Free abelian group

In mathematics, a free abelian group or free Z-module is an abelian group with a basis, or, equivalently, a free module over the integers.
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 a linear combination of basis elements with integer coefficients. For instance, the integers with addition form a free abelian group with basis {1}. Free abelian groups have properties which make them similar to vector spaces. They have applications in algebraic topology, where they are used to define chain groups, and in algebraic geometry, where they are used to define divisors. Integer lattices also form examples of free abelian groups, and lattice theory studies free abelian subgroups of real vector spaces.
The elements of a free abelian group with basis "B" may be described in several equivalent ways. These include formal sums over "B", which are expressions of the form formula_1 where each coefficient "ai" is a nonzero integer, each factor "bi" 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 "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 "B" to the integers with finitely many nonzero values; for this functional representation, the group operation is the pointwise addition of functions.
Every set "B" has a free abelian group with "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 group with basis "B" may be constructed as a direct sum of copies of the additive group of the integers, with one copy per member of "B". Alternatively, the free abelian group with basis "B" may be described by a presentation with the elements of "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.
Examples and constructions.
Integers and lattices.
The integers, under the addition operation, form a free abelian group with the basis {1}. Every integer "n" is a linear combination of basis elements with integer coefficients: namely, "n" = "n" × 1, with the coefficient "n".
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 {(0,1), (1,0)}. Letting these basis vectors be denoted formula_2 and formula_3, the element (4,3) can be written
In this basis, there is no other way to write (4,3). However, with a different basis such as {(1,0),(1,1)}, where formula_6 and formula_7, it can be written as
More generally, every lattice forms a finitely-generated free abelian group. The "d"-dimensional integer lattice has a natural basis consisting of the positive integer unit vectors, but it has many other bases as well: if "M" is a "d" × "d" integer matrix with determinant ±1, then the rows of "M" 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.
Direct sums, direct products, and trivial group.
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 "d"-dimensional integer lattice, for instance, is isomorphic to the direct product of "d" copies of the integer group Z.
The trivial group {0} is also considered to be free abelian, with basis the empty set. It may be interpreted as a direct product of zero copies of Z.
For infinite families of free abelian groups, the direct product (the family of tuples of elements from each group, with pointwise addition) is not necessarily free abelian.
For instance the Baer–Specker group formula_9, an uncountable group formed as the direct product of countably many copies of formula_10, was shown in 1937 by Reinhold Baer to not be free abelian; Ernst Specker proved in 1950 that every countable subgroup of formula_9 is free abelian.
The direct sum of finitely many groups is the same as the direct product, but differs from the direct product on an infinite number of summands; its elements consist of tuples of elements from each group with all but finitely many of them equal to the identity element. As in the case of a finite number of summands, the direct sum of infinitely many free abelian groups remains free abelian, with a basis formed by (the images of) a disjoint union of the bases of the summands.
The tensor product of two free abelian groups is always free abelian, with a basis that is the Cartesian product of the bases for the two groups in the product.
Every free abelian group may be described as a direct sum of copies of formula_10, with one copy for each member of its basis. This construction allows any set "B" to become the basis of a free abelian group.
Integer functions and formal sums.
Given a set "B", one can define a group formula_13 whose elements are functions from "B" to the integers, where the parenthesis in the superscript indicates that only the functions with finitely many nonzero values are included.
If "f"("x") and "g"("x") are two such functions, then "f" + "g" is the function whose values are sums of the values in "f" and "g": that is, ("f" + "g")("x") = "f"("x") + "g"("x") . This pointwise addition operation gives formula_13 the structure of an abelian group.
Each element "x" from the given set "B" corresponds to a member of formula_13, the function "ex" for which "ex"("x") = 1 and for which "ex"("y") = 0 for all "y" ≠ "x".
Every function "f" in formula_13 is uniquely a linear combination of a finite number of basis elements:
Thus, these elements "ex" form a basis for formula_13, and formula_13 is a free abelian group.
In this way, every set "B" can be made into the basis of a free abelian group.
The free abelian group with basis "B" is unique up to isomorphism, and its elements are known as formal sums of elements of "B".
They may also be interpreted as the signed multisets of finitely many elements of "B".
