Finitely generated abelian group

This page describes a group property obtained as a conjunction (AND) of two (or more) more fundamental group properties: finitely generated group and Abelian group
View other group property conjunctions OR view all group properties

Definition

Symbol-free definition

A finitely generated Abelian group is a group satisfying the following equivalent conditions:

1. It is finitely generated and Abelian
2. It is a finitely generated module over $\mathbb{Z}$, the ring of integers
3. It is a direct product of finitely many cyclic groups

Equivalence of definitions

Further information: structure theorem for finitely generated Abelian groups

Relation with other properties

Stronger properties

• Finite Abelian group
• Cyclic group

Weaker properties

• Finitely presented group
• Slender group
• Finitely generated nilpotent group

Metaproperties

Subgroups

This group property is subgroup-closed, viz., any subgroup of a group satisfying the property also satisfies the property
View a complete list of subgroup-closed group properties

Any subgroup of a finitely generated Abelian group is finitely generated Abelian. This is related to the fact that since the ring of integers is Noetherian, any submodule of a finitely generated module over it is finitely generated.

In general, subgroups of finitely generated groups are not finitely generated; those finitely generated groups for which every subgroup is finitely generated, are termed sliender groups or Noetherian groups.

Quotients

This group property is quotient-closed, viz., any quotient of a group satisfying the property also has the property
View a complete list of quotient-closed group properties

Any quotient of a finitely generated Abelian group is finitely generated Abelian. In fact, both the property of being finitely gneerated and the property of being Abelian are preserved on passing to quotients.

Direct products

This group property is finite direct product-closed, viz the direct product of a finite collection of groups each having the property, also has the property
View other finite direct product-closed group properties

A finite direct product of finitely generated Abelian groups is finitely generated Abelian. In fact, both the property of being finitely generated and the property of being Abelian are closed upon taking finite direct products.