Finitely-generated Abelian Group
In abstract algebra, an abelian group (G,+) is called finitely generated if there exist finitely many elements x1,...,xs in G such that every x in G can be written in the form
- x = n1x1 + n2x2 + ... + nsxs
with integers n1,...,ns. In this case, we say that the set {x1,...,xs} is a generating set of G or that x1,...,xs generate G.
Clearly, every finite abelian group is finitely generated. The finitely generated abelian groups are of a rather simple structure and can be completely classified, as will be explained below.
Read more about Finitely-generated Abelian Group: Examples, Classification, Corollaries, Non-finitely Generated Abelian Groups
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