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| In [[abstract algebra]], a '''free abelian group''' or '''free Z-module''' is an [[abelian group]] with a [[Free module|basis]].
| | I like my hobby Australian Football League. Seems boring? Not at all!<br>I try to learn Hindi in my spare time.<br><br>Here is my web blog; programs to help find housing ([http://rkslawyer.com/ rkslawyer.com]) |
| That is, it is a [[Set (mathematics)|set]] together with an [[associative]], [[commutative]], and invertible [[binary operation]],
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| and its basis is a subset of its elements
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| such that every element of the group can be written in one and only one way as a [[linear combination]] of basis elements with [[integer]] coefficients, finitely many of which are nonzero. The elements of a free abelian group with basis ''B'' are also known as '''formal sums''' over ''B''. Informally, formal sums may also be seen as signed [[multiset]]s with elements in ''B''. Free abelian groups and formal sums have applications in [[algebraic topology]], where they are used to define [[Chain (algebraic topology)|chain groups]], and in [[algebraic geometry]], where they are used to define [[Divisor (algebraic geometry)|divisors]].
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| Every set ''B'' has a unique free abelian group with ''B'' as its basis. This group may be constructed as a [[direct sum]] of copies of the additive group of the integers, with one copy per member of ''B''. Alternatively, it may be constructed as the set of functions from ''B'' to the integers that have finitely many nonzero values, with pointwise addition, or by a [[presentation of a group|presentation]] with the elements of ''B'' as its generators and with the [[commutator]]s of pairs of members as its relators.
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| Free abelian groups have properties which make them similar to [[vector space]]s and allow a general abelian group to be understood as a [[quotient group|quotient]] of a free abelian group by "relations". Every free abelian group has a rank defined as the cardinality of a basis. The rank determines the group up to [[isomorphism]], and the elements of such a group can be written as finite formal sums of the basis elements. Every subgroup of a free abelian group is itself free abelian, which allows the description of a general abelian group as a [[cokernel]] of an injective [[group homomorphism|homomorphism]] between free abelian groups.
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| == Examples and constructions ==
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| ===Integers and lattices===
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| The [[integer]]s, 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''.
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| 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)}.<ref>{{citation|title=Symmetries|series=Springer undergraduate mathematics series|first=D. L.|last=Johnson|publisher=Springer|year=2001|isbn=9781852332709|page=193|url=http://books.google.com/books?id=BecLeCWOjI4C&pg=PA193}}.</ref> If we say <math>\ e_1 = (1,0)</math> and <math>\ e_2 = (0,1)</math>, then the element (4,3) can be written
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| :<math>\ (4,3) = 4 e_1 + 3 e_2 </math> where 'multiplication' is defined so that <math>\ 4 e_1 := e_1 + e_1 + e_1 + e_1 </math>.
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| In this basis, there is no other way to write (4,3), but with a different basis such as {(1,0),(1,1)}, where <math>\ f_1 = (1,0)</math> and <math>\ f_2 = (1,1)</math>, it can be written as
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| :<math>\ (4,3) = f_1 + 3 f_2</math>.
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| More generally, every [[Lattice (group)|lattice]] forms a [[Finitely-generated abelian group|finitely-generated]] free abelian group.<ref>{{citation|title=Advanced Number Theory with Applications|first=Richard A.|last=Mollin|publisher=CRC Press|year=2011|isbn=9781420083293|page=182|url=http://books.google.com/books?id=6I1setlljDYC&pg=PA182}}.</ref> The ''d''-dimensional integer lattice has a natural basis consisting of the positive integer [[unit vector]]s, 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.<ref>{{citation|title=Lattice Basis Reduction: An Introduction to the LLL Algorithm and Its Applications|first=Murray R.|last=Bremner|publisher=CRC Press|year=2011|isbn=9781439807026|page=6|url=http://books.google.com/books?id=i5AkDxkrjPcC&pg=PA6}}.</ref> For more on the two-dimensional case, see [[fundamental pair of periods]].
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| ===Direct sums, direct products, and trivial group===
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| The [[direct product of groups|direct product]] of two free abelian groups is itself free abelian, with basis the [[disjoint union]] of the bases of the two groups.<ref name="h74-ex5">{{harvtxt|Hungerford|1974}}, Exercise 5, p. 75.</ref> 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'''.
