# Noetherian group

From Groupprops

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 propertiesVIEW 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) |

## Contents

## 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

Metaproperty name | Satisfied? | Proof | Statement with symbols |
---|---|---|---|

subgroup-closed group property | Yes | Noetherianness is subgroup-closed | If is a Noetherian group and is a subgroup, then is Noetherian. |

quotient-closed group property | Yes | Noetherianness is quotient-closed | If is a Noetherian group and is a normal subgroup, the quotient group is Noetherian. |

extension-closed group property | Yes | Noetherianness is extension-closed | If is a normal subgroup of such that both and are Noetherian, then is also Noetherian. |

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

finite group | Group of finite max-length|FULL LIST, MORE INFO | |||

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

### 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 |

Polycyclic group | solvable group | Group in which every subgroup is finitely presented|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 | Noetherian not implies Artinian | Artinian not implies Noetherian | Group of finite max-length|FULL LIST, MORE INFO | Group in which no subgroup has an infinite minimal generating set, Group with no infinite minimal generating set, Minimax group|FULL LIST, MORE INFO |

Finitely presented group | has a finite presentation | Noetherian not implies finitely presented | finitely presented not implies Noetherian | Finitely generated abelian group, Finitely generated nilpotent group, Group in which every subgroup is finitely presented, Polycyclic group|FULL LIST, MORE INFO | |FULL LIST, MORE INFO |