Finitely-generated abelian group

A finitely generated abelian group is an abelian group (G, ), in which it finitely many elements, so that each in the form

Can be written, where are integers and the times linking is with itself. This representation need not be unique. We also say are the producers of G or generate G.

Every finite abelian group is obviously finitely generated. Finitely generated abelian groups are rather simple in nature and can be classified in a simple manner, as will be shown below.

Examples

• All finite groups are finitely generated. Therefore, finite abelian groups are finitely generated.
• The integers ( ℤ, ) is an infinite abelian group is finitely generated with 1 being the producer.
• Every direct sum of finitely many finitely generated abelian groups is a finitely generated abelian group again.

The additive group of rational numbers ( ℚ, ) is not finitely generated: one to choose a natural number which is relatively prime to the denominators of all; then can not be represented as integer linear combination of.

Classification

Each subgroup and factor group of a finitely generated abelian group is again finitely generated abelian. The finitely generated abelian groups together with the Gruppenmorphismen form an abelian category.

Note that not every abelian group generated by finite rank finite. ℚ for example, is not finitely generated of rank 1. Another example is the direct sum of infinitely many copies of ℤ 2, this is also not finitely generated of rank 0, but.

The fundamental theorem on finitely generated abelian groups states that every finitely generated abelian group G is a finite direct sum of cyclic groups whose order is a power of a prime number, and infinite cyclic groups isomorphic.

Finite Abelian Groups

• For each natural number with prime factorization exist exactly isomorphism of abelian groups of items. The function is the partition function, the result sequence is A000688 in OEIS.

• Any such abelian group with elements having a generating set of at most elements.

• Specifically, the following applies: If a square-free natural number, then every abelian group with elements is cyclic.