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 Noetherian or slender 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
Metaproperties
Relation with other properties
Stronger properties
Weaker properties
Property |
Meaning |
Proof of implication |
Proof of strictness (reverse implication failure) |
Intermediate notions
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Finitely generated group |
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Finitely generated Hopfian group|FULL LIST, MORE INFO
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Maximal-covering group |
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Group satisfying ascending chain condition on subnormal subgroups |
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Group satisfying subnormal join property |
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Group satisfying ascending chain condition on subnormal subgroups, Group satisfying generalized subnormal join property|FULL LIST, MORE INFO
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Group satisfying ascending chain condition on normal subgroups |
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Hopfian group |
every surjective endomorphism is an automorphism |
Noetherian implies Hopfian |
Hopfian not implies Noetherian |
Finitely generated Hopfian group, Group satisfying ascending chain condition on normal subgroups, Group satisfying ascending chain condition on subnormal subgroups|FULL LIST, MORE INFO
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Finitely generated Hopfian group |
finitely generated and Hopfian |
Noetherian implies Hopfian |
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Minimax group |
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Group in which every locally finite subgroup is finite |
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Conjunction with other properties
Related properties