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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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
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

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Finitely generated group				Finitely generated Hopfian 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				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
Minimax group				|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
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