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| The trivial group {0} is also considered to be free abelian, with basis the [[empty set]].<ref name="lee">{{citation|first=John M.|last=Lee|title=Introduction to Topological Manifolds|volume=202|series=Graduate Texts in Mathematics|publisher=Springer|edition=2nd|year=2010|isbn=9781441979407|contribution=Free Abelian Groups|pages=244–248|url=http://books.google.com/books?id=ZQVGAAAAQBAJ&pg=PA244}}.</ref> It may be interpreted as a direct product of zero copies of '''Z'''.
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| 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.<ref>{{harvtxt|Hungerford|1974}}, Exercise 5, p. 75.</ref>
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| For instance the [[Baer–Specker group]] <math>\mathbb{Z}^\mathbb{N}</math>, an uncountable group formed as the direct product of [[countably infinite|countably]] many copies of <math>\mathbb{Z}</math>, was shown in 1937 by [[Reinhold Baer]] to not be free abelian;<ref>{{citation
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| | last = Baer | first = Reinhold | author-link = Reinhold Baer
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| | doi = 10.1215/S0012-7094-37-00308-9
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| | issue = 1
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| | journal = Duke Mathematical Journal
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| | mr = 1545974
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| | pages = 68–122
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| | title = Abelian groups without elements of finite order
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| | volume = 3
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| | year = 1937}}.</ref> [[Ernst Specker]] proved in 1950 that every countable subgroup of <math>\mathbb{Z}^\mathbb{N}</math> is free abelian.<ref>{{citation
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| | last = Specker | first = Ernst | author-link = Ernst Specker
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| | journal = Portugaliae Math.
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| | mr = 0039719
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| | pages = 131–140
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| | title = Additive Gruppen von Folgen ganzer Zahlen
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| | volume = 9
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| | year = 1950}}.</ref>
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| The [[direct sum of groups|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.<ref name="h74-ex5"/>
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| The [[tensor product of modules|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.<ref>{{citation
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| | last = Corner | first = A. L. S.
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| | contribution = Groups of units of orders in $\Bbb Q$-algebras
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| | doi = 10.1515/9783110203035.9
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| | mr = 2513226
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| | pages = 9–61
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| | publisher = Walter de Gruyter, Berlin
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| | title = Models, modules and abelian groups
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| | year = 2008}}. See in particular the proof of Lemma H.4, [http://books.google.com/books?id=khekRRwz0x0C&pg=PA36 p. 36], which uses this fact.</ref>
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| Every free abelian group may be described as a direct sum of copies of <math>\mathbb{Z}</math>, with one copy for each member of its basis.<ref>{{citation|title=Homology|series=Classics in Mathematics|first=Saunders|last=Mac Lane|authorlink=Saunders Mac Lane|publisher=Springer|year=1995|isbn=9783540586623|page=93|url=http://books.google.com/books?id=pxRlrJn-WPgC&pg=PA93}}.</ref><ref name="kap"/> This construction allows any set ''B'' to become the basis of a free abelian group.<ref name="hungerford">{{citation|title=Algebra|volume=73|series=Graduate Texts in Mathematics|first=Thomas W.|last=Hungerford|authorlink=Thomas W. Hungerford|publisher=Springer|year=1974|isbn=9780387905181|pages=70–75|contribution=II.1 Free abelian groups|url=http://books.google.com/books?id=t6N_tOQhafoC&pg=PA70}}. See in particular Theorem 1.1, pp. 72–73, and the remarks following it.</ref>
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| ===Integer functions and formal sums===
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| Given a set ''B'', one can define a group <math>\mathbb{Z}^{(B)}</math> 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.
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| 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 <math>\mathbb{Z}^{(B)}</math> the structure of an abelian group.<ref name="joshi">{{citation|title=Applied Discrete Structures|first=K. D.|last=Joshi|publisher=New Age International|year=1997|isbn=9788122408263|pages=45–46|url=http://books.google.com/books?id=lxIgGGJXacoC&pg=PA45}}.</ref>
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| Each element ''x'' from the given set ''B'' corresponds to a member of <math>\mathbb{Z}^{(B)}</math>, the function ''e<sub>x</sub>'' for which ''e<sub>x</sub>''(''x'') = 1 and for which ''e<sub>x</sub>''(''y'') = 0 for all ''y'' ≠ ''x''.
