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Topic: Rank of an abelian group

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Free abelian group

Divisible group

Rank

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	Kids.Net.Au - Encyclopedia > Rank of an abelian group
		An Abelian group is often thought of as composed of its torsion part T, and its torsion-free part A/T. The t.f.
		Furthermore, for every integer n ≥ 3, there is a rank 2n-2 torsion-free abelian group that is simultaneously a sum of two indecomposable groups, and a sum of n indecomposable groups.
		When one allows infinite rank, one is treated to a group G contained in a group K such that K is indecomposable and is generated by G and a single element, and yet every nonzero direct summand of G has yet another nonzero direct summand.
	www.kids.net.au /encyclopedia-wiki/ra/Rank_of_an_abelian_group (548 words)

	NationMaster - Encyclopedia: Abelian group
		In mathematics, an abelian group, also called a commutative group, is a group (G, ) with the additional property that "" commutes: for all a and b in G, a * b = b * a.
		The theory of abelian groups is generally easier than that of their non-abelian counterparts, although infinite abelian groups are the subject of current research.
		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)}.
	www.nationmaster.com /encyclopedia/Abelian_group (376 words)

	NationMaster - Encyclopedia: 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.
		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 (753 words)

	Rank of an abelian group - Definition, explanation
		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.calsky.com /lexikon/en/txt/r/ra/rank_of_an_abelian_group.php (714 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)

	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 countable subgroup is free (abelian).
		This is version 10 of Baer-Specker group, born on 2005-08-24, modified 2005-10-24.
	planetmath.org /encyclopedia/SpeckerGroup.html (160 words)

	Science Fair Projects - Rank of an abelian group
		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.all-science-fair-projects.com /science_fair_projects_encyclopedia/Rank_of_an_abelian_group (809 words)

	Abelian variety at AllExperts
		A morphism of abelian varieties is a morphism of the underlying algebraic varieties that preserves the identity element for the group structure.
		A morphism of polarized abelian varieties is a morphism A → B of abelian varieties such that the pullback of the Riemann form on B to A is equivalent to the given form on A.
		An abelian scheme, sometimes called an abelian variety, over a base scheme S of relative dimension g is a proper, smooth group scheme over S whose geometric fibers are connected and of dimension g.
	en.allexperts.com /e/a/ab/abelian_variety.htm (1478 words)

	Abelian group Summary
		An area is reached that is defined as its limit by reducing the size of the rectangles and increasing their numbers toward infinity (called the limit).
		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)

	What IS a Lie Group?
		The An and Cn Lie Algebras are exceptional in the Landsberg-Manivel Projective Geometric Classification because the adjoint representation is not a fundamental representation for the An and Cn Lie Algebras.
		Are we happy with G2 as the automorphism group of the octonions, F4 as the isometry of the [octonion] projective plane, E6 (in a noncompact form) as the collineations of the same, and E7 resp.
		A Lie algebra is a logarithm of a Lie group, and a Lie group is an exponential of a Lie algebra.
	www.valdostamuseum.org /hamsmith/Lie.html (3638 words)

	Lie group Summary
		The Lorentz group and the Poincare group of isometries of spacetime are Lie groups of dimensions 6 and 10 that are used in special relativity.
		The group U(1)×SU(2)×SU(3) is a Lie group of dimension 1+3+8=12 that is the gauge group of the standard model, whose dimension corresponds to the 1 photon + 3 vector bosons + 8 gluons of the standard model.
		The group of smooth maps from a manifold to a finite dimensional group is called a gauge group, and is used in quantum field theory and Donaldson theory.
	www.bookrags.com /Lie_group (4005 words)

	Free Abelian Groups (Site not responding. Last check: )
		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)

	Math Forum Discussions
		Robinson's "A Course in the Theory of Groups" 2nd edition, pp.
		the rank of a torsion-free group G to be the number of elements r(G) in a maximal independent subset.
		The Math Forum is a research and educational enterprise of the Drexel School of Education.
	www.mathforum.org /kb/thread.jspa?messageID=4850002&tstart=0 (356 words)

	Seminar "Group theory and topology"
		Finitely generated groups G become geometric objects when endowed with the word metric (depending on the generating set): the distance between a and b from G is the length of the shortest word representing a^{-1}b.
		The existence of a bounded-simple 2-generated group, containing a free non-cyclic subgroup, and the existence of an infinite simple bounded-generated 2-generated group are proven.
		group G and every homomorphism from F to a free group of rank 2 extends to G. Then F is a retract of G.
	math.vanderbilt.edu /~msapir/altopfall02.html (942 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)

	20: Group Theory and Generalizations
		Group theory can be considered the study of symmetry: the collection of symmetries of some object preserving some of its structure forms a group; in some sense all groups arise this way.
		Groups acting on topological spaces are the basis of equivariant topology and homotopy theory in Algebraic Topology.
		Nielsen's theorem: subgroups of free groups are free.
	www.math.niu.edu /~rusin/known-math/index/20-XX.html (2774 words)

	Group Element Representation
		If a is an m×n matrix, the columns of a represent n generators in the free abelian group of rank m.
		Therefore, finding a minimum representation of a group element with respect to a fixed set of generators is np complete.
		Since the moduli are relatively prime, the finite group is isomorphic to a cyclic group of order l, where l is the product of the various primes.
	www.mathreference.com /lan-cx-np,gel.html (871 words)

