*** Welcome to piglix ***

Fundamental theorem of finitely generated abelian groups


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

with integers n1, ..., ns. In this case, we say that the set {x1, ..., xs} is a generating set of G or that x1, ..., xsgenerate 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.

There are no other examples (up to isomorphism). In particular, the group of rational numbers is not finitely generated: if are rational numbers, pick a natural number coprime to all the denominators; then cannot be generated by . The group of non-zero rational numbers is also not finitely generated. The groups of real numbers under addition and non-zero real numbers under multiplication are also not finitely generated.


...
Wikipedia

...