Noetherian group

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:

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

Relation with other properties

Stronger properties

• Finite group
• Finitely generated Abelian group

Weaker properties

• Finitely generated group
• Maximal-covering group
• Group satisfying ascending chain condition on subnormal subgroups
• Group satisfying subnormal join property
• Group satisfying ascending chain condition on normal subgroups
• Hopfian group: For proof of the implication, refer Slender implies Hopfian and for proof of its strictness (i.e. the reverse implication being false) refer Hopfian not implies slender.
• Direct product of finitely many indecomposable groups