Finitely generated abelian group

In abstract algebra, an abelian group ${\displaystyle (G,+)}$ is called finitely generated if there exist finitely many elements ${\displaystyle x_{1},\dots ,x_{s}}$ in ${\displaystyle G}$ such that every ${\displaystyle x}$ in ${\displaystyle G}$ can be written in the form ${\displaystyle x=n_{1}x_{1}+n_{2}x_{2}+\cdots +n_{s}x_{s}}$ for some integers ${\displaystyle n_{1},\dots ,n_{s}}$. In this case, we say that the set ${\displaystyle \{x_{1},\dots ,x_{s}\}}$ is a generating set of ${\displaystyle G}$ or that ${\displaystyle x_{1},\dots ,x_{s}}$ generate ${\displaystyle G}$. So, finitely generated abelian groups can be thought of as a generalization of cyclic groups.

Every finite abelian group is finitely generated. The finitely generated abelian groups can be completely classified.