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Topic: Abelian group

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Cyclic group

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Glossary of group theory

Group (mathematics)

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Homology (mathematics)

Sylow theorems

Multiplicative group of integers modulo n

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	NationMaster - Encyclopedia: Free abelian group
		A typical example of a free abelian group is the direct sum Z ⊕ Z of two copies of the infinite cyclic group Z; a basis is {(1,0),(0,1)}.
		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.
		All free abelian groups are torsion free, and all finitely generated torsion free abelian groups are free abelian.
	www.nationmaster.com /encyclopedia/Free-abelian-group (752 words)

	Abelian group - Wikipedia, the free encyclopedia
		The multiplicative notation is the usual notation for groups, while the additive notation is the usual notation for modules.
		A typical example is the classification of finitely generated abelian groups.
		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 (963 words)

	PlanetMath: elementary abelian group
		An elementary abelian group is an abelian group in which every non-trivial element has the same finite order.
		Cross-references: isomorphism, infinite, cardinal number, cyclic group, direct sum, isomorphic, field, vector space, exponent, non-abelian group, group, prime, easy to see, order, finite, non-trivial element, abelian group
		This is version 9 of elementary abelian group, born on 2004-12-12, modified 2006-03-15.
	www.planetmath.org /encyclopedia/BooleanGroup.html (178 words)

	Free abelian group - Wikipedia, the free encyclopedia
		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.
		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).
		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 (650 words)

	:::► Dictionary of Meaning www.mauspfeil.net ◄::: (Site not responding. Last check: )
		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)}.
		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).
		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'').
	www.mauspfeil.net /Free_abelian_group.html (699 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.
		One can verify directly that this has the appropriate universal property in relation to arbitrary functions on X with values in some abelian group A: namely unique extension to a homomorphism of the free group.
		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.
	www.ebroadcast.com.au /lookup/encyclopedia/fr/Free_abelian_group.html (107 words)

	Wikinfo \| Abelian group
		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.wikinfo.org /wiki.php?title=Abelian_group (638 words)

	PlanetMath: abelian group
		Theorem 2 Any subgroup of an abelian group is normal.
		Theorem 3 Quotient groups of abelian groups are also abelian.
		This is version 13 of abelian group, born on 2003-10-15, modified 2005-04-26.
	planetmath.org /encyclopedia/AbelianGroup2.html (138 words)

	The Abelian Group: Assessment
		Abelian certificates are given at the end of a course of study.
		At The Abelian Group, three levels of achievement are recognized, and they are signified by the color of the seal placed on the certificate of completion:
		(See www.ets.org) Because Abelian courses are "the real thing", not watered down in any way, students who pass Abelian tests have no trouble with the CLEP tests, or any other test of the same material.
	www.abeliangroup.com /assessment.html (297 words)

	Free Abelian Groups (Site not responding. Last check: )
		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.
		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.
	www.mathreference.com /grp-free,abel.html (495 words)

	Abelian group
		A group that is commutative – in other words, in which the result of multiplying one member of the group by another is independent of the order of multiplication.
		Abelian groups, named after Niels Abel, are of central importance in modern mathematics, most notably in algebraic topology.
		Examples of Abelian groups include the real numbers (with addition), the non-zero real numbers (with multiplication), and all cyclic groups, such as the integers (with addition).
	www.daviddarling.info /encyclopedia/A/Abelian_group.html (155 words)

	PlanetMath: elementary abelian group
		An elementary abelian group is an abelian group in which every non-trivial element has the same finite order.
		Cross-references: isomorphism, infinite, cardinal number, cyclic group, direct sum, isomorphic, field, vector space, exponent, group, prime, easy to see, order, finite, non-trivial element, abelian group
		This is version 9 of elementary abelian group, born on 2004-12-12, modified 2006-03-15.
	planetmath.org /encyclopedia/ElementaryAbelianGroup.html (176 words)

	Rank of an abelian group at opensource encyclopedia (Site not responding. Last check: )
		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.
		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.wiki.tatet.com /Rank_of_an_abelian_group.html (683 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 : Finitely generated Abelian group
		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 isn't 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.fastload.org /fi/Finitely_generated_Abelian_group.html (406 words)

	Elements of a Generic Abelian Group
		Given an abelian group A and an element e of the domain over which it is defined, return e as an element of A. If A is not a proper subset of its underlying domain, then this always succeeds.
		Given a generic abelian group A with generators e_1,..., e_n and a sequence Q = [a_1,..., a_n] of integers, construct the element a_1 e_1 +...
		An element of a generic abelian group A can be represented by a sequence of integers giving the coefficients of its linear combination of a given set S of elements of A. This S can be the reduced set of generators of A as obtained from the group structure computation.
	wwwmaths.anu.edu.au /research.programs/aat/htmlhelp/text344.htm (661 words)