For instance, in algebraic topology, chains are formal sums of simplices, and the chain group is the free abelian group whose elements are chains. In algebraic geometry, the divisors of a Riemann surface (a combinatorial description of the zeros and poles of meromorphic functions) form an uncountable free abelian group, consisting of the formal sums of points from the surface.
Presentation.
A presentation of a group is a set of elements that generate the group (all group elements are products of finitely many generators), together with "relators", products of generators that give the identity element. The free abelian group with basis "B" has a presentation in which the generators are the elements of "B", and the relators are the commutators of pairs of elements of "B". Here, the commutator of two elements "x" and "y" is the product "x"−1"y"−1"xy"; setting this product to the identity causes "xy" to equal "yx", so that "x" and "y" 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 is also finite. This fact, together with the fact that every subgroup of a free abelian group is free abelian (below) can be used to show that every finitely generated abelian group is finitely presented. For, if "G" is finitely generated by a set "B", it is a quotient of the free abelian group over "B" by a free abelian subgroup, the subgroup generated by the relators of the presentation of "G". But since this subgroup is itself free abelian, it is also finitely generated, and its basis (together with the commutators over "B") forms a finite set of relators for a presentation of "G".
Terminology.
Every abelian group may be considered as a module over the integers by considering the scalar multiplication of a group member by an integer defined as follows:
A free module is a module that can be represented as a direct sum over its base ring, so free abelian groups and free formula_21-modules are equivalent concepts: each free abelian group is (with the multiplication operation above) a free formula_21-module, and each free formula_21-module comes from a free abelian group in this way.
Unlike vector spaces, not all abelian groups have a basis, hence the special name for those that do. For instance, any torsion formula_24-module, and thus any finite abelian group, is not a free abelian group, because 0 may be decomposed in several ways on any set of elements which could be a candidate for a basis: formula_25 for some positive integer "n". On the other hand, many important properties of free abelian groups may be generalized to free modules over a principal ideal domain.
Note that a "free abelian" group is "not" a free group except in two cases: a free abelian group having an empty basis (rank 0, giving the trivial group) or having just 1 element in the basis (rank 1, giving the infinite cyclic group). Other abelian groups are not free groups because in free groups "ab" must be different from "ba" if "a" and "b" are different elements of the basis, while in free abelian groups they must be identical. Free groups are the free objects in the category of groups, that is, the "most general" or "least constrained" groups with a given number of generators, whereas free abelian groups are the free objects in the category of abelian groups. In the general category of groups, it is an added constraint to demand that "ab = ba", whereas this is a necessary property in the category of abelian groups.
Properties.
Universal property.
A free abelian group formula_26 with basis formula_27 has the following universal property: for every function formula_28 from formula_27 to an abelian group formula_30, there exists a unique group homomorphism from formula_26 to formula_30 which extends formula_28. By a general property of universal properties, this shows that "the" abelian group of base formula_27 is unique up to an isomorphism. Therefore, the universal property can be used as a definition of the free abelian group of base formula_27. The uniqueness of the group defined by this property shows that all the other definitions are equivalent.
Rank.
Every two bases of the same free abelian group have the same cardinality, so the cardinality of a basis forms an invariant of the group known as its rank.
In particular, a free abelian group is finitely generated if and only if its rank is a finite number "n", in which case the group is isomorphic to formula_36.
This notion of rank can be generalized, from free abelian groups to abelian groups that are not necessarily free. The rank of an abelian group "G" is defined as the rank of a free abelian subgroup "F" of "G" for which the quotient group "G"/"F" is a torsion group. Equivalently, it is the cardinality of a maximal subset of "G" that generates a free subgroup. Again, this is a group invariant; it does not depend on the choice of the subgroup.
Subgroups.
Every subgroup of a free abelian group is itself a free abelian group. This result of Richard Dedekind 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.
The proof needs the axiom of choice.
A proof using Zorn's lemma (one of many equivalent assumptions to the axiom of choice) can be found in Serge Lang's "Algebra". Solomon Lefschetz and Irving Kaplansky have claimed that using the well-ordering principle in place of Zorn's lemma leads to a more intuitive proof.
