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Topic: Divisible group

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Free abelian group

Rank of an abelian group

List of abstract algebra topics

	Springer Online Reference Works
		A quasi-cyclic group is the union of an ascending chain of cyclic groups
		Divisible group), and each divisible Abelian group is the direct sum of a set of groups that are isomorphic to the additive group of rational numbers and to quasi-cyclic groups for certain prime numbers
		A quasi-cyclic group coincides with its Frattini subgroup.
	eom.springer.de /Q/q076440.htm (294 words)

	PlanetMath: example of divisible group
		denote the group of rational numbers taking the operation to be addition.
		"example of divisible group" is owned by mathcam.
		This is version 1 of example of divisible group, born on 2003-07-23.
	planetmath.org /encyclopedia/ExampleOfDivisibleGroup.html (49 words)

	Applied Group Theory (Site not responding. Last check: 2007-10-18)
		Group theory is that branch of mathematics concerned with the study of groups.
		A subset H of a group (G,) which remains a group* when the operation * is restricted to H is called a subgroup of G. Given a set S of G. We denote by S the smallest subgroup of G containing S. Normal subgroup.
		It is the preimage of the identity in the codomain of a group homomorphism.
	applied-group.com.ru (508 words)

	PlanetMath: divisible group
		Every group is isomorphic to a subgroup of a divisible group.
		Any divisible abelian group is isomorphic to the direct sum of its torsion subgroup and
		This is version 4 of divisible group, born on 2003-07-23, modified 2003-07-24.
	planetmath.org /encyclopedia/DivisibleGroup.html (95 words)

	subgroups
		Since we are using the group operation of the "larger" group the associative property for all elements of the larger group is assumed.
		THEOREM: A finite subset of a group that is closed under the group operation is a subgroup of that group.
		The question we pose is "do the integers which are divisible by 5 form a subgroup of the additive group of integers?" We could check the three properties in the definition of a subgroup or we could use the theorem.
	members.tripod.com /~dogschool/subgroups.html (1418 words)

	Divisible group - Wikipedia, the free encyclopedia
		In group theory, a divisible group is an abelian group G such that for any positive integer n and any g in G, there exists y in G such that ny = g.
		Since a divisible group is an injective module, Tor(G) is a direct summand of G.
		As a quotient of a divisible group, G/Tor(G) is divisible.
	en.wikipedia.org /wiki/Divisible_group (285 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 (647 words)

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	www.mytravelgroup.com /AniteNextPage.asp?p=PLCLEGALDISCLAIMER (810 words)

	Emerging Research Fronts Comments by Gennian Ge
		The necessary conditions for the existence of a resolvable group divisible design with block size k have been proved to be sufficient for the case when k=3.
		Resolvable group divisible designs are a special class of group divisible designs whose blocks can be partitioned into parallel classes each of which partitions the point set of the design.
		Resolvable group divisible designs are useful in the constructions of both group divisible designs and other types of combinatorial designs.
	www.esi-topics.com /erf/2005/december05-GennianGe.html (434 words)

	Current Commutation Relations, Continuous Tensor Products and Infinitely Divisible Group Representations, by R. F. ... (Site not responding. Last check: 2007-10-18)
		In the trivial case when the group is R, then the generator of the group defines a random variable, and the cyclic vector defines a probability measure on the line; then the theory reduces to the concept of infinite-divisibility of this measure.
		The theory can be easily reformulated in terms of infinitely divisible positive-definite functions on groups, namely, the expectation of the unitary operators representing the group in the cyclic vector (known as the characteristic function of the cyclic representation), which is always a positive semi-definite continuous function on the group.
		The concept of infinite divisibility in probability was introduced by di Finetti, and the classification of these was achieved by Lévy.
	www.mth.kcl.ac.uk /~streater/infidivi.html (631 words)

	FMR lossary (Site not responding. Last check: 2007-10-18)
		A group G is tame in the strict sense if there is no bad field interpreted in it; in the narrow sense, if there is no bad field (K,T) such that K and T occur as definable sections, with the multiplication induced by conjugation in G. The narrow sense is sufficient.
		A torus is a definable abelian divisible group.
		A group is unipotent if it is definable, connected, and of bounded exponent, and is p-unipotent if in addition it is a p-group.
	www.rci.rutgers.edu /~cherlin/FMR/glossary.html (304 words)

	[No title]
		The group T acts on LX, and the fixed point manifold is ag* *ain X, considered as the subspace of constant loops.
		The quotient of a formal group by finite subgroup.
		Indeed [GKV, Gro94], one expects in general that there is an abelian group scheme G which is the mor* e fundamental object in T-equivariant E-theory, with ET() as structure sheaf.
	www.math.purdue.edu /research/atopology/Ando-Morava/amrrrfls.txt (2493 words)

	Lee Lady: Finite Rank Torsion Free Modules over Dedekind Domains (a book)
		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.
		Unlike the theory of torsion groups, the theory of finite rank torsion free modules is becoming something that fits in fairly well with the mainstream of commutative ring theory.
	www.math.hawaii.edu /~lee/book (629 words)

