: Mathematics: finitely generated abelian group

algebra > group theory > abelian group > finitely generated abelian group

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 ${\\displaystyle G}$ .
Every finite abelian group is finitely generated. The finitely generated abelian groups can be completely classified.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Finitely_generated_abelian_group)