Finitely generated abelian is subgroup-closed

This article gives the statement, and possibly proof, of a group property (i.e., finitely generated abelian group) satisfying a group metaproperty (i.e., subgroup-closed group property)
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This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., finitely generated abelian group) must also satisfy the second group property (i.e., Noetherian group)
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Statement

Any subgroup of a finitely generated abelian group is a finitely generated abelian group. In particular, any finitely generated abelian group is a Noetherian group -- every subgroup of it is also a Finitely generated group (?).