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# 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 (1,0) and (0,1) as its basis. Free abelian groups have properties which make them similar to vector spaces, and may equivalently be called free ${\displaystyle \mathbb {Z} }$-modules, 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 ${\displaystyle B}$ may be described in several equivalent ways. These include formal sums over ${\displaystyle B}$, which are expressions of the form ${\textstyle \sum a_{i}b_{i}}$ where each ${\displaystyle a_{i}}$ is a nonzero integer, each ${\displaystyle b_{i}}$ 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 ${\displaystyle B}$, 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 ${\displaystyle B}$ to the integers with finitely many nonzero values; for this functional representation, the group operation is the pointwise addition of functions.

Every set ${\displaystyle B}$ has a free abelian group with ${\displaystyle B}$ 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 ${\displaystyle B}$ may be constructed as a direct sum of copies of the additive group of the integers, with one copy per member of ${\displaystyle B}$. Alternatively, the free abelian group with basis ${\displaystyle B}$ may be described by a presentation with the elements of ${\displaystyle B}$ 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

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 ${\displaystyle \{1\}}$. Every integer ${\displaystyle n}$ is a linear combination of basis elements with integer coefficients: namely, ${\displaystyle n=n\times 1}$, with the coefficient ${\displaystyle n}$.[1] Similarly, the positive rational numbers, under multiplication, form a free abelian group with the prime numbers as their basis. By the fundamental theorem of arithmetic, every positive rational can be factorized uniquely into the product of finitely many primes or their inverses. In this example, the integer coefficients are the exponents of each prime in the factorization, and are positive for prime divisors of the numerator of the given rational number and negative for divisors of the denominator.[2]

The polynomials of a single variable ${\displaystyle x}$, with integer coefficients, form a free abelian group under addition, with the powers of ${\displaystyle x}$ as its generators. In fact, this is an isomorphic group to the multiplicative group of positive rational numbers, with the exponent of the ${\displaystyle i}$th prime number in the multiplicative group of the rationals corresponding to the coefficient of ${\displaystyle x^{i-1}}$ in the corresponding polynomial.[3]

The two-dimensional integer lattice ${\displaystyle \mathbb {Z} ^{2}}$, consisting of the points in the plane with integer Cartesian coordinates, forms a free abelian group under vector addition with the basis ${\displaystyle \{(1,0),(0,1)\}}$.[1] For example, letting these basis vectors be denoted ${\displaystyle \ e_{1}=(1,0)}$ and ${\displaystyle \ e_{2}=(0,1)}$, the element ${\displaystyle (4,3)}$ 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 ${\displaystyle (4,3)}$. However, with a different basis such as ${\displaystyle \{(1,0),(1,1)\}}$, 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.[4] The ${\displaystyle d}$-dimensional integer lattice ${\displaystyle \mathbb {Z} ^{d}}$ has a natural basis consisting of the positive integer unit vectors, but it has many other bases as well: if ${\displaystyle M}$ is a ${\displaystyle d\times d}$ integer matrix with determinant ${\displaystyle \pm 1}$, then the rows of ${\displaystyle M}$ form a basis, and conversely every basis of the integer lattice has this form.[5] For more on the two-dimensional case, see fundamental pair of periods.

## Constructions

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 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 pointwise 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.[6] More generally the direct product of any finite number of free abelian groups is free abelian. The ${\displaystyle d}$-dimensional integer lattice, for instance, is isomorphic to the direct product of ${\displaystyle d}$ copies of the integer group ${\displaystyle \mathbb {Z} }$. The trivial group ${\displaystyle \{0\}}$ is also considered to be free abelian, with basis the empty set.[7] It may be interpreted as a direct product of zero copies of ${\displaystyle \mathbb {Z} }$.[8]

For infinite families of free abelian groups, the direct product is not necessarily free abelian.[6] 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,[9] although Ernst Specker proved in 1950 that all of its countable subgroups are free abelian.[10] 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.[6]

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.[11][12] This construction allows any set ${\displaystyle B}$ to become the basis of a free abelian group.[13]

### Integer functions and formal sums

Given a set ${\displaystyle B}$, one can define a group ${\displaystyle \mathbb {Z} ^{(B)}}$ whose elements are functions from ${\displaystyle B}$ to the integers, where the parenthesis in the superscript indicates that only the functions with finitely many nonzero values are included. If ${\displaystyle f(x)}$ and ${\displaystyle g(x)}$ are two such functions, then ${\displaystyle f+g}$ is the function whose values are sums of the values in ${\displaystyle f}$ and ${\displaystyle g}$: that is, ${\displaystyle (f+g)(x)=f(x)+g(x)}$. This pointwise addition operation gives ${\displaystyle \mathbb {Z} ^{(B)}}$ the structure of an abelian group.[14]

