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Topic: Free abelian

	PlanetMath: free module
		This is equivalent to saying that the module has a free basis, i.e.
		This is version 5 of free module, born on 2002-01-05, modified 2006-07-24.
		The usual free module definition does not require the ring to be commutative.
	planetmath.org /encyclopedia/RankOfAFreeModule.html (194 words)

	Free abelian group - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-17)
		Note a point on terminology: a free abelian group is not the same as a free group that is abelian; in fact the only free groups that are abelian are those of rank 0 (the trivial group) and rank 1 (the infinite cyclic group).
		For every set B, there exists a free abelian group with basis B, and all such free abelian groups having B as basis are isomorphic.
		All free abelian groups are torsion free, and all finitely generated torsion free abelian groups are free abelian.
	en.wikipedia.org /wiki/Free_abelian_group (666 words)

	Abelian group - Wikipedia, the free encyclopedia
		In mathematics, an abelian group, also called a commutative group, is a group (G, ) such that a b = b * a for all a and b in G.
		Abelian groups are named after Niels Henrik Abel.
		This is a special application of the fundamental theorem of finitely generated abelian groups in the case when G has torsion-free rank equal to 0.
	en.wikipedia.org /wiki/Abelian_group (837 words)

	Dictionary of Meaning www.mauspfeil.net (Site not responding. Last check: 2007-10-17)
		A typical example of a free abelian group is the direct sum of groups direct sum '''Z''' ⊕ '''Z''' of two copies of the infinite cyclic group integer '''Z'''; a basis is {(1,0),(0,1)}.
		As a consequence, to every abelian group ''A'' there exists a short exact sequence :0 → ''G'' → ''F'' → ''A'' → 0 with ''F'' and ''G'' being free abelian (which means that ''A'' is isomorphic to the factor group ''F''/''G'').
		Free abelian groups are a special case of free modules, as abelian groups are nothing but module (mathematics) modules over the ring (mathematics) ring '''Z'''.
	www.mauspfeil.net /Free_abelian_group.html (694 words)

	Kids.Net.Au - Encyclopedia > Free abelian group (Site not responding. Last check: 2007-10-17)
		In abstract algebra, the free abelian group on a set X may be constructed as the abelian group of functions on X, taking integer values that are almost all zero.
		When X is finite of cardinality n the free abelian group on X is the same up to isomorphism as the product of n copies of the infinite cyclic group.
		This construction is a special case of the construction of free modules[?].
	www.kids.net.au /encyclopedia-wiki/fr/Free_abelian_group (132 words)

	Rank of an abelian group - Wikipedia, the free encyclopedia
		In mathematics, the rank, or torsion-free rank, of an abelian group measures how large a group is in terms of how large a vector space one would need to "contain" it; or alternatively how large a free abelian group it can contain as a subgroup.
		An abelian group is often thought of as composed of its torsion subgroup T, and its torsion-free part A/T.
		There are many abelian groups of rank 0, but the only torsion-free one is the trivial group {0}.
	en.wikipedia.org /wiki/Rank_of_an_abelian_group (686 words)

	Abelian group: Definition and links.
		In abstract algebra, an abelian group is a group (G, ) that is commutative, i.e., in which a b = b * a holds for all elements a and b in G.
		If a group is abelian, we usually write the operation as + instead of *, the identity element as 0 (often called the zero element in this context) and the inverse of the element a as -a.
		Any subgroup of an abelian group is normal, and hence factor groups can be formed freely.
	www.encyclopedian.com /ab/Abelian.html (457 words)

	Free abelian group
		In abstract algebra, the free abelian group on a set X may be constructed as the abelian group of functions on X, taking integer values that are almost all zero.
		When X is finite of cardinality n the free abelian group on X is the same up to isomorphism as the product of n copies of the infinite cyclic group.
		This construction is a special case of the construction of free modules[?].
	www.ebroadcast.com.au /lookup/encyclopedia/fr/Free_abelian_group.html (127 words)

	Abelian group - free-definition (Site not responding. Last check: 2007-10-17)
		In mathematics, an abelian group is a commutative group, i.e.
		Every field gives rise to two abelian groups in the same fashion -- the additive group of all elements, and the multiplicative group of nonzero elements.
		If f, g : G → H are two group homomorphisms between abelian groups, then their sum f + g, defined by (f + g)(x) = f(x) + g(x), is again a homomorphism.
	www.free-definition.com /Abelian-group.html (715 words)

