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Concept
Finitely generated abelian group
Summary
In abstract algebra, an abelian group is called finitely generated if there exist finitely many elements in such that every in can be written in the form for some integers . In this case, we say that the set is a generating set of or that generate .
Every finite abelian group is finitely generated. The finitely generated abelian groups can be completely classified.
The integers, , are a finitely generated abelian group.
The integers modulo , , are a finite (hence 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. The groups of real numbers under addition and non-zero real numbers under multiplication are also not finitely generated.
The fundamental theorem of finitely generated abelian groups can be stated two ways, generalizing the two forms of the fundamental theorem of finite abelian groups. The theorem, in both forms, in turn generalizes to the structure theorem for finitely generated modules over a principal ideal domain, which in turn admits further generalizations.
The primary decomposition formulation states that every finitely generated abelian group G is isomorphic to a direct sum of primary cyclic groups and infinite cyclic groups. A primary cyclic group is one whose order is a power of a prime. That is, every finitely generated abelian group is isomorphic to a group of the form
where n ≥ 0 is the rank, and the numbers q1, ..., qt are powers of (not necessarily distinct) prime numbers. In particular, G is finite if and only if n = 0. The values of n, q1, ..., qt are (up to rearranging the indices) uniquely determined by G, that is, there is one and only one way to represent G as such a decomposition.
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