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Finitely generated abelian group

In abstract algebra, an abelian group (G,+) is called finitely generated if there exists finitely many elements x1,...,xs in G such that every x in G can be written in the from
x = n1x1 + n2x2 + ... + nsxs
with integers n1,...,ns. In this case, we say that the set {x1,...,xs} is a generating set of G or that x1,...,xs 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 (Z,+) are a finitely generated abelian group
  • the integers modulo n Zn 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 x1,...,xs are rational numbers, pick a natural number w coprime to all the denominators; then 1/w cannot be generated by x1,...,xs.

Properties and Classification

Every finitely generated abelian group G is isomorphic to a direct product of the form

(Z)n × Zm1 × ... × Zmt
where n ≥ 0, and the numbers m1,...,mt are (not necessarily distinct) powers of prime numbers. The values of n, m1,...,mt are (up to order) uniquely determined by G; in particular, G is finite if and only if n = 0

Because of the general fact that Zm is isomorphic to the direct product of Zj and Zk if and only if j and k are coprime and m = jk, we can also write any abelian group G as a direct product of the form

(Z)n × Zk1 × ... × Zku
where k1 divides k2, which divides k3 and so on up to ku. Again, the numbers n and k1,...,ku are uniquely determined by G.

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.

Every finitely generated abelian group has finite rank equal to the number n from above. Expressing the theorem in general terms, it says a finitely-generated abelian group is the sum of a free abelian group and a finite abelian group, each of those being unique up to isomorphism. The rank is an isomorphism invariant.

The converse is not true however: there are many abelian groups of finite rank which are not 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 Z2 is another one.



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