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.
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}.
Every finitely generated abelian group G is isomorphic to a direct product of the form
Because of the general fact 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, we can also write any abelian group G as a direct product of the form
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 finitelygenerated 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 rank1 group Q is one example, and the rank0 group given by a direct sum of countably many copies of Z_{2} is another one.
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