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, ..., x_s in G such that every x in G can be written in the form

with integers n₁, ..., n_s. In this case, we say that the set {x₁, ..., x_s} is a generating set of G or that x₁, ..., x_sgenerate 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 ${\displaystyle \left(\mathbb {Q} ,+\right)}$ of rational numbers is not finitely generated: if ${\displaystyle x_{1},\ldots ,x_{n}}$ are rational numbers, pick a natural number ${\displaystyle k}$ coprime to all the denominators; then ${\displaystyle 1/k}$ cannot be generated by ${\displaystyle x_{1},\ldots ,x_{n}}$. The group ${\displaystyle \left(\mathbb {Q} ^{*},\cdot \right)}$ of non-zero rational numbers is also not finitely generated. The groups of real numbers under addition ${\displaystyle \left(\mathbb {R} ,+\right)}$ and non-zero real numbers under multiplication ${\displaystyle \left(\mathbb {R} ^{*},\cdot \right)}$ are also not finitely generated.

...
Wikipedia