	Free Abelian Groups (Site not responding. Last check: 2007-10-17)
		Thus the free abelian group is indeed a free object.
		Conversely, a free abelian group f′ on s is the image of a free group f on s, and the kernel includes, and is spanned by, the commutators xy = yx.
		The free group on 3 letters is not equivalent to the free group on 2 letters, or the free group on an infinite set of letters.
	www.mathreference.com /grp-free,abel.html (495 words)

	[No title] (Site not responding. Last check: 2007-10-17)
		If the free group (of rank n) is finitely generated by n Elements (say a,b,c,d,...) then the group consists of all expressions of these elements and their inverses and all distict expressions are unequal (because otherwise you would have an equation).
		A free abelian group is the same thing, except that the group is abelian- i.e.
		The free abelian group of rank n is isomorphic to Z^n.
	www.math.niu.edu /~rusin/known-math/98/free (365 words)

	Abelian group Summary
		The matrices, in contrast, do not form an abelian group under multiplication, for even when restricted to sets of invertible matrices, matrix multplication is generally non-communtative.
		This is a special application of the fundamental theorem of finitely generated abelian groups in the case when G has torsion-free rank equal to 0.
		The abelian group, together with group homomorphisms, form a category, the prototype of an abelian category.
	www.bookrags.com /Abelian_group (2196 words)

	PlanetMath: free group
		A group with only one element is a free group of rank 0, freely generated by the empty set.
		So the rank of a free group is a well-defined concept, and free groups of different ranks are non-isomorphic.
		This is version 38 of free group, born on 2002-02-25, modified 2007-01-08.
	planetmath.org /encyclopedia/Rank4.html (427 words)

	Abelian Group Theory papers of Andreas R. Blass
		Let G be an abelian group and let k be the smallest rank of any group whose direct sum with a free group is isomorphic to G. If k is uncountable, then G has k pairwise disjoint, non-free subgroups.
		We study, in the context of torsion-free abelian groups G, the sets that are maximal with respect to the property of freely generating a pure subgroup of G. We generalize many but not all of the familiar properties of basic subgroups to the subgroups generated by these "maximal pure independent" sets.
		Suppose G is an abelian group such that, for all countable subgroups C, the divisible part of the quotient G/C is countable.
	www.math.lsa.umich.edu /~ablass/abgp.html (1188 words)

	Finitely generated abelian group: Definition and links.
		Every subgroup and factor group of a finitely generated abelian group is again finitely generated abelian.
		Expressing the theorem in general terms, it says a finitely-generated abelian group is the sum of a free abelian group and a finite abelian group, each of those being unique up to isomorphism.
		The converse is not true however: there are many abelian groups of finite rank which are not finitely generated; the rank-1 group Q is one example, and the rank-0 group given by a direct sum of countably many copies of Z
	www.encyclopedian.com /fi/Finitely-generated-abelian-group.html (379 words)

	E.L. Lady -- PUBLICATIONS
		Abelian groups would seem to be among the very simplest of algebraic structures.
		In my opinion, the theory of finite rank torsion free abelian groups can be seen as a spectrum with quotient divisible groups (essentially the class dealt with in my splitting field papers) at one end of it and locally free groups at the other.
		The material on finite rank torsion free groups in Fuchs's book (1973) is completely outdated now and the only other exposition in book form is a set of notes by David Arnold (1982) which is in very rough form and in many places barely readable.
	www.math.hawaii.edu /~lee/biobib.html (1800 words)

	Construction of a Free Abelian Group and its Elements (Site not responding. Last check: 2007-10-17)
		Construct the free abelian group F on n generators, where n is a positive integer.
		defines F to be the free abelian group on two generators and assigns the names x and y to the generators.
		Given an abelian group A with generators e_1,..., e_r and a sequence Q = [a_1,..., a_r] of integers, construct the element a_1 e_1 +...
	www.umich.edu /~gpcc/scs/magma/text273.htm (275 words)

	[No title]
		If this is not the case, could someone please provide a >counterexample (I think I remember reading about some abelian group G such that >G is isomorphic to G oplus G oplus G but not to G oplus G, and if so that would >probably be a counterexample).
		What's known is that "all Abelian groups" is much too murky a family of objects to permit this kind of structural theorem.
		It is similarly true that one can make some headway characterizing Abelian groups with some other limiting condition, such as divisible groups, groups of bounded height, matrix groups, etc. Of course there's more to do in a field than simply find a structure theorem for the objects.
	www.math.niu.edu /~rusin/known-math/99/ab_gps (613 words)

