# Noetherian group

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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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:

1. Every subgroup is finitely generated
2. Any ascending chain of subgroups stabilizes after a finite length.
3. 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

Metaproperty name Satisfied? Proof Statement with symbols
subgroup-closed group property Yes Noetherianness is subgroup-closed If $G$ is a Noetherian group and $H \le G$ is a subgroup, then $H$ is Noetherian.
quotient-closed group property Yes Noetherianness is quotient-closed If $G$ is a Noetherian group and $H \le G$ is a normal subgroup, the quotient group $G/H$ is Noetherian.
extension-closed group property Yes Noetherianness is extension-closed If $H$ is a normal subgroup of $G$ such that both $H$ and $G/H$ are Noetherian, then $G$ is also Noetherian.

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
finitely generated abelian group finitely generated and abelian Finitely generated nilpotent group, Group in which every subgroup is finitely presented, Polycyclic group|FULL LIST, MORE INFO
finitely generated nilpotent group finitely generated and nilpotent Group in which every subgroup is finitely presented, Polycyclic group|FULL LIST, MORE INFO
supersolvable group has a normal series where all successive quotients are cyclic Polycyclic group|FULL LIST, MORE INFO
polycyclic group slender and solvable Group in which every subgroup is finitely presented|FULL LIST, MORE INFO

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Group satisfying ascending chain condition on subnormal subgroups |FULL LIST, MORE INFO
Group satisfying subnormal join property Group satisfying ascending chain condition on subnormal subgroups, Group satisfying generalized 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 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
Finitely generated Hopfian group finitely generated and Hopfian Noetherian implies Hopfian |FULL LIST, MORE INFO
Group in which every locally finite subgroup is finite |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 Finitely generated nilpotent group, Group in which every subgroup is finitely presented, Polycyclic group|FULL LIST, MORE INFO Residually cyclic group|FULL LIST, MORE INFO
Finitely generated nilpotent group nilpotent group Group in which every subgroup is finitely presented, Polycyclic group|FULL LIST, MORE INFO |FULL LIST, MORE INFO