slender and solvable | |FULL LIST, MORE INFO |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| Finitely generated group | |FULL LIST, MORE INFO | |||
| Maximal-covering group | |FULL LIST, MORE INFO | |||
| Group satisfying ascending chain condition on subnormal subgroups | |FULL LIST, MORE INFO | |||
| Group satisfying subnormal join property | |FULL LIST, MORE INFO | |||
| Group satisfying ascending chain condition on normal subgroups | |FULL LIST, MORE INFO | |||
| Hopfian group | every surjective endomorphism is an automorphism | Slender implies Hopfian | Hopfian not implies slender | |FULL LIST, MORE INFO |
| Finitely generated Hopfian group | finitely generated and Hopfian | Slender implies Hopfian | |FULL LIST, MORE INFO |
Conjunction with other properties
| Conjunction | Other component of conjunction | Intermediate notions between slender group and conjunction | Intermediate notions between other component and conjunction |
|---|---|---|---|
| Finitely generated abelian group | abelian group | |FULL LIST, MORE INFO | |FULL LIST, MORE INFO |
| Finitely generated nilpotent group | nilpotent group | |FULL LIST, MORE INFO | |FULL LIST, MORE INFO |
| Polycyclic group | solvable group | |FULL LIST, MORE INFO | |FULL LIST, MORE INFO |
Related properties
| Property | Meaning | Proof of one non-implication | Proof of other non-implication | Notions stronger than both | Notions weaker than both |
|---|---|---|---|---|---|
| Artinian group | any descending chain of subgroups stabilizes after finitely many steps | slender not implies Artinian | Artinian not implies slender | |FULL LIST, MORE INFO | |FULL LIST, MORE INFO |
| Finitely presentable group | has a finite presentation | slender not implies finitely presentable | finitely presentable not implies slender | |FULL LIST, MORE INFO | |FULL LIST, MORE INFO |