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Topic: Infinite cyclic group

	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 additive group Z 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 (1318 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.
		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).
		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 (641 words)

	[No title]
		Group theory is a powerful method for analyzing abstract and physical systems in which symmetry --the intrinsic property of an object to remain invariant under certain classes of transformations-- is present because the mathematical study of symmetry is systematized and formalized in group theory.
		A group that is not trivial is nontrivial.
		One example of a group of permutations that appears frequently in group theory is the symmetric group on n objects, i.e.
	www.math.harvard.edu /~knill/sofia/data/group.txt (4457 words)

	Cyclic group -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-08)
		Equivalently, an element a of a group G generates G precisely if the only ((mathematics) a subset (that is not empty) of a mathematical group) subgroup of G that contains a is G itself.
		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
		The (Framework consisting of an ornamental design made of strips of wood or metal) lattice of subgroups of Z is isomorphic to the dual of the lattice of natural numbers ordered by (The quality of being divisible; the capacity to be divided into parts or divided among a number of persons) divisibility.
	www.absoluteastronomy.com /encyclopedia/c/cy/cyclic_group.htm (1252 words)

	PlanetMath: free group
		A group with only one element is a free group of rank 0, freely generated by the empty set.
		Every group is a homomorphic image of some free group.
		This is version 32 of free group, born on 2002-02-25, modified 2004-10-04.
	planetmath.org /encyclopedia/Rank4.html (407 words)

	PlanetMath: cyclic group
		All cyclic groups of the same order are isomorphic to one other.
		proof that all subgroups of a cyclic group are cyclic
		This is version 11 of cyclic group, born on 2002-02-19, modified 2004-11-05.
	planetmath.org /encyclopedia/CyclicGroup.html (136 words)

	Cyclic group
		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.
		of all rotations of a circle is not cyclic.
		All cyclic groups are abelian, that is they are commutative.
	www.brainyencyclopedia.com /encyclopedia/c/cy/cyclic_group.html (836 words)

	[No title]
		generated by a single element, in the sense that the group has an element a (called a "generator" of the group) such that every element of the group is a power of a.
		is isomorphic to the group Z/nZ of integers
		field extension of a finite field is finite and cyclic; conversely, given a finite field F and a finite cyclic group G, there is a finite field extension of F whose Galois group is G.
	en-cyclopedia.com /wiki/Cyclic_group (794 words)

	Cyclical Ketogenic Diet -- Recommendations and Resources (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, when written multiplicatively, every element of the group is a power of ''a''.
		A cyclic nucleotide is any nucleotide in which the phosphate group is bonded to two of the sugar's hydroxyl groups, forming a cyclical or ring structure.
		In geometry, a cyclic quadrilateral is a quadrilateral whose vertices are all lie on a single circle.
	www.becomingapediatrician.com /health/38/cyclical-ketogenic-diet.html (815 words)

	Construction of an FP-Group
		The group G is defined by means of a presentation which consists of the relations for F (if any), together with the additional relations defined by the list R. The expression defining F may be either simply the name of a previously constructed group, or an expression defining an fp-group.
		Given a subgroup H of the group G, construct the quotient of G by the normal closure N of H. The quotient is formed by taking the presentation for G and including the generating words of H as additional relators.
		Groups that satisfy certain properties, such as being abelian or polycyclic, are known to possess presentations with respect to which the word problem is soluble.
	www.umich.edu /~gpcc/scs/magma/text295.htm (2326 words)

	Subgroups, Quotient Groups, Homomorphisms and Extensions
		The collection of words and groups specified by the list must all belong to the group G and H will be constructed as a subgroup of G. The generators of H consist of the words specified directly by terms of L[i] together with the stored generating words for any groups specified by terms of L[i].
		Construct the subgroup N of the polycyclic group G as the normal closure of the subgroup generated by the elements specified by the terms of the generator list L. The possible forms of a term L[i] of the generator list are the same as for the sub-constructor.
		Construct the quotient Q of the polycyclic group G by the normal subgroup N, where N is the smallest normal subgroup of G containing the elements specified by the terms of the generator list L. The possible forms of a term L[i] of the generator list are the same as for the sub-constructor.
	magma.maths.usyd.edu.au /magma/htmlhelp/text445.htm (1447 words)

	3-D Crystals VII
		And the group is infinite because repeated application of any element except the identity will never reach 0, which means that the period of every element except the identity is infinite.
		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.
		The group elements of the symmetry of a regular polygon are rotations and reflections and all possible combinations of them, for instance a rotation followed by a reflection in some reflection line (mirror line).
	home.hetnet.nl /~turing/d3_lattice_7.html (6225 words)