	Weyl Groups
		Note that the 48-element Double Binary Tetrahedral Group and the 96-element Double Binary Octahedral Group are not in 1-1 correspondence with the 72 root vectors of E6 and the 126 root vectors of E7.
		AN is based on the 2^(N+1)-dimensional graded extrerior (wedge) algebra of an (N+1)-dimensional vector space, each grade of degree k forming a ((N+1) k)-dimensional representation of SU(N).
		DN generates the Lie Group Spin(2N), which is the double-cover of the group of rotations in the 2N-dimensional vector grade-1 part of Cl(2N).
	valdostamuseum.org /hamsmith/Weyl.html (5287 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)

	Montreal Geometric Group Theory Seminar
		of equations with parameters in a free group.
		Since many questions about endomorphisms and automorphisms of groups, solving equations over groups, studying embeddings of a group into another group, etc. lead to actions of groups on the asymptotic cones, it is natural to consider actions of groups on tree-graded spaces.
		According to the definition, any acd group is a torsion-free abelian group of finite rank which has a completely decomposable subgroup of finite index.
	www.math.mcgill.ca /~wise/ggt/seminar0506.html (1390 words)

	Math Forum Discussions
		> the rank of a torsion-free group G to be the number of elements r(G) > in a maximal independent subset.
		is the one given by L. Fuchs (in "Infinite Abelian Groups" Vol.
		For a fixed prime p, the p rank, r_p, is the cardinality of a maximal
	www.mathforum.org /kb/thread.jspa?messageID=4850400&tstart=0 (480 words)

	[No title] (Site not responding. Last check: )
		Imagine an infinite graph whose incidence is described by a set of matrices with the positive entries.
		An expected significance may ideally result in a method allowing one to move back and forth between noncommutative algebras and "shapes" of geometric nature.
		In case our manifold is surface of genus g with m boundary components, we established an injective map between the "leaf space" of foliations and ordered abelian groups of rank 2g+m-1.
	www.math.ucalgary.ca /~nikolaev/statement.html (2616 words)

	Search Rank
		searching for something that is known to exist, such as an object that apoorly-organized person has stored, or that someone else has stored, or that someone has lost, or a missing person, such as a child that has run away or a victim of crime, an accident or a disaster
		Usually this is carried out by an officialof the same sex.
		Generally, rank is a system of hierarchy used to classify like things.
	www.altvetmed.com /face/16175-search-rank.html (514 words)

	Curves over the Rationals
		Given an elliptic curve E defined over Q and a prime number p, this function returns the local Tamagawa number of E at p, which is the index in E(Q_p) of the subgroup E^0(Q_p) of points with nonsingular reduction modulo p.
		The Mordell--Weil group of an elliptic curve E over the rationals is the finitely generated group of points of E with rational coordinates.
		Given an elliptic curve E defined over Q, this function returns an abelian group A isomorphic to the torsion subgroup of the Mordell--Weil group, and a map from this abstract group A to the elliptic curve providing the isomorphism.
	www.math.lsu.edu /magma/text1220.htm (5136 words)

	The Math Forum - Math Library - Group Theory
		Group theory can be considered the study of symmetry: the collection of symmetries of some object preserving some of its structure forms a group; in some sense all groups arise this way.
		Group theory takes an abstract approach, dealing with many mathematical systems at once and requiring only that a mathematical system obey a few simple rules, seeking then to find properties common to all systems that obey these few rules.
		A short article designed to provide an introduction to finite groups - their internal properties: all those results about group theory for which a consideration of the order of elements is a central part of the question.
	mathforum.org /library/topics/group_theory (2240 words)

	commalg.org - the center for commutative algebra
		Then the ideal of essential classes in the mod-p cohomology ring of G is a Cohen-Macaulay module whose Krull dimension is the p-rank of the centre of G. This basically answers in the affirmative a question posed by J. Carlson.
		As a byproduct, an explicit example of a vector bundle on a curve is given whose $n$-th iterated Frobenius pullback is not semistable, while its $(n-2)$-nd such pullback is semistable, where $n>1$ is arbitrary.
		Abstract: Let A be an integer (d x n) matrix, and assume that the convex hull conv(A) of its columns is a simplex of dimension d-1.
	www.commalg.org /preprints/2004_02.shtml (1837 words)

	Abelian groups (Site not responding. Last check: )
		The free abelian group on k generators is isomorphic to Z^k, the direct product of k copies of the integers.
		r is called the rank of the abelian group, and q_i the invariants.
		The rank and invariants are uniquely determined by the group.
	www.math.usf.edu /~eclark/algctlg/abelian_gp.html (122 words)

	[No title] (Site not responding. Last check: )
		Let A be the free abelian group of rank 2 with generators a,b.
		Form the split extension G of A by C with this action, so A is a normal subgroup of G with G/A isomorphic C. This is a finitely presented group with first Betti number 0 and second Betti number 1.
		An example of a finitely generated torsion-free virtually abelian group (all finitely generated virtually abelian groups are finitely presented) with first Betti number 0 and second Betti number 1 is given in Example 4.7 on p.
	www.lehigh.edu /~dmd1/pl213 (198 words)

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