In abstract algebra, an abelian group (G,+) is called finitely generated if there exist finitely many elements x_{1},...,x_{s} in G such that every x in G can be written in the form
with integers n_{1},...,n_{s}. In this case, we say that the set {x_{1},...,x_{s}} is a generating set of G or that x_{1},...,x_{s} generate G.
Clearly, every finite abelian group is finitely generated. The finitely generated abelian groups are of a rather simple structure and can be completely classified, as will be explained below.
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Examples
 the integers (Z,+) are a finitely generated abelian group
 the integers modulo n Z_{n} are a finitely generated abelian group
 any direct sum of finitely many finitely generated abelian groups is again finitely generated abelian
There are no other examples (up to isomorphism). The group (Q,+) of rational numbers is not finitely generated: if x_{1},...,x_{s} are rational numbers, pick a natural number w coprime to all the denominators; then 1/w cannot be generated by x_{1},...,x_{s}.
Classification
The fundamental theorem of finitely generated abelian groups (which is a special case of the structure theorem for finitely generated modules over a principal ideal domain) can be stated two ways (analogously with PIDs):
Primary decomposition
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 the rank n ≥ 0, and the numbers q_{1},...,q_{t} are powers of (not necessarily distinct) prime numbers. In particular, G is finite if and only if n = 0. The values of n, q_{1},...,q_{t} are (up to rearranging the indices) uniquely determined by G.
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