Free abelian group

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 and as its basis. Free abelian groups have properties which make them similar to vector spaces, and may equivalently be called free 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 may be described in several equivalent ways. These include formal sums which are expressions of the form where each is a nonzero integer, each 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 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 to the integers with finitely many nonzero values; for this functional representation, the group operation is the pointwise addition of functions.
Every set has a free abelian group with 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 may be constructed as a direct sum of copies of the additive group of the integers, with one copy per member Alternatively, the free abelian group with basis may be described by a presentation with the elements of 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.

Definition and examples

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 of its elements and a binary operation conventionally denoted as an additive group by the symbol that obey the following properties:

The operation is commutative and associative, meaning for all elements and Therefore, when combining two or more elements of using this operation, the ordering and grouping of the elements does not affect the result.
contains an identity element 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 prime number in the multiplicative group of the rationals as instead giving the coefficient of in the corresponding polynomial, or vice versa. For instance the rational number has exponents of for the first three prime numbers and would correspond in this way to the polynomial having the same coefficients 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.

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. For example, if one replaces any element of a basis by its inverse, one gets another basis. As a more elaborated example, the two-dimensional integer lattice consisting of the points in the plane with integer Cartesian coordinates, forms a free abelian group under vector addition with the basis For this basis, the element can be written where 'multiplication' is defined so that, for instance, There is no other way to write in the same basis. However, with a different basis such as it can be written as Generalizing this example, every lattice forms a finitely-generated free abelian group. The integer lattice has a natural basis consisting of the positive integer unit vectors, but it has many other bases as well: if is a integer matrix with then the rows of form a basis, and conversely every basis of the integer lattice has this form. For more on the two-dimensional case, see fundamental pair of periods.

Constructions

Every set can be the basis of a free abelian group, which is unique up to group isomorphisms. 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.

Products and sums

The direct product of groups consists of tuples of an element from each group in the product, with componentwise addition. The direct product of two free abelian groups is itself free abelian, with basis the disjoint union of the bases of the two groups. More generally the direct product of any finite number of free abelian groups is free abelian. The integer lattice, for instance, is isomorphic to the direct product of copies of the integer The trivial group is also considered to be free abelian, with basis the empty set. It may be interpreted as an empty product, the direct product of zero copies
For infinite families of free abelian groups, the direct product is not necessarily free abelian. For instance the Baer–Specker group an uncountable group formed as the direct product of countably many copies was shown in 1937 by Reinhold Baer to not be free abelian, although Ernst Specker proved in 1950 that all of its countable subgroups are free abelian. Instead, to obtain a free abelian group from an infinite family of groups, the direct sum rather than the direct product should be used. The direct sum and direct product are the same when they are applied to finitely many groups, but differ on infinite families of groups. In the direct sum, the elements are again tuples of elements from each group, but with the restriction that all but finitely many of these elements are the identity for their group. The direct sum of infinitely many free abelian groups remains free abelian. It has a basis consisting of tuples in which all but one element is the identity, with the remaining element part of a basis for its group.
Every free abelian group may be described as a direct sum of copies with one copy for each member of its basis. This construction allows any set to become the basis of a free abelian group.

Integer functions and formal sums

Given a one can define a group whose elements are functions from to the integers, where the parenthesis in the superscript indicates that only the functions with finitely many nonzero values are included.
If and are two such functions, then is the function whose values are sums of the values in that is, This pointwise addition operation gives the structure of an abelian group.
Each element from the given set corresponds to a member the function for which and for which for
Every function in is uniquely a linear combination of a finite number of basis elements:
Thus, these elements form a basis and is a free abelian group.
In this way, every set can be made into the basis of a free abelian group.
The elements of may also be written as formal sums, expressions in the form of a sum of finitely many terms, where each term is written as the product of a nonzero integer with a distinct member These expressions are considered equivalent when they have the same terms, regardless of the ordering of terms, and they may be added by forming the union of the terms, adding the integer coefficients to combine terms with the same basis element, and removing terms for which this combination produces a zero coefficient. They may also be interpreted as the signed multisets of finitely many elements

Presentation

A presentation of a group is a set of elements that generate the group, together with "relators", products of generators that give the identity element. 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. The free abelian group with basis has a presentation in which the generators are the elements and the relators are the commutators of pairs of elements Here, the commutator of two elements and is the product setting this product to the identity causes to so that and commute. More generally, if all pairs of generators commute, then all pairs of products of generators also commute. Therefore, the group generated by this presentation is abelian, and the relators of the presentation form a minimal set of relators needed to ensure that it is abelian.
When the set of generators is finite, the presentation of a free abelian group is also finite, because there are only finitely many different commutators to include in the presentation. This fact, together with the fact that every subgroup of a free abelian group is free abelian can be used to show that every finitely generated abelian group is finitely presented. For, if is finitely generated by a it is a quotient of the free abelian group over by a free abelian subgroup, the subgroup generated by the relators of the presentation But since this subgroup is itself free abelian, it is also finitely generated, and its basis (together with the commutators forms a finite set of relators for a presentation