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 isomorphic to an external 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

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Finite abelian group	finite and abelian	follows from finite implies finitely generated	the group of integers is a finitely generated abelian group that is not finite	|FULL LIST, MORE INFO
Cyclic group	generated by one element	cyclic implies abelian	abelian not implies cyclic, even in the finite case	|FULL LIST, MORE INFO
Finitely generated free abelian group	direct product of finitely many copies of the group of integers			|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Group in which every subgroup is finitely presented				|FULL LIST, MORE INFO
Finitely presented group	has a finite presentation	follows from structure theorem for finitely generated abelian groups		|FULL LIST, MORE INFO
Noetherian group (also called slender group)	every subgroup is finitely generated	finitely generated abelian implies Noetherian (equivalently, ring of integers is Noetherian)		|FULL LIST, MORE INFO
Residually finite group	every non-identity element is outside a normal subgroup of finite index	finitely generated abelian implies residually finite		|FULL LIST, MORE INFO
Finitely generated residually finite group	both a finitely generated group and a residually finite group	finitely generated abelian implies residually finite		|FULL LIST, MORE INFO
Hopfian group	every surjective endomorphism is an automorphism	finitely generated abelian implies Hopfian		|FULL LIST, MORE INFO
Finitely generated nilpotent group	both a finitely generated group and a nilpotent group	abelian implies nilpotent	nilpotent not implies abelian (counterexamples are there in the finite case)	|FULL LIST, MORE INFO
Finitely generated solvable group	both a finitely generated group and a solvable group			|FULL LIST, MORE INFO

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.