Free abelian group

In abstract algebra, a free abelian group or free Z-module is an abelian group with a basis. Being an abelian group means that it is a set together with an associative, commutative, and invertible binary operation. Conventionally, this operation is thought of as addition and its inverse is thought of as subtraction on the group elements. A basis is a subset of the elements such that every group element can be found by adding or subtracting a finite number of basis elements, and such that, for every group element, its expression as a linear combination of basis elements is unique. For instance, the integers under addition form a free abelian group with basis {1}. Addition of integers is commutative, associative, and has subtraction as its inverse operation, each integer can be formed by using addition or subtraction to combine some number of copies of the number 1, and each integer has a unique representation as an integer multiple of the number 1. Integer lattices also form examples of free abelian groups.

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.

The elements of a free abelian group with basis B may be represented by expressions of the form $\sum a_i b_i$ where each coefficient a_i is a nonzero integer, each factor b_i is a distinct basis element, and the sum has finitely many terms. These expressions, and the group elements they represent, are also known as formal sums over B. 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 the elements of a free abelian group is as the functions from B to the integers that have finitely many nonzero values, with pointwise addition of these functions as the group operation.

For every set B there is 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 element by element, 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. Every free abelian group has a rank defined as 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.

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)}.^[1] If we say $\ e_1 = (1,0)$ and $\ e_2 = (0,1)$, then the element (4,3) can be written

$\ (4,3) = 4 e_1 + 3 e_2$ where 'multiplication' is defined so that $\ 4 e_1 := e_1 + e_1 + e_1 + e_1.$

In this basis, there is no other way to write (4,3), but with a different basis such as {(1,0),(1,1)}, where $\ f_1 = (1,0)$ and $\ f_2 = (1,1)$, it can be written as

$\ (4,3) = f_1 + 3 f_2.$

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.

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.^[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 $\mathbb{Z}^\mathbb{N}$, an uncountable group formed as the direct product of countably many copies of $\mathbb{Z}$, was shown in 1937 by Reinhold Baer to not be free abelian;^[6] Ernst Specker proved in 1950 that every countable subgroup of $\mathbb{Z}^\mathbb{N}$ 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 $\mathbb{Z}$, 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]

Integer functions and formal sums

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

Each element x from the given set B corresponds to a member of $\mathbb{Z}^{(B)}$, 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 $\mathbb{Z}^{(B)}$ is uniquely a linear combination of a finite number of basis elements:

$f=\sum_{\{x\mid f(x)\ne 0\}} f(x) e_x$

Thus, these elements e_x form a basis for $\mathbb{Z}^{(B)}$, and $\mathbb{Z}^{(B)}$ 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]

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⁻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]

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:^[17]

$\begin{align} 0\,x&=0\\ 1\,x&=x\\ n\,x&= x+ (n-1)\,x \qquad \text{if} \quad n>1\\ n\,x&= -((-n)\,x) \qquad \text{if} \quad n<0 \end{align}$

A free module is a module that can be represented as a direct sum over its base ring, so free abelian groups and free $\mathbb Z$-modules are equivalent concepts: each free abelian group is (with the multiplication operation above) a free $\mathbb Z$-module, and each free $\mathbb Z$-module comes from a free abelian group in this way.^[18]

Unlike vector spaces, not all abelian groups have a basis, hence the special name for those that do. For instance, any torsion $\mathbb Z$-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: $0 = 0\,b = n\,b$ 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.^[19]

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).^[5][20] 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.^[21] 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

If F is a free abelian group with basis B, then we have the following universal property: for every arbitrary function f from B to some abelian group A, there exists a unique group homomorphism from F to A which extends f.^[5] By a general property of universal properties, this shows that "the" abelian group of base B is unique up to an isomorphism. This allows to use this universal property as a definition of the free abelian group of base B and shows that all the other definitions are equivalent.^[11]

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.^[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 $\mathbb{Z}^n$.

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]

Subgroups

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.

Theorem: Let $F$ be a free abelian group and let $G\subset F$ be a subgroup. Then $G$ is a free abelian group.

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 groups, the proof is easier, and leads to a more precise result.

Theorem: Let $G$ be a subgroup of a finitely generated free abelian group $F$. Then $G$ is free and there exists a basis $(e_1, \ldots, e_n)$ of $F$ and positive integers $d_1|d_2|\ldots|d_k$ (that is, each one divides the next one) such that $(d_1e_1,\ldots, d_ke_k)$ is a basis of $G.$ Moreover, the sequence $d_1,d_2,\ldots,d_k$ depends only on $F$ and $G$ and not on the particular basis $(e_1, \ldots, e_n)$ 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 r ≤ k, the greatest common divisor of the minors of rank r of the matrix is not changed during the Smith normal form computation and is the product $d_1\cdots d_r$ at the end of the computation.^[30]

Torsion and divisibility

All free abelian groups are torsion-free, meaning that there is no group element (non-identity) x and nonzero integer n such that 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 Q provides an example of a torsion-free (but not finitely generated) abelian group that is not free abelian.^[32] One reason that Q is not free abelian is that it is divisible, meaning that, for every element x of Q and every nonzero integer n, it is possible to express x as a scalar multiple ny of another element y. 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]

Relation to arbitrary abelian groups

Given an arbitrary abelian group A, there always exists a free abelian group F and a surjective group homomorphism from F to A. One way of constructing a surjection onto a given group A is to let $F=\mathbb{Z}^{(A)}$ be the free abelian group over A, represented as the set of functions from A to the integers with finitely many nonzeros. Then a surjection can be defined from the representation of members of F as formal sums of members of A:

$f=\sum_{\{x\mid f(x)\ne 0\}} f(x) e_x \mapsto \sum_{\{x\mid f(x)\ne 0\}} f(x) x,$

where the first sum is in F and the second sum is in A.^[23][34] This construction can be seen as an instance of the universal property: this surjection is the unique group homomorphism which extends the function $e_x\mapsto x$.

