Free abelian group

Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible.

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.

Lattice theory studies free abelian subgroups of real vector spaces.

Instead of constructing it by describing its individual elements, a free abelian group with basis

symbol (although it need not be the usual addition of numbers) that obey the following properties: A basis is a subset

[2] 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.

, under the usual addition operation, form a free abelian group with the basis

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.

, with integer coefficients, form a free abelian group under polynomial addition, with the powers of

A bijective homomorphism is called an isomorphism, and its existence demonstrates that these two groups have the same properties.

[5] Although the representation of each group element in terms of a given basis is unique, a free abelian group has generally more than one basis, and different bases will generally result in different representations of its elements.

, consisting of the points in the plane with integer Cartesian coordinates, forms a free abelian group under vector addition with the basis

has a natural basis consisting of the positive integer unit vectors, but it has many other bases as well: if

The free abelian group for a given basis set can be constructed in several different but equivalent ways: as a direct sum of copies of the integers, as a family of integer-valued functions, as a signed multiset, or by a presentation of a group.

to the integers, where the parenthesis in the superscript indicates that only the functions with finitely many nonzero values are included.

The elements of a group defined in this way are equivalence classes of sequences of generators and their inverses, under an equivalence relation that allows inserting or removing any relator or generator-inverse pair as a contiguous subsequence.

[19] The modules over the integers are defined similarly to vector spaces over the real numbers or rational numbers: they consist of systems of elements that can be added to each other, with an operation for scalar multiplication by integers that is compatible with this addition operation.

[21] As well as the direct sum, another way to combine free abelian groups is to use the tensor product of

This result of Richard Dedekind[32] was a precursor to the analogous Nielsen–Schreier theorem that every subgroup of a free group is free, and is a generalization of the fact that every nontrivial subgroup of the infinite cyclic group is infinite cyclic.

[25] A proof using Zorn's lemma (one of many equivalent assumptions to the axiom of choice) can be found in Serge Lang's Algebra.

[33] Solomon Lefschetz and Irving Kaplansky argue that using the well-ordering principle in place of Zorn's lemma leads to a more intuitive proof.

[14] In the case of finitely generated free abelian groups, the proof is easier, does not need the axiom of choice, and leads to a more precise result.

[34] A constructive proof of the existence part of the theorem is provided by any algorithm computing the Smith normal form of a matrix of integers.

, which can be described concretely (for a specific basis of the free automorphism group) as the set of

This result depends on the structure of involutions of free abelian groups, the automorphisms that are their own inverse.

If these multisets are interpreted as members of a free abelian group over the complex numbers, then the product or quotient of two rational functions corresponds to the sum or difference of two group members.

There are different definitions of divisors, but in general they form an abstraction of a codimension-one subvariety of an algebraic variety, the set of solution points of a system of polynomial equations.

In the case where the system of equations has one degree of freedom (its solutions form an algebraic curve or Riemann surface), a subvariety has codimension one when it consists of isolated points, and in this case a divisor is again a signed multiset of points from the variety.

[48] The meromorphic functions on a compact Riemann surface have finitely many zeros and poles, and their divisors form a subgroup of a free abelian group over the points of the surface, with multiplication or division of functions corresponding to addition or subtraction of group elements.

To be a divisor, an element of the free abelian group must have multiplicities summing to zero, and meet certain additional constraints depending on the surface.

A lattice in the Euclidean plane . Adding any two blue lattice points produces another lattice point; the group formed by this addition operation is a free abelian group.
The rational function has a zero of order four at 0 (the black point at the center of the plot), and simple poles at the four complex numbers and (the white points at the ends of the four petals). It can be represented (up to a scalar) by the divisor where is the basis element for a complex number in a free abelian group over the complex numbers.