Finitely-generated abelian group

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 formAs far as the fundamental theorem on finite abelian groups is concerned, it is not clear how far back in time one needs to go to trace its origin. ... it took a long time to formulate and prove the fundamental theorem in its present form ... 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, ..., xs generate G.