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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,...,x_s in G such that every x in G can be written in the from

x = n₁x₁ + n₂x₂ + ... + n_sx_s

with integers n₁,...,n_s. In this case, we say that the set {x₁,...,x_s} is a generating set of G or that x₁,...,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 (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 isn't finitely generated: if x₁,...,x_s are rational numbers, pick a natural number w coprime to all the denominators; then 1/w cannot be generated by x₁,...,x_s.

Properties and Classification

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

(Z)ⁿ × Z_m₁ × ... × Z_{m_t}

where n ≥ 0, and the numbers m₁,...,m_t are (not necessarily distinct) powers of prime numbers. The values of n, m₁,...,m_t 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 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

(Z)ⁿ × Z_k₁ × ... × Z_{k_u}

where k₁ divides k₂, which divides k₃ and so on up to k_u. Again, the numbers n and k₁,...,k_u 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 isn't 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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