	Construction of Free Abelian Group and its Elements (Site not responding. Last check: 2007-10-17)
		Construct the free abelian group F on n generators, where n is a positive integer.
		defines F to be the free abelian group on two generators and assigns the names x and y to the generators.
		Given an abelian group A with generators e_1,..., e_r and a sequence Q = [a_1,..., a_r] of integers, construct the element a_1 * e_1 +...
	www.math.uiuc.edu /Software/magma/text188.html (252 words)

	Maybe this Explains the Economic Cycle... best Free Abelian Group (Site not responding. Last check: 2007-10-17)
		In abstract algebra, a free abelian group is an abelian group that has a "basis" in the...
		Free Abelian Group -- from MathWorld Free Abelian Group -- from MathWorld A free Abelian group is a group G with a subset which generates the group G with the only relation being ab = ba.
		In a torsion free abelian group A every subgroup G is contained in a unique...
	ascot.pl /th/Fourier3/Free-Abelian-Group.htm (468 words)

	Lee Lady: Finite Rank Torsion Free Modules over Dedekind Domains (a book)
		Kaplansky, in his "little red book", asserted that abelian group theory is really the study of modules over principal ideal domains, and since then most abelian group theorists tend to feel more at home with commutative ring theory than with group theory in general.
		The theory of finite rank torsion free abelian groups is full of results that depend on countability, or on having characteristic zero, or working over a ring whose quotient field is a perfect field, as well as proofs using quite specialized results from number theory.
		And one becomes more aware of the fact that the theory of finite rank torsion free abelian groups is moving away from abelian group theory in general in much the same fashion that abelian group theory has moved away from general group theory.
	www.math.hawaii.edu /~lee/book (629 words)

	Whitehead problem - Wikipedia, the free encyclopedia
		(A, Z) = 0 can be equivalently formulated as follows: whenever B is an abelian group and f : B → A is a surjective group homomorphism whose kernel is isomorphic to the group of integers Z, then there exists a group homomorphism g : A → B with fg = id
		Various similar independence statements were proved and it was realized more and more that the theory of abelian groups depends very sensitively on the underlying set theory.
		Shelah: "Whitehead groups may not be free, even assuming CH.
	www.wikipedia.org /wiki/Whitehead_problem (307 words)

	VITA
		Abelian groups, especially torsion-free abelian groups, and related topics in algebra, such as modules over discrete valuation rings, orders, and pullback rings and representations of finite partially ordered sets.
		Abelian groups A such that Hom(A,-) preserves direct sums of copies of A, Pacific J. Math., 56 (1975), 7-20 (with C. Murley).
		Torsion free abelian groups of finite rank projective as modules over their endomorphism rings, J. of Alg., 71 (1981), 1-10 (with R. Pierce, J. Reid, C. Vinsonhaler, and W. Wickless).
	www3.baylor.edu /~David_Arnold/Arnold_vita.html (1263 words)

	Cancellative Superposition Decides the Theory of Divisible Torsion-Free Abelian Groups - Waldmann (ResearchIndex) (Site not responding. Last check: 2007-10-17)
		Abstract: In divisible torsion-free abelian groups, the eciency of the cancellative superposition calculus can be greatly increased by combining it with a variable elimination algorithm that transforms every clause into an equivalent clause without unshielded variables.
		This is not surprising since abelian groups are of course ubiquitous in many applications of (semi)automated reasoning.
		6.5%: A Superposition Calculus for Divisible Torsion-Free Abelian..
	citeseer.ist.psu.edu /waldmann99cancellative.html (526 words)

	Normal Structure and Characteristic Subgroups
		Returns a normal series of G, the factors of which are either elementary abelian p-groups or free abelian groups.
		Returns a normal series of G, the factors of which are either elementary abelian p-groups which are semisimple as GF(p)[G]-modules or free abelian groups which are semisimple as Q[G]-modules.
		Returns a sequence containing the invariants of the maximal abelian quotient G/G^prime of the group G. Each infinite cyclic factor of G/G^prime is represented by zero.
	www.umich.edu /~gpcc/scs/magma/text370.htm (829 words)

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