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For questions regarding finitely generated groups, modules, and other algebraic structures. A structure is called finitely generated if there exists a finite subset that generates it.

For questions regarding finitely generated groups and other algebraic structures. A structure is called finitely generated if there exists a finite subset that generates it.

The structure of finitely generated abelian groups in particular is easily described. Many theorems that are true for finitely generated groups fail for groups in general. It has been proven that if a finite group is generated by a subset S, then each group element may be expressed as a word from the alphabet S of length less than or equal to the order of the group.