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Topic: Infinite cyclic

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	Cyclic group - Wikipedia, the free encyclopedia
		In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a "generator" of the group) such that, when written multiplicatively, every element of the group is a power of a (or na when the notation is additive).
		The cycle graphs of finite cyclic groups are all n-sided polygons with the elements at the vertices.
		Similarly, the endomorphism ring of the infinite cyclic group is isomorphic to the ring Z, and its automorphism group is isomorphic to the group of units of the ring Z, i.e.
	en.wikipedia.org /wiki/Cyclic_group (1287 words)

	Cyclic group (Site not responding. Last check: 2007-10-08)
		In mathematics, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element ''a (called a "generator" of the group) such that all elements of the group are powers of a.
		Cyclic Vomiting Syndrome CVS is a childhood disorder characterized by bouts of vomiting that last from a few hours to several days.
		Cyclic Vomiting Syndrome CVS is an uncommon, unexplained disorder of children and some adults that is characterized by recurrent, prolonged episodes of severe nausea, vomiting and prostration with no apparent cause.
	www.serebella.com /encyclopedia/article-Cyclic_group.html (1197 words)

	PlanetMath: cyclic ring
		A ring is a http://planetmath.org/encyclopedia/SpecialLinearGroup.html cyclic ring f its additive group is cyclic.
		Thus, any infinite cyclic ring that has zero divisors is a zero ring.
		This is version 15 of cyclic ring, born on 2003-03-10, modified 2003-04-02.
	planetmath.org /encyclopedia/CyclicRing3.html (331 words)

	PlanetMath: characteristic of a cyclic ring
		The characteristic of an infinite cyclic ring is 0.
		A finite ring is cyclic if and only if its order and characteristic are equal.
		This is version 4 of characteristic of a cyclic ring, born on 2003-03-11, modified 2004-04-30.
	www.planetmath.org /encyclopedia/Characteristic2.html (154 words)

	Cyclic group -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-08)
		The cyclic groups are the simplest groups and they are completely classified: for any positive (Any of the natural numbers (positive or negative) or zero) integer n, there is a cyclic group C
		Every cyclic group is (Click link for more info and facts about isomorphic) isomorphic to one of these.
		The finite cyclic groups can be introduced as a series of (Click link for more info and facts about symmetry group) symmetry groups, or as the groups of rotations of a regular (Click link for more info and facts about n-gon) n-gon: for example C
	www.absoluteastronomy.com /encyclopedia/c/cy/cyclic_group.htm (1252 words)

	2 (Site not responding. Last check: 2007-10-08)
		The speed of the primary cyclic pattern is absolute and constant, while the speeds of the secondary cyclic wave patterns are relative and variable.
		Subatomic particles, atoms, molecules, and all the entities in the universe are transitory states of simple or complex wave patterns of motion generated by the virtually infinite interweavings and intertransformations of the formal gestalt wave patterns of the universe.
		As the photon is the phantom or the transitory state of the primordial cyclic pattern of thought-wave, so are all of the subatomic or atomic particles phantoms or transitory states of various compound cyclic thought-wave patterns.
	www.philosophy.org /webarticles/text/third18.htm (587 words)

	Integer - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-08)
		As a group under addition, Z is a cyclic group, since every nonzero integer can be written as a finite sum 1 + 1 +...
		In fact, Z under addition is the only infinite cyclic group, in the sense that any infinite cyclic group is isomorphic to Z.
		Integer datatypes are typically implemented using a fixed number of bits, and even variable-length representations eventually run out of storage space when trying to represent especially large numbers.
	www.bucyrus.us /project/wikipedia/index.php/Integer (854 words)

	[No title]
		cyclic group +------------------------------------------------------------ If G is a group and if x is an element of G, the cyclic group langle x rangle generated by x is the set of all powers of x: langle x rangle = ldots, x^-2, x^-1, 1, x, x^2, ldots.
		Note that if langle x rangle is infinite cyclic then langle x rangle is isomorphic to Z^+, the integers under addition.
		If langle x rangle is a cyclic group of order n, then langle x rangle is isomorphic to Z/n, where Z/n is the group satisfying the following properties (we use additive notation as opposed to multiplicative notation for the law of composition of Z/n): 1.
	www.math.harvard.edu /~knill/sofia/data/group.txt (4457 words)

	Search Results for infinite (Site not responding. Last check: 2007-10-08)
		Hill was the first to use infinite determinants to study the orbit of the Moon in On the part of the motion of the lunar perigee which is a function of the mean motion of the sun and moon.
		Similarly, the infinite cyclic group generated by a spiral similarity is illustrated by the Nautilus shell and by the arrangement of florets in a sunflower.
		A Tomasini Bassols, Aporias, antinomies and the infinite : Russell's critique of Zeno and Kant, Mathesis.
	www-history.mcs.st-and.ac.uk /Search/historysearch.cgi?SUGGESTION=infinite&CONTEXT=1 (11508 words)

	Search results
		The infinite Whitehead group of a forward tame CW complex is described algebraically as a relative Whitehead group.
		The infinite torsion of a proper homotopy equivalence is related to the locally finite finiteness obstruction at infinity.
		Chapter 13, Infinite cyclic covers, proves that a connected infinite cyclic cover $\overline W$ of a connected compact ANRW has two ends $\overline W\sp +$, $\overline W\sp -$, and establishes a duality between the two types of tameness: $\overline W\sp +$ is forward tame if and only if $\overline W\sp -$ is reverse tame.
	www.maths.ed.ac.uk /~aar/books/munkholm.htm (1788 words)

