# Finitely generated abelian group

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

*x*=*n*_{1}*x*_{1}+*n*_{2}*x*_{2}+ ... +*n*_{s}*x*_{s}

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.

## 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:^{[1]} 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 (**R**, +) and real numbers under multiplication (**R**, ×) are also not finitely generated.^{[1]}^{[2]}

## 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 principal ideal domains):

### 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*.

### Invariant factor decomposition

We can also write any finitely generated abelian group *G* as a direct sum of the form

where *k*_{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 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.

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: 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.

## Non-finitely generated abelian groups

Note that not every abelian group of finite rank is finitely generated; the rank 1 group is one counterexample, and the rank-0 group given by a direct sum of countably infinitely many copies of is another one.

## See also

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

## Notes

## References

- Silverman, Joseph H.; Tate, John Torrence (1992).
*Rational points on elliptic curves*. Undergraduate Texts in Mathematics. Springer. ISBN 978-0-387-97825-3. - de la Harpe, Pierre (2000).
*Topics in geometric group theory*. Chicago lectures in mathematics. University of Chicago Press. ISBN 978-0-226-31721-2.