Each element ${\displaystyle x}$ from the given set ${\displaystyle B}$ corresponds to a member of ${\displaystyle \mathbb {Z} ^{(B)}}$, the function ${\displaystyle e_{x}}$ for which ${\displaystyle e_{x}(x)=1}$ and for which ${\displaystyle e_{x}(y)=0}$ for all ${\displaystyle y\neq x}$. Every function ${\displaystyle f}$ 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 ${\displaystyle e_{x}}$ form a basis for ${\displaystyle \mathbb {Z} ^{(B)}}$, and ${\displaystyle \mathbb {Z} ^{(B)}}$ is a free abelian group. In this way, every set ${\displaystyle B}$ can be made into the basis of a free abelian group.[14]

The elements of ${\displaystyle \mathbb {Z} ^{(B)}}$ 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 of ${\displaystyle B}$. 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.[2] They may also be interpreted as the signed multisets of finitely many elements of ${\displaystyle B}$.[15]

### 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 ${\displaystyle B}$ has a presentation in which the generators are the elements of ${\displaystyle B}$, and the relators are the commutators of pairs of elements of ${\displaystyle B}$. Here, the commutator of two elements ${\displaystyle x}$ and ${\displaystyle y}$ is the product ${\displaystyle x^{-1}y^{-1}xy}$; setting this product to the identity causes ${\displaystyle xy}$ to equal ${\displaystyle yx}$, so that ${\displaystyle x}$ and ${\displaystyle y}$ 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.[16]

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 (below) can be used to show that every finitely generated abelian group is finitely presented. For, if ${\displaystyle G}$ is finitely generated by a set ${\displaystyle B}$, it is a quotient of the free abelian group over ${\displaystyle B}$ by a free abelian subgroup, the subgroup generated by the relators of the presentation of ${\displaystyle G}$. But since this subgroup is itself free abelian, it is also finitely generated, and its basis (together with the commutators over ${\displaystyle B}$) forms a finite set of relators for a presentation of ${\displaystyle G}$.[17]

## As a module

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. Every abelian group may be considered as a module over the integers, with a scalar multiplication operation defined as follows:[18]

 ${\displaystyle 0\,x=0}$ ${\displaystyle 1\,x=x}$ ${\displaystyle n\,x=x+(n-1)\,x,\quad }$ if ${\displaystyle \quad n>1}$ ${\displaystyle n\,x=-((-n)\,x),}$ if ${\displaystyle \quad n<0}$

However, unlike vector spaces, not all abelian groups have a basis, hence the special name for those that do. A free module is a module that can be represented as a direct sum over its base ring, so free abelian groups and free ${\displaystyle \mathbb {Z} }$-modules are equivalent concepts: each free abelian group is (with the multiplication operation above) a free ${\displaystyle \mathbb {Z} }$-module, and each free ${\displaystyle \mathbb {Z} }$-module comes from a free abelian group in this way.[19] As well as the direct sum, another way to combine free abelian groups is to use the tensor product of ${\displaystyle \mathbb {Z} }$-modules. 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.[20]

Many important properties of free abelian groups may be generalized to free modules over a principal ideal domain. 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.[21] Additionally, the theorem that every projective ${\displaystyle \mathbb {Z} }$-module is free generalizes in the same way.[22]

## Properties

### Universal property

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 ${\displaystyle f}$.[2][7] 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.[13]

It is because of this universal property that free abelian groups are called "free": they are the free objects in the category of abelian groups, and the map from a basis to its free abelian group is a functor from sets to abelian groups, adjoint to the forgetful functor from abelian groups to sets.[23] However, a free abelian group is not a free group except in two cases: a free abelian group having an empty basis (rank zero, giving the trivial group) or having just one element in the basis (rank one, giving the infinite cyclic group).[7][24] Other abelian groups are not free groups because in free groups ${\displaystyle ab}$ must be different from ${\displaystyle ba}$ if ${\displaystyle a}$ and ${\displaystyle b}$ are different elements of the basis, while in free abelian groups the two products must be identical for all pairs of elements. In the general category of groups, it is an added constraint to demand that ${\displaystyle ab=ba}$, whereas this is a necessary property in the category of abelian groups.[25]