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| Every function ''f'' in <math>\mathbb{Z}^{(B)}</math> is uniquely a linear combination of a finite number of basis elements:
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| :<math>f=\sum_{\{x\mid f(x)\ne 0\}} f(x) e_x</math>
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| Thus, these elements ''e<sub>x</sub>'' form a basis for <math>\mathbb{Z}^{(B)}</math>, and <math>\mathbb{Z}^{(B)}</math> is a free abelian group.
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| In this way, every set ''B'' can be made into the basis of a free abelian group.<ref name="joshi"/>
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| The free abelian group with basis ''B'' is unique up to isomorphism, and its elements are known as '''formal sums''' of elements of ''B''.
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| They may also be interpreted as the signed [[multiset]]s of finitely many elements of ''B''.
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| For instance, in [[algebraic topology]], [[Chain (algebraic topology)|chains]] are formal sums of [[simplex|simplices]], and the chain group is the free abelian group whose elements are chains.<ref>{{citation|title=Dictionary of Classical and Theoretical Mathematics|volume=3|series=Comprehensive Dictionary of Mathematics|editor1-first=Catherine|editor1-last=Cavagnaro|editor2-first=William T., II|editor2-last= Haight|publisher=CRC Press|year=2001|isbn=9781584880509|page=15|url=http://books.google.com/books?id=ljvmahfSDtwC&pg=PA15}}.</ref> In [[algebraic geometry]], the [[Divisor (algebraic geometry)|divisors]] of a [[Riemann surface]] (a combinatorial description of the zeros and poles of [[meromorphic function]]s) form an uncountable free abelian group, consisting of the formal sums of points from the surface.<ref>{{citation|title=Algebraic Curves and Riemann Surfaces|volume=5|series=Graduate Studies in Mathematics|publisher=American Mathematical Society|year=1995|isbn=9780821802687|first=Rick|last=Miranda|page=129|url=http://books.google.com/books?id=qjg6GOQaHNEC&pg=PA129}}.</ref>
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| ===Presentation===
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| The free abelian group with basis ''B'' has a [[presentation of a group|presentation]] in which the generators are the elements of ''B'', and the relators are the [[commutator]]s of pairs of elements of ''B''.<ref>{{harvtxt|Hungerford|1974}}, Exercise 3, p. 75.</ref>
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| This fact, together with the fact that every subgroup of a free abelian group is free abelian ([[#Subgroups|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''.<ref>{{harvtxt|Johnson|2001}}, p. 71.</ref>
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| == Terminology ==
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| Every abelian group may be considered as a [[Module (mathematics)|module]] over the integers by considering the scalar multiplication of a group member by an integer defined as follows:<ref>{{citation|title=Algebra|first1=Vivek|last1=Sahai|first2=Vikas|last2=Bist|publisher=Alpha Science Int'l Ltd.|year=2003|isbn=9781842651575|page=152|url=http://books.google.com/books?id=VsoyRX_nHLkC&pg=PA152}}.</ref>
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| :<math>\begin{align}
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| 0\,x&=0\\
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| 1\,x&=x\\
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| n\,x&= x+ (n-1)\,x \qquad \text{if} \quad n>1\\
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| (-n)\,x&= -(n\,x) \qquad \text{if} \quad n<0
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| \end{align}</math>
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| A [[free module]] is a module that can be represented as a direct sum over its base ring, so free abelian groups and free <math>\Z</math>-modules are equivalent concepts: each free abelian group is (with the multiplication operation above) a free <math>\Z</math>-module, and each free <math>\Z</math>-module comes from a free abelian group in this way.<ref>{{citation|title=Advanced Modern Algebra|first=Joseph J.|last=Rotman|publisher=American Mathematical Society|isbn=9780821884201|page=450|url=http://books.google.com/books?id=RGzK_DOTijsC&pg=PA450}}.</ref>
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| Unlike [[vector space]]s, not all abelian groups have a basis, hence the special name for those that do. For instance, any [[torsion (algebra)|torsion <math>\Z</math>-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: <math>0 = 0\,b = n\,b</math> for some positive integer ''n''. On the other hand, many important property of free abelian groups may be generalized to free modules over a [[principal ideal domain]].<ref>For instance, submodules of free modules over principal ideal domains are free, a fact that {{harvtxt|Hatcher|2002}} writes allows for "automatic generalization" of homological machinery to these modules. Additionally, the theorem that every projective <math>\Z</math>-module is free generalizes in the same way {{harv|Vermani|2004}}. {{citation|title=Algebraic Topology|first=Allen|last=Hatcher|publisher=Cambridge University Press|year=2002|isbn=9780521795401|page=196|url=http://books.google.com/books?id=BjKs86kosqgC&pg=PA196}}. {{citation|title=An Elementary Approach to Homological Algebra|series=Monographs and Surveys in Pure and Applied Mathematics|first=L. R.|last=Vermani|publisher=CRC Press|year=2004|isbn=9780203484081|page=80|url=http://books.google.com/books?id=P27AtdajYRgC&pg=PA80}}.</ref>
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| 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]]).<ref name="lee"/><ref>{{harvtxt|Hungerford|1974}}, Exercise 4, p. 75.</ref> 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 group]]s are the [[free object]]s 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]];<ref>{{harvtxt|Hungerford|1974}}, p. 70.</ref> 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.