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. If formula_37 is a subgroup of a finitely generated free abelian group formula_26, then formula_37 is free and there exists a basis formula_40 of formula_26 and positive integers formula_42 (that is, each one divides the next one) such that formula_43 is a basis of formula_44 Moreover, the sequence formula_45 depends only on formula_26 and formula_37 and not on the particular basis formula_40 that solves the problem.
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. Uniqueness follows from the fact that, for any formula_49, the greatest common divisor of the minors of rank formula_50 of the matrix is not changed during the Smith normal form computation and is the product formula_51 at the end of the computation.
As every finitely generated abelian group is the quotient of a finitely generated free abelian group by a submodule, the fundamental theorem of finitely generated abelian groups is a corollary of the above result.
Torsion and divisibility.
All free abelian groups are torsion-free, meaning that there is no group element (non-identity) formula_52 and nonzero integer formula_53 such that formula_54.
Conversely, all finitely generated torsion-free abelian groups are free abelian. The same applies to flatness, since an abelian group is torsion-free if and only if it is flat.
The additive group of rational numbers formula_55 provides an example of a torsion-free (but not finitely generated) abelian group that is not free abelian. One reason that formula_55 is not free abelian is that it is divisible, meaning that, for every element formula_57 and every nonzero integer formula_53, it is possible to express formula_52 as a scalar multiple formula_60 of another element formula_61. In contrast, non-zero free abelian groups are never divisible, because it is impossible for any of their basis elements to be nontrivial integer multiples of other elements.
Relation to other abelian groups.
Given an arbitrary abelian group formula_30, there always exists a free abelian group formula_26 and a surjective group homomorphism from formula_26 to formula_30. One way of constructing a surjection onto a given group formula_30 is to let formula_67 be the free abelian group over formula_30, represented as formal sums. Then a surjection can be defined 
by mapping formal sums in formula_26 to the corresponding sums of members of formula_30. That is, the surjection maps
where formula_72 is the integer coefficient of basis element formula_73 in a given formal sum,
the first sum is in formula_26, and the second sum is in formula_30. This surjection is the unique group homomorphism which extends the function formula_76, and so its construction can be seen as an instance of the universal property.
When formula_26 and formula_30 are as above, the kernel formula_37 of the surjection from formula_26 to formula_30 is also free abelian, as it is a subgroup of formula_26 (the subgroup of elements mapped to the identity).
Therefore, these groups form a short exact sequence
in which formula_26 and formula_37 are both free abelian and formula_30 is isomorphic to the factor group formula_87. This is a free resolution of formula_30. Furthermore, assuming the axiom of choice, the free abelian groups are precisely the projective objects in the category of abelian groups.
Applications.
Algebraic topology.
In algebraic topology, a formal sum of formula_89-dimensional simplices is called a formula_89-chain, and the free abelian group having a collection of formula_89-simplices as its basis is called a chain group. The simplices are generally taken from some topological space, for instance as the set of formula_89-simplices in a simplicial complex, or the set of singular formula_89-simplices in a manifold. Any formula_89-dimensional simplex has a boundary that can be represented as a formal sum of formula_95-dimensional simplices, and the universal property of free abelian groups allows this boundary operator to be extended to a group homomorphism from formula_89-chains to formula_95-chains. The system of chain groups linked by boundary operators in this way forms a chain complex, and the study of chain complexes forms the basis of homology theory.
Algebraic geometry and complex analysis.
Every rational function over the complex numbers can be associated with a signed multiset of complex numbers formula_98, the zeros and poles of the function (points where its value is zero or infinite). The multiplicity formula_99 of a point in this multiset is its order as a zero of the function, or the negation of its order as a pole.
Then the function itself can be recovered from this data, up to a scalar factor, as
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. Thus, the multiplicative group of rational functions can be factored into the multiplicative group of complex numbers (the associated scalar factors for each function) and the free abelian group over the complex numbers. The rational functions that have a nonzero limiting value at infinity (the meromorphic functions on the Riemann sphere) form a subgroup of this group in which the sum of the multiplicities is zero.
This construction has been generalized, in algebraic geometry, to the notion of a divisor. 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. The meromorphic functions on a compact Riemann surface have finitely many zeros and poles, and their divisors can again be represented as elements of a free abelian group, with multiplication or division of functions corresponding to addition or subtraction of group elements. However, in this case there are additional constraints on the divisor beyond having zero sum of multiplicities.