### 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.[26][27] Two free abelian groups are isomorphic if and only if they have the same rank.[2] A free abelian group is finitely generated if and only if its rank is a finite number ${\displaystyle n}$, 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 ${\displaystyle G}$ is defined as the rank of a free abelian subgroup ${\displaystyle F}$ of ${\displaystyle G}$ for which the quotient group ${\displaystyle G/F}$ is a torsion group. Equivalently, it is the cardinality of a maximal subset of ${\displaystyle G}$ that generates a free subgroup. Again, this is a group invariant; it does not depend on the choice of the subgroup.[28]

### Subgroups

Every subgroup of a free abelian group is itself a free abelian group. This result of Richard Dedekind[29] 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.[23] A proof using Zorn's lemma (one of many equivalent assumptions to the axiom of choice) can be found in Serge Lang's Algebra.[30] 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.[12]

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.[31] 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.[32] 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.[33]

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. This theorem states that every finitely generated abelian group is a direct sum of cyclic groups.

### Torsion and divisibility

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.[7][34]

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.[35] 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.[36]

### Relation to other groups

If a free abelian group is a quotient of two groups ${\displaystyle A/B}$, then ${\displaystyle A}$ is the direct sum ${\displaystyle B\oplus A/B}$.[2]

Given an arbitrary abelian group ${\displaystyle A}$, there always exists a free abelian group ${\displaystyle F}$ and a surjective group homomorphism from ${\displaystyle F}$ to ${\displaystyle A}$. One way of constructing a surjection onto a given group ${\displaystyle A}$ is to let ${\displaystyle F=\mathbb {Z} ^{(A)}}$ be the free abelian group over ${\displaystyle A}$, represented as formal sums. Then a surjection can be defined by mapping formal sums in ${\displaystyle F}$ to the corresponding sums of members of ${\displaystyle A}$. That is, the surjection maps

${\displaystyle \sum _{\{x\mid a_{x}\neq 0\}}a_{x}e_{x}\mapsto \sum _{\{x\mid a_{x}\neq 0\}}a_{x}x,}$
where ${\displaystyle a_{x}}$ is the integer coefficient of basis element ${\displaystyle e_{x}}$ in a given formal sum, the first sum is in ${\displaystyle F}$, and the second sum is in ${\displaystyle A}$.[27][37] This surjection is the unique group homomorphism which extends the function ${\displaystyle e_{x}\mapsto x}$, and so its construction can be seen as an instance of the universal property.

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

${\displaystyle 0\to G\to F\to A\to 0}$
in which ${\displaystyle F}$ and ${\displaystyle G}$ are both free abelian and ${\displaystyle A}$ is isomorphic to the factor group ${\displaystyle F/G}$. This is a free resolution of ${\displaystyle A}$.[38] Furthermore, assuming the axiom of choice,[39] the free abelian groups are precisely the projective objects in the category of abelian groups.[2][40]

## Applications

### Algebraic topology

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.[41] 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.[42]

### Algebraic geometry and complex analysis

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.[43]

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.[44] 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.[43]

### Group rings

The group ring ${\displaystyle \mathbb {Z} [G]}$, for any group ${\displaystyle G}$, is ring whose additive group is the free abelian group over ${\displaystyle G}$.[45] When ${\displaystyle G}$ is finite and abelian, the multiplicative group of units in ${\displaystyle \mathbb {Z} [G]}$ has the structure of a direct product of a finite group and a finitely generated free abelian group.[46][47]