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| == Properties ==
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| ===Universal property===
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| 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''.<ref name="lee"/> 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.<ref name="hungerford"/>
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| ===Rank===
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| Every two bases of the same free abelian group have the same [[cardinality]], so the cardinality of a basis forms an [[Invariant (mathematics)|invariant]] of the group known as its rank.<ref>{{harvtxt|Hungerford|1974}}, Theorem 1.2, p. 73.</ref><ref name="hm"/>
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| In particular, a free abelian group is [[finitely generated module|finitely generated]] if and only if its rank is a finite number ''n'', in which case the group is isomorphic to <math>\mathbb{Z}^n</math>.
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| 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 element|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.<ref>{{citation|title=An Introduction to Algebraic Topology|volume=119|series=Graduate Texts in Mathematics|first=Joseph J.|last=Rotman|publisher=Springer|year=1988|isbn=9780387966786|pages=61–62|url=http://books.google.com/books?id=waq9mwUmcQgC&pg=PA61}}.</ref>
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| ===Subgroups===
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| Every subgroup of a free abelian group is itself a free abelian group. This result of [[Richard Dedekind]]<ref>{{Citation|title=Topics in the Theory of Group Presentations|volume=42|series=London Mathematical Society lecture note series|first=D. L.|last=Johnson|publisher=Cambridge University Press|year=1980|isbn=978-0-521-23108-4|page=9}}.</ref> 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 [[fundamental theorem of cyclic groups|every nontrivial subgroup of the infinite cyclic group is infinite cyclic]].
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| '''[[Theorem]]:''' ''Let <math>F</math> be a free abelian group and let <math>G\subset F</math> be a subgroup. Then <math>G</math> is a free abelian group.''
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| The proof needs the [[axiom of choice]].<ref>{{harvtxt|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 <math>\left(\mathbb{Z}^{(A)}\right)^n</math>, 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. {{citation
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| | last = Blass | first = Andreas | authorlink = Andreas Blass
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| | doi = 10.1090/S0002-9947-1979-0542870-6
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| | journal = Transactions of the American Mathematical Society
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| | jstor = 1998165
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| | mr = 542870
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| | pages = 31–59
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| | title = Injectivity, projectivity, and the axiom of choice
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| | volume = 255
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| | year = 1979}}.</ref>
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| A proof using [[Zorn's lemma]] (one of many equivalent assumptions to the axiom of choice) can be found in [[Serge Lang]]'s ''Algebra''.<ref>Appendix 2 §2, page 880 of {{Lang Algebra|edition=3r}}.</ref> [[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.<ref name="kap">{{citation|title=Set Theory and Metric Spaces|volume=298|series=AMS Chelsea Publishing Series|first=Irving|last=Kaplansky|authorlink=Irving Kaplansky|publisher=American Mathematical Society|year=2001|isbn=9780821826942|pages=124–125|url=http://books.google.com/books?id=1XFDM75VK5MC&pg=PA124}}.</ref>
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| In the case of finitely generated free groups, the proof is easier, and leads to a more precise result.