When F and A are as above, the kernel G of the surjection from F to A is also free abelian, as it is a subgroup of F (the subgroup of elements mapped to the identity). Therefore, these groups form a short exact sequence

0 → G → F → A → 0

in which F and G are both free abelian and A is isomorphic to the factor group F/G. This is a free resolution of A.^[35] Furthermore, assuming the axiom of choice,^[36] the free abelian groups are precisely the projective objects in the category of abelian groups.^[37]

Applications

Algebraic topology

In algebraic topology, a formal sum of k-dimensional simplices is called a k-chain, and the free abelian group having a collection of 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 k-simplices in a simplicial complex, or the set of singular k-simplices in a manifold. Any k-dimensional simplex has a boundary that can be represented as a formal sum of (k − 1)-dimensional simplices, and the universal property of free abelian groups allows this boundary operator to be extended to a group homomorphism from k-chains to (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]

Algebraic geometry and complex analysis

Every rational function over the complex numbers can be associated with a signed multiset of complex numbers c_i, the zeros or poles of the function (points where its value is zero or infinite). The multiplicity 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

$f(q)=\prod (q-c_i)^{m_i}.$

If these multisets are interpreted as members of a free abelian group, with the set of complex numbers as its basis, 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 meromorphic rational functions 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 Riemann surface are again determined up to a scalar factor by their divisors, but in this case there are additional constraints on the divisor beyond having zero sum of multiplicities.^[39]

References

1. Johnson, D. L. (2001), Symmetries, Springer undergraduate mathematics series, Springer, p. 193, ISBN 9781852332709.
2. Mollin, Richard A. (2011), Advanced Number Theory with Applications, CRC Press, p. 182, ISBN 9781420083293.
3. Bremner, Murray R. (2011), Lattice Basis Reduction: An Introduction to the LLL Algorithm and Its Applications, CRC Press, p. 6, ISBN 9781439807026.
4. Hungerford (1974), Exercise 5, p. 75.
5. Lee, John M. (2010), "Free Abelian Groups", Introduction to Topological Manifolds, Graduate Texts in Mathematics 202 (2nd ed.), Springer, pp. 244–248, ISBN 9781441979407.
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15. Hungerford (1974), Exercise 3, p. 75.
16. Johnson (2001), p. 71.
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19. For instance, submodules of free modules over principal ideal domains are free, a fact that Hatcher (2002) writes allows for "automatic generalization" of homological machinery to these modules. Additionally, the theorem that every projective $\Z$-module is free generalizes in the same way (Vermani 2004). Hatcher, Allen (2002), Algebraic Topology, Cambridge University Press, p. 196, ISBN 9780521795401. Vermani, L. R. (2004), An Elementary Approach to Homological Algebra, Monographs and Surveys in Pure and Applied Mathematics, CRC Press, p. 80, ISBN 9780203484081.
20. Hungerford (1974), Exercise 4, p. 75.
21. Hungerford (1974), p. 70.
22. Hungerford (1974), Theorem 1.2, p. 73.
23. Hofmann, Karl H.; Morris, Sidney A. (2006), The Structure of Compact Groups: A Primer for Students - A Handbook for the Expert, De Gruyter Studies in Mathematics 25 (2nd ed.), Walter de Gruyter, p. 640, ISBN 9783110199772.
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26. Blass (1979), Example 7.1, provides a model of set theory, and a non-free projective abelian group P in this model that is a subgroup of a free abelian group $\left(\mathbb{Z}^{(A)}\right)^n$, where A is a set of atoms and n is a finite integer. He writes that this model makes the use of choice essential in proving that every projective group is free; by the same reasoning it also shows that choice is essential in proving that subgroups of free groups are free. Blass, Andreas (1979), "Injectivity, projectivity, and the axiom of choice", Transactions of the American Mathematical Society 255: 31–59, doi:10.1090/S0002-9947-1979-0542870-6, JSTOR 1998165, MR 542870.
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28. Hungerford (1974), Theorem 1.6, p. 74.
29. Johnson (2001), pp. 71–72.
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31. Hungerford (1974), Exercise 9, p. 75.
32. Hungerford (1974), Exercise 10, p. 75.
33. Hungerford (1974), Exercise 4, p. 198.
34. Hungerford (1974), Theorem 1.4, p. 74.
35. Vick, James W. (1994), Homology Theory: An Introduction to Algebraic Topology, Graduate Texts in Mathematics 145, Springer, p. 70, ISBN 9780387941264.
36. The theorem that free abelian groups are projective is equivalent to the axiom of choice; see Moore, Gregory H. (2012), Zermelo's Axiom of Choice: Its Origins, Development, and Influence, Courier Dover Publications, p. xii, ISBN 9780486488417.
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