## References

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2. Fuchs, László (2015), "Section 3.1: Freeness and projectivity", Abelian Groups, Springer Monographs in Mathematics, Cham: Springer, pp. 75–80, doi:10.1007/978-3-319-19422-6, ISBN 978-3-319-19421-9, MR 3467030
3. ^ Bradley, David M. (2005), Counting the positive rationals: A brief survey, arXiv:math/0509025
4. ^ Mollin, Richard A. (2011), Advanced Number Theory with Applications, CRC Press, p. 182, ISBN 9781420083293
5. ^ Bremner, Murray R. (2011), Lattice Basis Reduction: An Introduction to the LLL Algorithm and Its Applications, CRC Press, p. 6, ISBN 9781439807026
6. ^ a b c Hungerford (1974), Exercise 5, p. 75.
7. ^ a b c d Lee, John M. (2010), "Free Abelian Groups", Introduction to Topological Manifolds, Graduate Texts in Mathematics, 202 (2nd ed.), Springer, pp. 244–248, ISBN 9781441979407
8. ^ As stated explicitly, for instance, in Hartley, Brian; Turull, Alexandre (1994), "On characters of coprime operator groups and the Glauberman character correspondence", Journal für die Reine und Angewandte Mathematik, 451: 175–219, doi:10.1515/crll.1994.451.175/html, MR 1277300, proof of Lemma 2.3: "the trivial group is the direct product of the empty family of groups"
9. ^ Baer, Reinhold (1937), "Abelian groups without elements of finite order", Duke Mathematical Journal, 3 (1): 68–122, doi:10.1215/S0012-7094-37-00308-9, hdl:10338.dmlcz/100591, MR 1545974
10. ^ Specker, Ernst (1950), "Additive Gruppen von Folgen ganzer Zahlen", Portugaliae Math., 9: 131–140, MR 0039719
11. ^ Mac Lane, Saunders (1995), Homology, Classics in Mathematics, Springer, p. 93, ISBN 9783540586623
12. ^ a b Kaplansky, Irving (2001), Set Theory and Metric Spaces, AMS Chelsea Publishing Series, 298, American Mathematical Society, pp. 124–125, ISBN 9780821826942
13. ^ a b Hungerford, Thomas W. (1974), "II.1 Free abelian groups", Algebra, Graduate Texts in Mathematics, 73, Springer, pp. 70–75, ISBN 9780387905181. See in particular Theorem 1.1, pp. 72–73, and the remarks following it.
14. ^ a b Joshi, K. D. (1997), Applied Discrete Structures, New Age International, pp. 45–46, ISBN 9788122408263
15. ^ van Glabbeek, Rob; Goltz, Ursula; Schicke-Uffmann, Jens-Wolfhard (2013), "On characterising distributability", Logical Methods in Computer Science, 9 (3): 3:17, 58, arXiv:1309.3883, doi:10.2168/LMCS-9(3:17)2013, MR 3109601
16. ^ Hungerford (1974), Exercise 3, p. 75.
17. ^ Johnson, D. L. (2001), Symmetries, Springer undergraduate mathematics series, Springer, p. 71, ISBN 9781852332709
18. ^ Sahai, Vivek; Bist, Vikas (2003), Algebra, Alpha Science Int'l Ltd., p. 152, ISBN 9781842651575
19. ^ Rotman, Joseph J., Advanced Modern Algebra, American Mathematical Society, p. 450, ISBN 9780821884201
20. ^ Corner, A. L. S. (2008), "Groups of units of orders in Q-algebras", Models, modules and abelian groups, Walter de Gruyter, Berlin, pp. 9–61, doi:10.1515/9783110203035.9, MR 2513226. See in particular the proof of Lemma H.4, p. 36, which uses this fact.
21. ^ Hatcher, Allen (2002), Algebraic Topology, Cambridge University Press, p. 196, ISBN 9780521795401
22. ^ Vermani, L. R. (2004), An Elementary Approach to Homological Algebra, Monographs and Surveys in Pure and Applied Mathematics, CRC Press, p. 80, ISBN 9780203484081
23. ^ a b 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 0542870. For the connection to free objects, see Corollary 1.2. Example 7.1 provides a model of set theory without choice, and a non-free projective abelian group ${\displaystyle P}$ in this model that is a subgroup of a free abelian group ${\textstyle {\bigl (}\mathbb {Z} ^{(A)}{\bigr )}^{n}}$, where ${\displaystyle A}$ is a set of atoms and ${\displaystyle n}$ is a finite integer. Blass 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.
24. ^ Hungerford (1974), Exercise 4, p. 75.
25. ^ Hungerford (1974), p. 70.
26. ^ Hungerford (1974), Theorem 1.2, p. 73.
27. ^ a b 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
28. ^ Rotman, Joseph J. (1988), An Introduction to Algebraic Topology, Graduate Texts in Mathematics, 119, Springer, pp. 61–62, ISBN 9780387966786
29. ^ Johnson, D. L. (1980), Topics in the Theory of Group Presentations, London Mathematical Society lecture note series, 42, Cambridge University Press, p. 9, ISBN 978-0-521-23108-4
30. ^ Appendix 2 §2, page 880 of Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556, Zbl 0984.00001
31. ^ Hungerford (1974), Theorem 1.6, p. 74.
32. ^ Johnson (2001), pp. 71–72.
33. ^ Norman, Christopher (2012), "1.3 Uniqueness of the Smith Normal Form", Finitely Generated Abelian Groups and Similarity of Matrices over a Field, Springer undergraduate mathematics series, Springer, pp. 32–43, ISBN 9781447127307
34. ^ Hungerford (1974), Exercise 9, p. 75.
35. ^ Hungerford (1974), Exercise 10, p. 75.
36. ^ Hungerford (1974), Exercise 4, p. 198.
37. ^ Hungerford (1974), Theorem 1.4, p. 74.
38. ^ Vick, James W. (1994), Homology Theory: An Introduction to Algebraic Topology, Graduate Texts in Mathematics, 145, Springer, p. 70, ISBN 9780387941264
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