A finitely generated abelian group has a finite generating set. Equivalently, it is the cokernel of a homomorphism between finitely generated free abelian groups.
The rank of a finitely generated abelian group is the number of infinite cyclic factors in its invariant-factor decomposition, equivalently .
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A finitely generated abelian group is a specific type of group in abstract algebra that has some important properties. 1. **Group**: An abelian group is a set equipped with an operation (often called addition) that satisfies four properties: closure, associativity, identity, and inverses. Additionally, an abelian group is commutative, meaning that the order in which you combine elements does not matter (i.e.