	3-D Crystals VII
		When we have some infinite pattern, and when we shift this pattern a certain distance along some direction, and when, after having done so, the pattern is precisely mapped onto itself, the pattern is said to possess translational symmetry.
		Infinite groups can also be generated by elements each of which has a finite period, for instance the reflections in two parallel mirrors, and indeed a reflection has period 2.
		Because the subgroup H is cyclic, multiplication with one of its elements does not lead us outside this subgroup, so all cosets of this subgroup with respect to elements of this subgroup are identical to the subgroup itself.
	home.hetnet.nl /~turing/d3_lattice_7.html (6225 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		-1 generates a 2 element cyclic group, 2 generates an infinite cyclic group and the whole group is their direct product.
		Yes, the identity and reflection through x axis form one cyclic factor and the identity together with y reflection forms another.
		The identity and (12)(34) form one cyclic factor, the identity and (13)(24) form another, in a decomposition as required.
	www.maths.warwick.ac.uk /~moody/sheet82004solutions (110 words)

	Extended Glossary (Site not responding. Last check: 2007-10-08)
		As abstract groups, the infinite cyclic subgroups within the modular group are all isomorphic.
		However, for the infinite cyclic groups generated by elements with trace values equal to two, there is just a single conjugacy equivalence class, while for each trace value greater than two there are more than one conjugacy equivalence class.
		The three possible types of actions are referred to as elliptic/rotations, as for the first two family members of {f_n=-1/z+n}, parabolic/infinite cyclic, as for the third member of the family, and hyperbolic/pushing forward, as for all other family members.
	www.xula.edu /math/faculty/McCreary/modularGarden/extendedGlossary.htm (1586 words)

	"The Endless Universe: Introduction to the Cyclic Universe" by Paul J. Steinhardt, Ph.D.
		Cyclic Universe: The Big Bang was not the beginning of time because the universe undergoes endless cycles of evolution in infinite space and time.
		The cyclic model builds on lessons learned from the ekpyrotic example to produce a picture with remarkable predictive and explanatory power.
		The Cyclic Universe builds upon this model, which is based on the idea that our hot big bang universe was created from the collision of two three-dimensional worlds moving along a hidden, extra dimension.
	www.actionbioscience.org /newfrontiers/steinhardt.html (1928 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-10-08)
		Date: 03/11/2003 at 08:51:24 From: Doctor Jacques Subject: Re: Cyclic Groups Hi Karen, I'm afraid you cannot say "n is the order of the infinite group" : if the group is infinite, its order is infinite by definition, and you cannot use it as a number since infinity is not a number.
		If any of these is the infinite cyclic group, it is isomorphic to Z under addition, and Z contains an infinite number of subgroups (all nZ for integer n).
		This means that all the cyclic subgroups of G are finite.
	mathforum.org /library/drmath/view/62388.html (326 words)

	Motifs (Site not responding. Last check: 2007-10-08)
		The circular patterns are generated by dihedral and cyclic symmetries and have an infinite wedge as their motif.
		The linear patterns are known as the freize groups and have as motif either an infinite strip, or half of an infinite strip.
		The infinite cyclic and dihedral symmetries are examples, as is the symmetry generated by a single glide reflection.
	comp.uark.edu /~strauss/symmetry.unit/sym.1.4.5.html (250 words)

	Talk:Cyclic group - InformationBlast
		for the finite cyclic groups (where of course J is used for the infinite cyclic group).
		This is something a lot of algebra textbooks do with respect to the cyclic groups, and it drives me nuts.
		In Examples of cyclic groups: The group of rotations in a circle, S1, is not a cyclic group.
	www.informationblast.com /Talk:Cyclic_group.html (997 words)

	Cyclic subgroups, Cauchy's Theorem.
		CS1 : All elements of a finite group will be generating, or belong to, a cyclic subgroup.
		Elements in infinite groups may have finite orders, but if there is no positive integer n such that a
		The existence of such cyclic subgroup, for some orders, is true as well.
	hemsidor.torget.se /users/m/mauritz/math/alg/cyclsub.htm (320 words)

	MATH 371: Abstract Algebra (Site not responding. Last check: 2007-10-08)
		In the case of infinite cyclic groups, how many distinct elements are in a given group?
		What is the relationship between the generator of a subgroup of a cyclic group and the generator of the original cyclic group?
		Given a generator of a cyclic group, g, the order of this cyclic group, n, and k, where k divides n, how do you form the subgroup of order k?
	www.math.umd.edu /~cnglover/mhabsalg.htm (856 words)

	Passman's Abstracts (Site not responding. Last check: 2007-10-08)
		In the case of infinite groups, this problem has been studied with reasonable success during the past 45 years, and our goal here is to survey what is known.
		In particular, if G is generated by the infinite support of L, then we prove that C is homogeneous.
		Then we show that there are two cyclic subgroups X and Y of G of prime power order, and two special units u in KX and v in KY, where KX is the K-linear span of X in R, such that the group generated by u and v contains a nonabelian free group.
	math.wisc.edu /~passman/abstracts.html (3362 words)

	Topfest 99 -- Abstracts
		More generally, in a closed, orientable, irreducible 3-manifold with cyclic fundamental group, knots which are round, in the sense that their exteriors are solid tori, can be characterized among all knots with irreducible exterior in terms of their essential surfaces or their boundary slopes.
		Indeed, it seems possible that in a manifold with cyclic fundamental group, round knots can be characterized by some property, stated in terms of essential surfaces or boundary slopes, which is "ubiquitous" in the sense that every closed, irreducible, orientable 3-manifold contains a knot having the property in question.
		A further consequence will be that there are infinitely many almost free actions by SO(3) on the 7-sphere which preserve the Hopf fibration over the 4-sphere and which do not extend to the 8-disc.
	www.math.cornell.edu /~festival/1999/99abstracts.html (1415 words)

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