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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“There are many examples of women that have excelled in learning, and even in war, but this is no reason we should bring ‘em all up to Latin and Greek or else military discipline, instead of needle-work and housewifry.”
—Bernard Mandeville (1670–1733)
“No rules exist, and examples are simply life-savers answering the appeals of rules making vain attempts to exist.”
—André Breton (1896–1966)
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—G.C. (Georg Christoph)