In mathematics, a free abelian group is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis, also called an integral basis, is a subset such that every element of the group can be uniquely expressed as an integer combination of finitely many basis elements. For instance, the two-dimensional integer lattice forms a free abelian group, with coordinatewise addition as its operation, and with the two points (1, 0) and (0, 1) as its basis. Free abelian groups have properties which make them similar to vector spaces, and may equivalently be called free ${\displaystyle \mathbb {Z} }$-modules, the free modules over the integers. Lattice theory studies free abelian subgroups of real vector spaces. In algebraic topology, free abelian groups are used to define chain groups, and in algebraic geometry they are used to define divisors.

The elements of a free abelian group with basis ${\displaystyle B}$ may be described in several equivalent ways. These include formal sums over ${\displaystyle B}$, which are expressions of the form ${\textstyle \sum a_{i}b_{i}}$ where each ${\displaystyle a_{i}}$ is a nonzero integer, each ${\displaystyle b_{i}}$ is a distinct basis element, and the sum has finitely many terms. Alternatively, the elements of a free abelian group may be thought of as signed multisets containing finitely many elements of ${\displaystyle B}$, with the multiplicity of an element in the multiset equal to its coefficient in the formal sum. Another way to represent an element of a free abelian group is as a function from ${\displaystyle B}$ to the integers with finitely many nonzero values; for this functional representation, the group operation is the pointwise addition of functions.

Every set ${\displaystyle B}$ has a free abelian group with ${\displaystyle B}$ as its basis. This group is unique in the sense that every two free abelian groups with the same basis are isomorphic. Instead of constructing it by describing its individual elements, a free abelian group with basis ${\displaystyle B}$ may be constructed as a direct sum of copies of the additive group of the integers, with one copy per member of ${\displaystyle B}$. Alternatively, the free abelian group with basis ${\displaystyle B}$ may be described by a presentation with the elements of ${\displaystyle B}$ as its generators and with the commutators of pairs of members as its relators. The rank of a free abelian group is the cardinality of a basis; every two bases for the same group give the same rank, and every two free abelian groups with the same rank are isomorphic. Every subgroup of a free abelian group is itself free abelian; this fact allows a general abelian group to be understood as a quotient of a free abelian group by "relations", or as a cokernel of an injective homomorphism between free abelian groups. The only free abelian groups that are free groups are the trivial group and the infinite cyclic group.

A free abelian group is an abelian group that has a basis. Here, being an abelian group means that it is described by a set ${\displaystyle S}$ of its elements and a binary operation on ${\displaystyle S}$, conventionally denoted as an additive group by the ${\displaystyle +}$ symbol (although it need not be the usual addition of numbers) that obey the following properties:

A basis is a subset ${\displaystyle B}$ of the elements of ${\displaystyle S}$ with the property that every element of ${\displaystyle S}$ may be formed in a unique way by choosing finitely many basis elements ${\displaystyle b_{i}}$ of ${\displaystyle B}$, choosing a nonzero integer ${\displaystyle k_{i}}$ for each of the chosen basis elements, and adding together ${\displaystyle k_{i}}$ copies of the basis elements ${\displaystyle b_{i}}$ for which ${\displaystyle k_{i}}$ is positive, and ${\displaystyle -k_{i}}$ copies of ${\displaystyle -b_{i}}$ for each basis element for which ${\displaystyle k_{i}}$ is negative. As a special case, the identity element can always be formed in this way as the combination of zero basis elements, according to the usual convention for an empty sum, and it must not be possible to find any other combination that represents the identity.

The integers ${\displaystyle \mathbb {Z} }$, under the usual addition operation, form a free abelian group with the basis ${\displaystyle \{1\}}$. The integers are commutative and associative, with 0 as the additive identity and with each integer having an additive inverse, its negation. Each non-negative ${\displaystyle x}$ is the sum of ${\displaystyle x}$ copies of ${\displaystyle 1}$, and each negative integer ${\displaystyle x}$ is the sum of ${\displaystyle -x}$ copies of ${\displaystyle -1}$, so the basis property is also satisfied.

An example where the group operation is different from the usual addition of numbers is given by the positive rational numbers ${\displaystyle \mathbb {Q} ^{+}}$, which form a free abelian group with the usual multiplication operation on numbers and with the prime numbers as their basis. Multiplication is commutative and associative, with the number ${\displaystyle 1}$ as its identity and with ${\displaystyle 1/x}$ as the inverse element for each positive rational number ${\displaystyle x}$. The fact that the prime numbers forms a basis for multiplication of these numbers follows from the fundamental theorem of arithmetic, according to which every positive integer can be factorized uniquely into the product of finitely many primes or their inverses. If ${\displaystyle q=a/b}$ is a positive rational number expressed in simplest terms, then ${\displaystyle q}$ can be expressed as a finite combination of the primes appearing in the factorizations of ${\displaystyle a}$ and ${\displaystyle b}$. The number of copies of each prime to use in this combination is its exponent in the factorization of ${\displaystyle a}$, or the negation of its exponent in the factorization of ${\displaystyle b}$.

The polynomials of a single variable ${\displaystyle x}$, with integer coefficients, form a free abelian group under polynomial addition, with the powers of ${\displaystyle x}$ as a basis. As an abstract group, this is the same as (an isomorphic group to) the multiplicative group of positive rational numbers. One way to map these two groups to each other, showing that they are isomorphic, is to reinterpret the exponent of the ${\displaystyle i}$th prime number in the multiplicative group of the rationals as instead giving the coefficient of ${\displaystyle x^{i-1}}$ in the corresponding polynomial, or vice versa. For instance the rational number ${\displaystyle 5/27}$ has exponents of ${\displaystyle 0,-3,1}$ for the first three prime numbers ${\displaystyle 2,3,5}$ and would correspond in this way to the polynomial ${\displaystyle -3x+x^{2}}$ having the same coefficients ${\displaystyle 0,-3,1}$ for its constant, linear, and quadratic terms. Because these mappings merely reinterpret the same numbers, they define a bijection between the elements of the two groups. And because the group operation of multiplying positive rationals acts additively on the exponents of the prime numbers, in the same way that the group operation of adding polynomials acts on the coefficients of the polynomials, these maps preserve the group structure; they are homomorphisms. A bijective homomorphism is called an isomorphism, and its existence demonstrates that these two groups have the same properties.