Finitely generated abelian group

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 defining ingredient::external direct product of finitely many cyclic groups.

Metaproperties
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.

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.

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.