Free abelian group

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 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)}.[1] Letting these basis vectors be denoted ${\displaystyle \ e_{1}=(1,0)}$ and ${\displaystyle \ e_{2}=(0,1)}$, the element (4,3) can be written

${\displaystyle \ (4,3)=4e_{1}+3e_{2}}$ where 'multiplication' is defined so that ${\displaystyle \ 4e_{1}:=e_{1}+e_{1}+e_{1}+e_{1}.}$

In this basis, there is no other way to write (4,3). However, with a different basis such as {(1,0),(1,1)}, where ${\displaystyle \ f_{1}=(1,0)}$ and ${\displaystyle \ f_{2}=(1,1)}$, it can be written as

${\displaystyle \ (4,3)=f_{1}+3f_{2}.}$

More generally, every lattice forms a finitely-generated free abelian group.[2] 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.[3] For more on the two-dimensional case, see fundamental pair of periods.

The direct product of two free abelian groups is itself free abelian, with basis the disjoint union of the bases of the two groups.[4] 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.[5] 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.[4] For instance the Baer–Specker group ${\displaystyle \mathbb {Z} ^{\mathbb {N} }}$, an uncountable group formed as the direct product of countably many copies of ${\displaystyle \mathbb {Z} }$, was shown in 1937 by Reinhold Baer to not be free abelian;[6] Ernst Specker proved in 1950 that every countable subgroup of ${\displaystyle \mathbb {Z} ^{\mathbb {N} }}$ is free abelian.[7] 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.[4]

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.[8]

Every free abelian group may be described as a direct sum of copies of ${\displaystyle \mathbb {Z} }$, with one copy for each member of its basis.[9][10] This construction allows any set B to become the basis of a free abelian group.[11]

Given a set B, one can define a group ${\displaystyle \mathbb {Z} ^{(B)}}$ 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 ${\displaystyle \mathbb {Z} ^{(B)}}$ the structure of an abelian group.[12]

Each element x from the given set B corresponds to a member of ${\displaystyle \mathbb {Z} ^{(B)}}$, the function e_x for which e_x(x) = 1 and for which e_x(y) = 0 for all y ≠ x. Every function f in ${\displaystyle \mathbb {Z} ^{(B)}}$ is uniquely a linear combination of a finite number of basis elements:

${\displaystyle f=\sum _{\{x\mid f(x)\neq 0\}}f(x)e_{x}}$

Thus, these elements e_x form a basis for ${\displaystyle \mathbb {Z} ^{(B)}}$, and ${\displaystyle \mathbb {Z} ^{(B)}}$ is a free abelian group. In this way, every set B can be made into the basis of a free abelian group.[12]

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.[13] 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.[14]

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⁻¹y⁻¹xy; 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.[15]

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.[16]

A free abelian group ${\displaystyle F}$ with basis ${\displaystyle B}$ has the following universal property: for every function ${\displaystyle f}$ from ${\displaystyle B}$ to an abelian group ${\displaystyle A}$, there exists a unique group homomorphism from ${\displaystyle F}$ to ${\displaystyle A}$ which extends ${\displaystyle f}$.[5] By a general property of universal properties, this shows that "the" abelian group of base ${\displaystyle B}$ is unique up to an isomorphism. Therefore, the universal property can be used as a definition of the free abelian group of base ${\displaystyle B}$. The uniqueness of the group defined by this property shows that all the other definitions are equivalent.[11]

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.[22][23] 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 ${\displaystyle \mathbb {Z} ^{n}}$.

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.[24]

Every subgroup of a free abelian group is itself a free abelian group. This result of Richard Dedekind[25] 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.[26] A proof using Zorn's lemma (one of many equivalent assumptions to the axiom of choice) can be found in Serge Lang's Algebra.[27] 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.[10]

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 ${\displaystyle G}$ is a subgroup of a finitely generated free abelian group ${\displaystyle F}$, then ${\displaystyle G}$ is free and there exists a basis ${\displaystyle (e_{1},\ldots ,e_{n})}$ of ${\displaystyle F}$ and positive integers ${\displaystyle d_{1}|d_{2}|\ldots |d_{k}}$ (that is, each one divides the next one) such that ${\displaystyle (d_{1}e_{1},\ldots ,d_{k}e_{k})}$ is a basis of ${\displaystyle G.}$ Moreover, the sequence ${\displaystyle d_{1},d_{2},\ldots ,d_{k}}$ depends only on ${\displaystyle F}$ and ${\displaystyle G}$ and not on the particular basis ${\displaystyle (e_{1},\ldots ,e_{n})}$ that solves the problem.[28] 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.[29] Uniqueness follows from the fact that, for any ${\displaystyle r\leq k}$, the greatest common divisor of the minors of rank ${\displaystyle r}$ of the matrix is not changed during the Smith normal form computation and is the product ${\displaystyle d_{1}\cdots d_{r}}$ at the end of the computation.[30]

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.

All free abelian groups are torsion-free, meaning that there is no group element (non-identity) ${\displaystyle x}$ and nonzero integer ${\displaystyle n}$ such that ${\displaystyle nx=0}$. Conversely, all finitely generated torsion-free abelian groups are free abelian.[5][31] 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 ${\displaystyle \mathbb {Q} }$ provides an example of a torsion-free (but not finitely generated) abelian group that is not free abelian.[32] One reason that ${\displaystyle \mathbb {Q} }$ is not free abelian is that it is divisible, meaning that, for every element ${\displaystyle x\in \mathbb {Q} }$ and every nonzero integer ${\displaystyle n}$, it is possible to express ${\displaystyle x}$ as a scalar multiple ${\displaystyle ny}$ of another element ${\displaystyle y=x/n}$. 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.[33]

In algebraic topology, a formal sum of ${\displaystyle k}$-dimensional simplices is called a ${\displaystyle k}$-chain, and the free abelian group having a collection of ${\displaystyle k}$-simplices as its basis is called a chain group. The simplices are generally taken from some topological space, for instance as the set of ${\displaystyle k}$-simplices in a simplicial complex, or the set of singular ${\displaystyle k}$-simplices in a manifold. Any ${\displaystyle k}$-dimensional simplex has a boundary that can be represented as a formal sum of ${\displaystyle (k-1)}$-dimensional simplices, and the universal property of free abelian groups allows this boundary operator to be extended to a group homomorphism from ${\displaystyle k}$-chains to ${\displaystyle (k-1)}$-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.[38]

The rational function ${\displaystyle z^{4}/(z^{4}-1)}$ 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 ${\displaystyle \pm 1}$ and ${\displaystyle \pm i}$ (the white points at the ends of the four petals). It can be represented (up to a scalar) by the divisor ${\displaystyle 4e_{0}-e_{1}-e_{-1}-e_{i}-e_{-i}}$ where ${\displaystyle e_{z}}$ is the basis element for a complex number ${\displaystyle z}$ in a free abelian group over the complex numbers.

Every rational function over the complex numbers can be associated with a signed multiset of complex numbers ${\displaystyle c_{i}}$, the zeros and poles of the function (points where its value is zero or infinite). The multiplicity ${\displaystyle m_{i}}$ 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

${\displaystyle f(q)=\prod (q-c_{i})^{m_{i}}.}$

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.[39]

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.[39]