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| '''Theorem:''' ''Let <math>G</math> be a subgroup of a finitely generated free abelian group <math>F</math>. Then <math>G</math> is free and there exists a basis <math>(e_1, \ldots, e_n)</math> of <math>F</math> and positive integers <math>d_1|d_2|\ldots|d_k</math> (that is, each one divides the next one) such that <math>(d_1e_1,\ldots, d_ke_k)</math> is a basis of <math>G.</math> Moreover, the sequence <math>d_1,d_2,\ldots,d_k</math> depends only on <math>F</math> and <math>G</math> and not on the particular basis <math>(e_1, \ldots, e_n)</math> that solves the problem.''<ref>{{harvtxt|Hungerford|1974}}, Theorem 1.6, p. 74.</ref>
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| 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.<ref>{{harvtxt|Johnson|2001}}, pp. 71–72.</ref> 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 <math>d_1\cdots d_r</math> at the end of the computation.<ref>{{citation|title=Finitely Generated Abelian Groups and Similarity of Matrices over a Field|series=Springer undergraduate mathematics series|first=Christopher|last=Norman|publisher=Springer|year=2012|isbn=9781447127307|contribution=1.3 Uniqueness of the Smith Normal Form|pages=32–43}}.</ref>
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| ===Torsion and divisibility===
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| All free abelian groups are [[torsion (algebra)|torsion-free]], meaning that there is no group element ''x'' and nonzero integer ''n'' such that ''nx'' = 0.
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| Conversely, all finitely generated torsion-free abelian groups are free abelian.<ref name="lee"/><ref name="h-ex9">{{harvtxt|Hungerford|1974}}, Exercise 9, p. 75.</ref> The same applies to [[flat module|flatness]], since an abelian group is torsion-free if and only if it is flat.
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| The additive group of [[rational number]]s '''Q''' provides an example of a torsion-free (but not finitely generated) abelian group that is not free abelian.<ref>{{harvtxt|Hungerford|1974}}, Exercise 10, p. 75.</ref> One reason that '''Q''' is not free abelian is that it is [[divisible group|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.<ref>{{harvtxt|Hungerford|1974}}, Exercise 4, p. 198.</ref>
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| ==Relation to arbitrary abelian groups==
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| 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 <math>F=\mathbb{Z}^{(A)}</math> 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'':
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| :<math>f=\sum_{\{x\mid f(x)\ne 0\}} f(x) e_x \mapsto \sum_{\{x\mid f(x)\ne 0\}} f(x) x,</math>
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| where the first sum is in ''F'' and the second sum is in ''A''.<ref name="hm">{{citation|title=The Structure of Compact Groups: A Primer for Students - A Handbook for the Expert|volume=25|series=De Gruyter Studies in Mathematics|first1=Karl H.|last1=Hofmann|first2=Sidney A.|last2=Morris|edition=2nd|publisher=Walter de Gruyter|year=2006|isbn=9783110199772|page=640|url=http://books.google.com/books?id=YvcRi0x67mgC&pg=PA640}}.</ref><ref>{{harvtxt|Hungerford|1974}}, Theorem 1.4, p. 74.</ref> This construction can be seen as an instance of the universal property: this surjection is the unique group homomorphism which extends the function <math>e_x\mapsto x</math>.
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| When ''F'' and ''A'' are as above, the [[Kernel (algebra)|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).
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| Therefore, these groups form a [[short exact sequence]]
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| :0 → ''G'' → ''F'' → ''A'' → 0
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| 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''.<ref>{{citation|title=Homology Theory: An Introduction to Algebraic Topology|volume=145|series=Graduate Texts in Mathematics|first=James W.|last=Vick|publisher=Springer|year=1994|isbn=9780387941264|page=70|url=http://books.google.com/books?id=5Bq8YlLrNc8C&pg=PA70}}.</ref> Furthermore, assuming the [[axiom of choice]],<ref>The theorem that free abelian groups are projective is equivalent to the axiom of choice; see {{citation|title=Zermelo's Axiom of Choice: Its Origins, Development, and Influence|first=Gregory H.|last=Moore|publisher=Courier Dover Publications|year=2012|isbn=9780486488417|page=xii|url=http://books.google.com/books?id=3RLGKcEjVIoC&pg=PR12}}.</ref> the free abelian groups are precisely the [[projective module|projective objects]] in the [[category of abelian groups]].<ref>{{citation | author=Phillip A. Griffith | title=Infinite Abelian group theory | series=Chicago Lectures in Mathematics | publisher=University of Chicago Press | year=1970 | isbn=0-226-30870-7 |page=18}}.</ref>
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| ==References==
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| {{Reflist|30em}}
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| {{DEFAULTSORT:Free Abelian Group}}
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| [[Category:Abelian group theory]]
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| [[Category:Properties of groups]]
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| [[Category:Free algebraic structures]]
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