	Isomorphism
		The idea behind an isomorphism is to realize that two groups are structurally the same even though the names and notation for the elements are different.
		There is only one infinite cyclic group up to isomorphism, namely the integers under addition.
		To show that two groups are not isomorphic, we need to exhibit a structural property of one group not shared by the other.
	www.math.csusb.edu /notes/advanced/algebra/gp/node19.html (393 words)

	NTU Info Centre: Classifying space (Site not responding. Last check: 2007-10-08)
		In mathematics, a classifying space in homotopy theory of a discrete group G is, roughly speaking, a path connected topological space X such that the fundamental group of X is isomorphic to G and the higher homotopy groups of X are trivial.
		A more formal statement takes into account that G may be a topological group (not simply a discrete group), and that group actions of G are taken to be continuous; in the absence of continuous actions the classifying space concept can be dealt with, in homotopy terms, via the Eilenberg-MacLane space construction.
		Since group cohomology can (in many cases) be defined by the use of classifying spaces, they can also be seen as foundational in much homological algebra.
	www.nowtryus.com /article:Classifying_space (750 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.
		The group is the direct sum of these.
		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)

	MUG: group package (7.1.98)
		It seems it computes whether two elements a and b of a permutation group G of degree n are conjugate in G, but in fact, it only tells whether they are conjugate in the symmetric group S_n.
		Since the command "permrep" which computes the permutation representation of a group defined by generators and relations on some quotient by a subgroup names the generators, one is obliged to transform the data.
		For the command "galois", I had to make by hand a small procedure to transform the output into a (subgroup of a) permutation group, which is necessary if you want to use it.
	www.math.rwth-aachen.de /mapleAnswers/html/462.html (232 words)

	Untitled Document (Site not responding. Last check: 2007-10-08)
		Abstract: The quandle homology theory is generalized to the case when the coefficient groups admit the structure of Alexander quandles, by including an action of the infinite cyclic group in the boundary operator.
		The theory is parallel to that of group cohomology and group extensions.
		Third, cocycles with group actions on coefficient groups are used to define quandle cocycle invariants for both classical knots and knotted surfaces.
	www.math.usf.edu /~emohamed/articles.htm (613 words)

	INTEGER WEALTHY AND WISE FACT FINDER (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.boostmoney.com /integer (723 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		Every abelian group apart from the 2-torsion >>groups has an automorphism of order 2 and 2-torsion abelian >>groups of order >= 4 also have automorphisms of order 2.
		All subgroups of a free abelian group are free abelian, so a subgroup of ZxZ is isomorphic to 1, Z or ZxZ.
		Let a and b in G map onto the two generators of ZxZ, and let c = [b,a] = b^-1 a^-1 b a be the commutator.
	www.math.niu.edu /~rusin/known-math/01_incoming/autG_free (286 words)

	Math 441 Final Exam 1
		(b) an infinite Abelian group that is not cyclic.
		(d) a finite abelain group that is not cyclic.
		Let H and K be cyclic subgroups of an abelian group G, with H=10 and K=14.
	www.andrews.edu /~ohy/math441/Final1.htm (183 words)

	Passman's Abstracts
		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.
		Specifically, we show that certain basic results which hold when G is a polycyclic-by-finite group with no finite normal subgroups need not hold in the case of group algebras of finite groups.
		Here we begin the analysis in the case where the abelian group A is the additive group of a finite-dimensional vector space V over a locally finite field F of prime characteristic p, and the automorphism group G is a simple infinite absolutely irreducible subgroup of GL(V).
	math.wisc.edu /~passman/abstracts.html (3362 words)

	Divisor (Site not responding. Last check: 2007-10-08)
		The meet operation ^ is given by the greatest common divisor and the join operation v by the least common multiple.
		This lattice is isomorphic to the dual of the lattice of subgroups of the infinite cyclic group Z.
		If an integer n is written in base b, and d is an integer with b ≡ 1 (mod d), then n is divisible by d if and only if the sum of its digits is divisible by d.
	hallencyclopedia.com /Divisor (1252 words)

	[No title]
		At each step, you can use the HNN extension construction to embed G_k in a group G_{k+1}, in which any two elements of G_k are conjugate provided only that their orders are equal (it may be, however, that two elements of G_{k+1} with infinite orders are not conjugate in G_{k+1}).
		You probably mixed two results here: every group is embeddable into a 2-generated group and every torsion free group is embeddable into a group with 2 conjugacy classes.
		No infinite groups with two generators and two conjugacy classes are known.
	www.math.niu.edu /~rusin/known-math/95/finite.conj (2102 words)

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