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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
_{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.## Examples

## Classification

### 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 such group is isomorphic to one of the form
_{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.
### Invariant factor decomposition

We can also write any finitely generated abelian group G as a direct sum of the form
_{1} divides k_{2}, which divides k_{3} and so on up to k_{u}. Again, the rank n and the invariant factors k_{1},...,k_{u} are uniquely determined by G (here with a unique order).
### Equivalence

These statements are equivalent because of the Chinese remainder theorem, which here states that Z_{m} is isomorphic to the direct product of Z_{j} and Z_{k} if and only if j and k are coprime and m = jk.
## Corollaries

Stated differently the fundamental theorem says that a finitely-generated abelian group is the direct sum of a free abelian group of finite rank and a finite abelian group, each of those being unique up to isomorphism. The finite abelian group is just the torsion subgroup of G. The rank of G is defined as the rank of the torsion-free part of G; this is just the number n in the above formulas.## Non-finitely generated abelian groups

Note that not every abelian group of finite rank is finitely generated; the rank-1 group Q is one example, and the rank-0 group given by a direct sum of countably many copies of Z_{2} is another one.
## See also

- x = n
_{1}x_{1}+ n_{2}x_{2}+ ... + n_{s}x_{s}

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.

- 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. 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}.

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):

- $mathbb\{Z\}^n\; oplus\; mathbb\{Z\}\_\{q\_1\}\; oplus\; cdots\; oplus\; mathbb\{Z\}\_\{q\_t\}$

- $mathbb\{Z\}^n\; oplus\; mathbb\{Z\}\_\{k\_1\}\; oplus\; cdots\; oplus\; mathbb\{Z\}\_\{k\_u\}$

A corollary to the fundamental theorem is that every finitely generated torsion-free abelian group is free abelian. The finitely generated condition is essential here: Q is torsion-free but not free abelian.

Every subgroup and factor group of a finitely generated abelian group is again finitely generated abelian. The finitely generated abelian groups, together with the group homomorphisms, form an abelian category which is a Serre subcategory of the category of abelian groups.

- The Jordan-Hölder theorem is a non-abelian generalization

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Last updated on Monday October 06, 2008 at 00:44:27 PDT (GMT -0700)

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