	Euler systems for higher K-theory of number fields, by Grzegorz Banaszak and Wojciech Gajda (Site not responding. Last check: 2007-10-18)
		Recall that for an abelian, p-torsion group A its group of divisible elements is by definition the intersection of the subgroups p^kA, for k>0.
		In the case of n even positive the group of divisible elements of the p-torsion part of K_{2n}(Q) should vanish, what is equivalent to the classical conjecture of Kummer and Vandiver on the class numbers of cyclotomic fields cf.
		The formula states that the number of divisible elements in the p-torsion part of the group K_{2n}(F), for n even positive and F an abelian, totally real number field equals the index of the group of cyclotomic elements of Deligne and Soule in the etale K-group K^{et}_{2n+1}(Z[{1/p}]).
	www.math.uiuc.edu /K-theory/0070 (294 words)

	HJM, Vol. 31, No. 1, 2005
		A reduced abelian group G is qd if it contains a finite rank free subgroup F such that G/F is a divisible torsion group.
		The Groups of Strong Symmetric Genus 4, pp.
		We establish that, for any locally compact subgroup H of a topological group G, the natural quotient mapping of G onto the quotient space G/H is locally perfect, that is, the restriction of it to the closure of some open set is an open mapping with compact fibers.
	www.math.uh.edu /~hjm/Vol31-1.html (1971 words)

	Niven Lectures (Site not responding. Last check: 2007-10-18)
		A monodromy group is the group of symmetries of a family of objects which are "locally isomorphic".
		In many situations one would like to show that the monodromy group is "as large as possible", subject to the obvious constraints.
		A leaf in M is the locus corresponding to a fixed isomorphism class of polarized p-divisible group with prescribed endomorphisms.
	www.math.ubc.ca /~reichst/niven.html (451 words)

	The Galois Group of a Polynomial
		If f/k is a field extension, its galois group is the group of automorphisms of f that fix k.
		If p(x) is an irreducible polynomial with coefficients in k, it also has a galois group, namely the galois group of its splitting field.
		Since the size of the orbit, times the size of the stabilizing subgroup, gives the size of g, the order of the galois group is divisible by n, where n is the number of roots in p(x).
	www.mathreference.com /fld-slv,gpx.html (666 words)

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		Located near many East Valley parks and sports venues, our group lodging in Mesa, Arizona is perfect for sports fans or sports teams coming to Mesa.
	www.dobsonranchinn.com /groups.htm (812 words)

	[No title]
		The torsion subgroup is the set of sequences of elements in these groups for which the order of the terms in the sequence stays bounded.
		The existence of a guaranteed splitting is equivalent to the vanishing of the cohomology group H^2(G/T, T), but inasmuch as it is not possible (as far as I know!) to "classify" all torsion-free groups it is unlikely one could hope to compute their cohomology.
		But G itself is not a divisible group, so there is no nonzero homomorphism from Q to G (there is no possible image for 1).
	www.math.niu.edu /~rusin/known-math/94/split-abel (1283 words)

	[SONATA] 10 Nearfields, planar nearrings and weakly divisible nearrings
		The (additive) group reduct of a finite nearfield is necessarily elementary abelian.
		All finite integral planar nearrings are weakly divisible.
		A finite Ferrero pair is a pair of groups (N,
	www-groups.dcs.st-and.ac.uk /gap/Manuals/pkg/sonata/htm/ref/CHAP010.htm (873 words)

	AMCA: o-Automorphisms of o-Groups of Finite Rank by Ramiro H. Lafuente-Rodriguez (Site not responding. Last check: 2007-10-18)
		A group G is divisible if for every g in G and every natural number n, there exists x in G such that x
		The classical unanswered question in the theory of totally ordered groups (o-groups) is whether or not every o-group can be embedded in a divisible o-group.
		W.C.Holland proved in 1961 the existence of an example of an o-group of archimedean rank 3 that is not embeddable in a divisible o-group of archimedean rank 3.
	at.yorku.ca /c/a/i/n/13.htm (218 words)

	[No title]
		Divisible Difference Sets with Multiplier minus one, (with D.Jungnickel, A.Pott), J. Algebra, 133(1),(1990), 35-62.
		Some constructions of group divisible designs with singer groups, (with A.Pott), Discrete math, 97, (1991), 39-45.
		On a construction of group divisible designs due to Sinha and Kageyama, (with A.Pott), J.Comb Information and System Sciences, 17 (1992), 148-152.
	www.math.wright.edu /people/KT_Arasu/HOME.HTM (944 words)

	Performance Advantages
		Groups are likely to outperform individuals when a task is divisible, or able to be broken down into specific subtasks and roles, but problems can arise when unitary due to ….
		Ex.) Groups without assigned roles often focus on shared information, not on the unique information each person brings to the table as in juries
		-They sense the group’s position and adjust their own attitudes even further in that direction to appear to be "good" group members
	www.umich.edu /~psychol/380/thompson/exam3rvw.html (646 words)

	Problem H - Divisible Group Sums (Site not responding. Last check: 2007-10-18)
		Given a list of N numbers you will be allowed to choose any M of them.
		You will have to determine how many of these chosen groups have a sum, which is divisible by D.
		Then for each query in the set print the query number followed by the number of desired groups.
	acm.uva.es /p/v106/10616.html (178 words)

	Weekly Calendar (Site not responding. Last check: 2007-10-18)
		Holland proved in 1961 the existence of an example of an o-group of archimedean rank 3 that is not embeddable in a divisible o-group of achimedean rank 3.
		We will discuss a more general case, when the o-group G is an o-extension of R by R, and prove that o-Aut(G) is divisible when the extension is not central.
		G can be embedded in an o-group of archimedean rank at most 2n where a has a square root.
	www-math.bgsu.edu /oldcalendars/2002-02-25.html (249 words)

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