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Topic: Free abelian group

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	Station Information - Free abelian group
		In abstract algebra, a free abelian group is an abelian group that has a "basis" in the sense that every element of the group can be written in one and only one way as a finite linear combination of elements of the basis, with integer coefficients.
		Note a point on terminology: a free abelian group is not the same as a free group that is abelian; in fact most free groups are not abelian.
		Free abelian groups are a special case of free modules, as abelian groups are nothing but modules over the ring Z.
	www.stationinformation.com /encyclopedia/f/fr/free_abelian_group.html (330 words)

	Abelian group - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-21)
		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.
		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.newlenox.us /project/wikipedia/index.php/Abelian_group (824 words)

	Encyclopedia: Free group
		A free group on a two-element subset S occurs in the proof of the Banach-Tarski paradox and is described there.
		The free group on S is characterized by the following universal property: if G is any group and In various branches of mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique morphism under certain conditions.
		Free groups are thus instances of the more general concept of free objects in category theory.
	www.nationmaster.com /encyclopedia/Free-group (1670 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.
		This shows that a single group can have all possible combinations of a given number of building blocks, so that even if one were to know complete decompositions of two torsion-free groups, one would not be sure that they were not isomorphic.
	www.wikipedia.org /wiki/Rank_of_an_abelian_group (665 words)

	Abelian group: Definition and Links by Encyclopedian.com - All about Abelian group (Site not responding. Last check: 2007-10-21)
		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.
		The abelian groups, together with group homomorphisms, form a category, the prototype of an abelian category.
	www.encyclopedian.com /ab/Abelian.html (461 words)

	Encyclopedia: Finitely-generated abelian group
		The fundamental theorem of finitely generated abelian groups states that every finitely generated abelian group G is isomorphic to a direct sum of primary and infinite cyclic groups.
		Stated differently the fundamental theorem says that a finitely-generated abelian group is the direct sum of a free abelian group of finite rank and a finite abelian group, each of those being unique up to isomorphism.
		Every subgroup and factor group of a finitely generated abelian group is again finitely generated abelian.
	www.nationmaster.com /encyclopedia/Finitely_generated-abelian-group (491 words)

	Injective cogenerator - Wikipedia, the free encyclopedia
		For example, the integers are a generator of the category of abelian groups (since every abelian group is a quotient of a free abelian group).
		As an example of a cogenerator in the same category, we have Q/Z, the rationals modulo the integers, which is a divisible abelian group.
		Finding a generator of an abelian category allows one to express every object as a quotient of a direct sum of copies of the generator.
	en.wikipedia.org /wiki/Injective_cogenerator (551 words)

	Abelian Group Theory papers of Andreas R. Blass
		We are concerned with subgroups of P that are free abelian groups.
		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 (1114 words)

	Free group -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-21)
		A free group on a two-element subset S occurs in the proof of the (Click link for more info and facts about Banach-Tarski paradox) Banach-Tarski paradox and is described there.
		Free groups are thus instances of the more general concept of (Click link for more info and facts about free object) free objects in (Click link for more info and facts about category theory) category theory.
		If F is a free group on S and also on T, then S and T have the same (Click link for more info and facts about cardinality) cardinality.
	www.absoluteastronomy.com /encyclopedia/f/fr/free_group.htm (733 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)

	PlanetMath: Baer-Specker group
		The Baer-Specker group is an important example of a torsion-free abelian group whose rank is infinite.
		It is not a free abelian group, but any of its subgroup whose cardinality is less than
		This is version 9 of Baer-Specker group, born on 2005-08-24, modified 2005-08-26.
	planetmath.org /encyclopedia/BaerSpeckerGroup.html (159 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		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).
		Particularly, the group with one generator is isomorphic to Z. In a sense, a free group is the set of all words that you can write using an alphabet consisting of the generators and their inverses.
		The free abelian group of rank n is isomorphic to Z^n.
	www.math.niu.edu /~rusin/known-math/98/free (365 words)

	Whitehead problem - Wikipedia, the free encyclopedia
		The affirmative answer for countable groups was already found in the 1950s.
		Progress for larger groups was slow, and the problem was considered one of the most important ones in algebra for many years.
		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.
	www.wikipedia.org /wiki/Whitehead_problem (286 words)

	Station Information - Free group
		Note that the notion of free group is different from the notion free abelian group: in this case the order in the product matters.
		Any group G is a quotient group of some free group F(S).
		With this understanding, the fundamental group of every connected graph is free.
	www.stationinformation.com /encyclopedia/f/fr/free_group.html (508 words)

