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.
Read more about this topic: Finitely-generated Abelian Group
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—Bernard Mandeville (1670–1733)
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“No rules exist, and examples are simply life-savers answering the appeals of rules making vain attempts to exist.”
—André Breton (1896–1966)