Finitely-generated Abelian Group - Examples

Examples

  • the integers are a finitely generated abelian group
  • the integers modulo, are a finitely generated abelian group
  • any direct sum of finitely many finitely generated abelian groups is again a finitely generated abelian group
  • every lattice forms a finitely-generated free abelian group

There are no other examples (up to isomorphism). In particular, the group of rational numbers is not finitely generated: if are rational numbers, pick a natural number coprime to all the denominators; then cannot be generated by . The group of non-zero rational numbers is also not finitely generated.

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