	EVERY FINITE ABELIAN GROUP IS A BRAUER GROUP. (Site not responding. Last check: )
		Since every finite abelian group G is a direct sum of cyclic groups we see that any G is the Brauer group of the three-dimensional noetherian ring A (CRTIMES)(,(//C)) (B(,n(,1)) (CRPLUS)...(CRPLUS)B(,n(,r))) for suitable choices of n(,i).
		Using cohomological techniques we investigate the Brauer group of Y x (,k)' where Y is a scheme over a field k.
		We give sufficient conditions on Y so that the Brauer group of Y x ' is equal to the Brauer group of Y. In particular, equality holds if Y has dimension one over k and the characteristic of k is zero.
	digitalcommons.fau.edu /faculty_dissertations/AAI8110805 (314 words)

	Construction of a Generic Abelian Group
		Typically one reason for not computing the structure of the group is that it may be expensive to do so and may actually not be required for further operations (like finding the order or the discrete logarithm of an element of the group).
		The first requires that the order of the abelian group is known, or that it can be computed; this then allows to construct each of the p-Sylow subgroups of the group.
		Construct the generic abelian group A over the domain U. The domain U can be an aggregate (set, indexed set or sequence) of elements or it can be `anything', for example, an elliptic curve, a jacobian, a magma of quadratic forms.
	www.math.niu.edu /help/math/magmahelp/text363.html (813 words)

	Rushanan, Joseph John (1986-05-22) Topics in integral matrices and Abelian group codes. ...
		We consider the SNF of a matrix [...] to be the ratio of two [...]-modules-a finitely generated abelian group; this is called the Smith group of A. The Smith group provides a unified setting to present both new and old results.
		The old results discussed are the interlacing of the SNF in the case of augmented matrices and the symmetries of the SNF for certain combinatorial matrices.
		We take abelian group codes to be ideals in the group ring [...], where [...] is a finite abelian group of odd order [...] and [...] is a finite field with characteristic relatively prime to [...].
	etd.caltech.edu /etd/available/etd-05022003-113743 (356 words)

	ipedia.com: Abelian group Article (Site not responding. Last check: )
		a group such that a * b = b * a for all a and b in G. Abelian groups are named after Niels Henrik Abel.
		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 the matrix is a symmetric matrix), if and only if the group is abelian.
	www.ipedia.com /abelian_group.html (772 words)

	Construction of a Generic Abelian Group
		Typically, one reason for not computing the structure of the group is that it may be expensive to do so and may actually not be required for further operations such as finding the order or the discrete logarithm of an element of the group.
		Construct the generic abelian group A over the domain U. The domain U can be an aggregate (set, indexed set or sequence) of elements or it can be any structure, for example, an elliptic curve, a jacobian, a magma of quadratic forms.
		This may be useful as it removes the need to compute the order of the group, should this be required by some group structure computation, or to solve the discrete logarithm problem.
	wwwmaths.anu.edu.au /research.programs/aat/htmlhelp/text343.htm (813 words)

	Math Forum - Ask Dr. Math
		According to the Sylow Theorems, the group must contain elements of orders 2 and 3.
		It must also have order 3, but it can't equal b, because the group must be nonabelian.
		Now the six group elements are e, a, b, ba, b^2, and b^2a.
	mathforum.org /library/drmath/view/51668.html (177 words)

	S.O.S. Mathematics CyberBoard :: View topic - Abelian group and automorphisms
		Posted: Sat, 23 Sep 2006 01:02:44 GMT Post subject: Abelian group and automorphisms
		If we have an abelian group where all the elements have order 2, how many automorphisms (other than the identity) does this group have?
		Powered by phpBB © 2001, 2005-2007 phpBB Group.
	www.sosmath.com /CBB/viewtopic.php?t=25116 (155 words)

	The FFT on an Abelian Group
		So far, we have only shown that the DFT can be thought of as an operation on a finite Abelian group.
		A character on a group will also be a character on a subgroup of that group.
		Since we know that there is a one-to-one correlation between a group's characters and its elements, some of the characters of the group must be identical on the subgroup (as long as the subgroup has fewer elements than the group).
	www.fou.uib.no /fd/1996/h/413003/node124.html (196 words)

	Invariants of an Abelian Group (Site not responding. Last check: )
		The invariants of the abelian group G. Each infinite cyclic factor is represented by zero.
		The torsion invariants of the abelian group G. PrimaryInvariants(A) : GrpAb -> [ RngIntElt ]
		The primary invariants of the abelian group G. pPrimaryInvariants(A, p) : GrpAb, RngIntElt -> [ RngIntElt ]
	www.math.ufl.edu /help/magma/text214.html (77 words)

	Codes Closed under Arbitrary Abelian Group of PermutationsePrints@IISc - Open Access Archive of IISc Research ...
		, closed under arbitrary abelian group G of permutations with exponent relatively prime to q, called G-invariant codes, is investigated using a transform domain approach.
		In particular, this general approach unveils algebraic structure of quasicyclic codes, abelian codes, cyclic codes, and quasi-abelian codes with restriction on G to appropriate special cases.
		The number of G-invariant self-dual codes for any abelian group G is found.
	eprints.iisc.ernet.in /archive/00002310 (262 words)

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