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In abstract algebra, an abelian group {\displaystyle } is called finitely generated if there exist finitely many elements x 1 , … , x s {\displaystyle x_{1},\dots ,x_{s}} in G {\displaystyle G} such that every x {\displaystyle x} in G {\displaystyle G} can be written in the form x = n 1 x 1 + n 2 x 2 + ⋯ + n s x s {\displaystyle x=n_{1}x_{1}+n_{2}x_{2}+\cdots +n_{s}x_{s}} for some integers n 1 , … , n s {\displaystyle n_{1},\dots ,n_{s}}. In this case, we say that the set { x 1 , … , x s } {\displaystyle \{x_{1},\dots ,x_{s}\}} is a generating set of G {\displaystyle G} or that x 1 , … , x s {\displaystyle x_{1},\dots ,x_{s}} generate G {\displaystyle G}.
Every finite abelian group is finitely generated. The finitely generated abelian groups can be completely classified.