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
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A finitely generated abelian group is a group satisfying the following equivalent conditions:
- It is finitely generated and abelian.
- It is a finitely generated module over , the ring of integers.
- 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
|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|
This group property is subgroup-closed, viz., any subgroup of a group satisfying the property also satisfies the property
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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.
This group property is quotient-closed, viz., any quotient of a group satisfying the property also has the property
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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.
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
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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.