In abstract algebra, an abelian group (G,+) is called finitely generated if there exist finitely many elements x₁,...,x_s in G such that every x in G can be written in the form Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. ... In mathematics, an abelian group, also called a commutative group, is a group (G, *) such that a * b = b * a for all a and b in G. Abelian groups are named after Niels Henrik Abel. ...

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. The integers consist of the positive natural numbers (1, 2, 3, …) the negative natural numbers (−1, −2, −3, ...) and the number zero. ... In abstract algebra, a generating set of a group is a subset S such that every element of G can be expressed as the product of finitely many elements of S and their inverses. ...

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 is not 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. The integers consist of the positive natural numbers (1, 2, 3, …) the negative natural numbers (−1, −2, −3, ...) and the number zero. ... Modular arithmetic is a system of arithmetic for integers, sometimes referred to as clock arithmetic, where numbers wrap around after they reach a certain value (the modulus). ... In group theory, a group G is called the direct sum of a set of subgroups {Hi} if each Hi is a normal subgroup of G each distinct pair of subgroups has trivial intersection, and G = <{Hi}>; in other words, G is generated by the subgroups {Hi}. If G is... In mathematics, a rational number (or informally fraction) is a ratio of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is... In mathematics, the integers a and b are said to be coprime or relatively prime if and only if they have no common factor other than 1 and −1, or equivalently, if their greatest common divisor is 1. ...

Classification

The fundamental theorem of finitely generated abelian groups states that every finitely generated abelian group G is isomorphic to a direct sum of primary and infinite cyclic groups. A primary cyclic group is one whose order is a power of a prime. That is, every such group is isomorphic to one of the form In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ... In mathematics, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that every element of the group is a power of a. ... In group theory, the term order is used in two closely related senses: the order of a group is its cardinality, i. ... In mathematics, a prime number, or prime for short, is a natural number greater than one and whose only distinct positive divisors are one and itself. ...

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 In mathematics, the term up to xxxx is used to describe a situation in which members of an equivalence class can be regarded as a single entity for some purpose. ...

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 In mathematics, the integers a and b are said to be coprime or relatively prime if and only if they have no common factor other than 1 and −1, or equivalently, if their greatest common divisor is 1. ...

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. In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ...

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. In abstract algebra, a free abelian group is an abelian group that has a basis in the sense that every element of the group can be written in one and only one way as a finite linear combination of elements of the basis, with integer coefficients. ... In mathematics, the rank, or torsion-free rank, of an abelian group measures how large a group is in terms of how large a vector space one would need to contain it; or alternatively how large a free abelian group it can contain as a subgroup. ... In group theory, the torsion subgroup of an abelian group A is the subgroup of A consisting of all elements that have finite order. ...

Note that not every abelian group of finite rank is 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. In mathematics the term countable set is used to describe the size of a set, e. ...

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: Q is torsion-free but not free abelian. A theorem is a statement which can be proven true within some logical framework. ...

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. In mathematics, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group operation... In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is a group that intuitively collapses the normal subgroup N to the identity element. ... Given two groups (G, *) and (H, ·), a group homomorphism from (G, *) to (H, ·) is a function h : G -> H such that for all u and v in G it holds that h(u * v) = h(u) · h(v) From this property, one can deduce that h maps the identity element... In mathematics, an abelian category is a certain kind of category in which morphisms and objects can be added and in which kernels and cokernels exist and have nice properties. ...