	Free Abelian Groups (Site not responding. Last check: 2007-10-21)
		The free abelian group on s is equal to the direct sum of copies of Z - one copy for each generator in s.
		Thus the free abelian group is indeed a free object.
		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 (476 words)

	Free group (Site not responding. Last check: 2007-10-21)
		The Nielsen-Schreier theorem states that any subgroup of a free group is free.
		This was proved first, for groups free on a finite set, by Jakob Nielsen.
		This article is licensed under the GNU Free Documentation License, which means that you can copy and modify it as long as the entire work (including additions) remains under this license.
	www.sciencedaily.com /encyclopedia/free_group (670 words)

	Construction of Free Abelian Group and its Elements (Site not responding. Last check: 2007-10-21)
		Construct the free abelian group F on n generators, where n is a positive integer.
		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 +...
		Let A be an abelian group with basis e_1,..., e_r and suppose x is an element of A, where x = a_1 * e_1 +...
	www.math.uiuc.edu /Software/magma/text188.html (252 words)

	Magnus
		Since each such group is given by finitely many generators and finitely many defining relators in the category of abelian groups, i.e., as a quotient of a free abelian group of finite rank by a subgroup generated by finitely many elements, such a decomposition can be obtained by Gaussian Elimination.
		The isolator of a given subgroup is the pre-image of the torsion subgroup of the factor group of the given group by the subgroup.
		Automorphisms of the free abelian group of rank n are in one-to-one correspondence with matrices from the group GL_n(Z).
	zebra.sci.ccny.cuny.edu /web/caiss/MAGNUS/website/pages/checkinabeliangroup.html (1968 words)

	Free abelian group - Definition up Erdmond.Com (Site not responding. Last check: 2007-10-21)
		Unlike vector_spaces, not all abelian groups have a basis, hence the special name for those that do.
		Note a point on terminology: a free abelian group is not the same as a free_group that is abelian; in fact most free_groups are not abelian.
		Free abelian groups are a special case of free_modules, as abelian groups are nothing but modules over the ring Z.
	www.erdmond.com /Free_abelian_group.html (289 words)

	Which Equation Explains the Market Cycle ? best Abelian Group (Site not responding. Last check: 2007-10-21)
		In abstract algebra, an abelian group is a group (G, *) that is commutative, i.e...
		Abelian Group -- from MathWorld Abelian Group -- from MathWorld A group for which the elements commute (i.e., AB = BA for all elements A and B) is called an Abelian group.
		In mathematics, an abelian group is a commutative group, i.e...
	ascot.pl /th/Fourier1/Abelian-Group.htm (460 words)

	Finitely generated abelian group: Definition and Links by Encyclopedian.com - All about Finitely generated abelian group (Site not responding. Last check: 2007-10-21)
		Every finitely generated abelian group has finite rank equal to the number n from above.
		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 (385 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)

	Free article - Free freedom libre free will free software gratis free in beer Free - What-Means.com (Site not responding. Last check: 2007-10-21)
		This is a disambiguation page; that is, one that points to other pages that might otherwise have the same name.
		Free article - Free definition - what means Free
		All text is available under the terms of the GNU Free Documentation License.
	www.what-means.com /encyclopedia/Free (101 words)

	Further Results (Site not responding. Last check: 2007-10-21)
		Proposition 4.5 If a Hausdorff abelian topological group G is uniformly free from small subgroups then it is topologically isomorphic to a quotient group of a subgroup of a Banach space.
		The class of all topological groups which are topologically isomorphic to a quotient group of a subgroup of a product of Banach spaces is the class of all abelian topological groups (Corollary 3.4).
		The class of all topological groups (respectively, Hausdorff topological groups) topologically isomorphic to a quotient group of a subgroup of a Banach space is the class of all pseudo-metrizable abelian topological groups (respectively, metrizable abelian topological groups).
	cedir.uow.edu.au /Projects/math_test/node4.html (625 words)

	Group Theory Publications
		Monoid kernels and profinite topologies on the free Abelian group, Bull.
		A sampler of a topological approach to inverse semigroups (pdf) (abstract) is to appear in the proceedings of the Thematic Trimester on Automata, Languages, and Semigroups in Coimbra 2001.
		The geometry of profinite graphs with applications to free groups and finite monoids with Karl Auinger (pdf) (abstract).
	www.fc.up.pt /cmup/bsteinbg/pub3.html (223 words)

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