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Topic: Cyclic group

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List of small groups

Multiplicative group of integers modulo n

Abelian group

Klein group

Primitive root modulo n

Examples of groups

Totient function

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	Cyclic group - Wikipedia
		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.
		All subgroups and factor groups of cyclic groups are cyclic.
		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.
	wikipedia.findthelinks.com /cy/Cyclic_group.html (700 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.
		All subgroups and factor groups of cyclic groups are cyclic.
		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.
	www.ebroadcast.com.au /lookup/encyclopedia/cy/Cyclic_group.html (691 words)

	PlanetMath: cyclic group
		A group is said to be cyclic if it is generated by a single element.
		Note that the isomorphisms mentioned in the previous paragraph imply that all cyclic groups of the same order are isomorphic to one another.
		This is version 18 of cyclic group, born on 2002-02-19, modified 2007-06-13.
	planetmath.org /encyclopedia/CyclicGroup.html (242 words)

	PlanetMath: cyclic ring
		See Also: cyclic group, proof of the converse of Lagrange's theorem for finite cyclic groups, criterion for cyclic rings to be principal ideal rings, the multiplicative identity of a cyclic ring must be a generator
		This is version 29 of cyclic ring, born on 2003-03-10, modified 2007-06-01.
		There are at least two articles by the name of "cyclic", and, in my article, cyclic rings, I would like the word "cyclic" to link with the article on cyclic groups; however, it links to the article on cyclic quadrilaterals instead.
	planetmath.org /encyclopedia/CyclicRing3.html (395 words)

	Springer Online Reference Works
		Cyclic group) are Abelian, in particular, the additive group of integers.
		A free Abelian group is a direct sum of infinite cyclic groups.
		Owing to their relative simplicity and to the fact that they have been very thoroughly studied (which is confirmed, for instance, by the solvability of the elementary theory of Abelian groups), and to the availability of a sufficient variety of objects, Abelian groups serve as a constant source of examples in various fields of mathematics.
	eom.springer.de /a/a010230.htm (1358 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)

	Monoids and Groups. Group Theory and Symmetries - Numericana
		The centralizer in a group G of a subset E consists of all the elements of G which commute with every element of E. It is a subgroup of G. The centralizer in G of G itself is the center of G (it's the intersection of all centralizers in G).
		The alternating group is the derived subgroup of the symmetric group: A
		The derived subgroup of the Quaternion group is {+1,-1}.
	home.att.net /~numericana/answer/groups.htm (5181 words)

	Group theory terms
		Not all subsets of a group are subgroups.
		A cyclic group is one that is generated by one element.
		Group theory is the study of symmetry in the abstract.
	groupexplorer.sourceforge.net /help/rf-groupterms.html (3947 words)

	Cyclic group - Wikipedia, the free encyclopedia
		Every cyclic group is isomorphic to the group { 0, 1, 2,..., n − 1 } under addition modulo n, or Z, the additive group of all of integers.
		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 (1327 words)

	Cyclic Groups and Subgroups
		Groups that can be generated in their entirety from one member are called cyclic groups.
		Thus for a cyclic group we have the definition that all the elements may be generated from a single element together with its inverse.
		All infinite cyclic groups are isomorphic to the additive group of integers.
	members.tripod.com /~dogschool/cyclic.html (2281 words)

	group - Information from Reference.com
		Groups are thus essential abstractions in branches of physics involving symmetry principles, such as relativity, quantum mechanics, and particle physics.
		A cyclic group is a group whose elements may be generated by successive composition of the operation defining the group being applied to a single element of that group.
		Quotient group: Given a group G and a normal subgroup N, the quotient group is the set of cosets of G/N together with the operation (gN)(hN)=ghN.
	www.reference.com /browse/group (3530 words)

	Group theory
		Möbius in 1827, although he was completely unaware of the group concept, began to classify geometries using the fact that a particular geometry studies properties invariant under a particular group.
		Hölder was to prove it in the context of abstract groups in 1889.
		At that time the only known groups were groups of permutations and even this was a radically new area, yet Cayley defines an abstract group and gives a table to display the group multiplication.
	www-groups.dcs.st-and.ac.uk /~history/HistTopics/Development_group_theory.html (1464 words)

	Good Math has moved to ScienceBlogs: Cyclic Groups
		Cyclic groups are interesting buggers: they express the symmetry of rotation in 2 dimensions, and the symmetric properties of addition in integers.
		A cyclic group of size N (often written Z_N) is a set of N values with a "circular" relationship.
		There's plenty more to say about it, but group theory isn't my specialty, and since the next couple of weeks are going to be very busy for me, both at work and at home, so I'm not going to have the time to write about something that requires so much background reading.
	goodmath.blogspot.com /2006/04/cyclic-groups.html (850 words)

	Infinite cyclic Wikipedia, Flickr, Delicious Bash at Bashr.com (Site not responding. Last check: 2007-11-06)
		Since the groups are abelian they are often written additively, and denoted by Zn; however, this notation is often avoided by number theorists because it conflicts or is easily confused with the usual notation for p-adic number rings or localisation at a prime ideal.
		Hence, cyclic groups are one of the simplest groups to study and a number of nice properties are known.
		Note that the group S1 of all rotations of a circle (the circle group) is not cyclic, since it is not even countable.
	www.bashr.com /en_bio_pics/Infinite_cyclic (1467 words)

	Math Forum - Ask Dr. Math
		A cyclic group is a group generated by a single element (for example a).
		If a cyclic group is finite, there is a smallest integer n such that a^n = e, and the group consists of the elements: = {e, a, a^2,..., a^(n-1)} These n elements are all different, and the size of the group is therefore n.
		As the group is Abelian, x^(mn) = (a^m)*b^(-n) = e, and k divides mn.
	mathforum.org /library/drmath/view/62340.html (1114 words)

	Group Library
		This group is the group of symmetries of a non-square rectangle.
		This group is the group of symmetries of a tetrahedron.
		This group is the group of symmetries of a cube, and also the group of symmetries of an octagon.
	www.platosheaven.com /groupexplorer_v1/groups.html (1656 words)

	Group (mathematics) - Wikipedia, the free encyclopedia
		Groups are thus essential abstractions in branches of physics involving symmetry principles, such as relativity, quantum mechanics, and particle physics.
		A cyclic group is a group all of whose elements may be generated by successive composition of the operation defining the group applied to a single element of that group.
		However, a group constructed from cyclic subgroups is itself not necessarily a cyclic group, e.g., a Klein group is not a cyclic group even though it is constructed from two copies of the cyclic group of order 2.
	en.wikipedia.org /wiki/Group_(mathematics) (2760 words)

	Mathematical Group Theory
		The group is equivalent to the addition of integers modulo 3; the sum of two numbers is replaced by the remainder when the sum is divided by 3.
		A subgroup of a group is a subset of the elements of the group that satisfy the conditions for a group with respect to the group operation.
		The number of distinct cosets of a subgroup H within a group G is denoted by the symbol [G:H].
	www.applet-magic.com /groups.htm (480 words)

	Abelian group Summary
		Such groups are generally easier to understand, although large infinite abelian groups remain a subject of current research.
		The multiplicative notation is the usual notation for groups, while the additive notation is the usual notation for modules.
		Subgroups, factor groups, and direct sums of abelian groups are again abelian.
	www.bookrags.com /Abelian_group (2196 words)

	[No title]
		Every group G has at least two subgroups: 1) The whole group G (since a group G is regarded as a subgroup of itself), and 2) The group consisting of the identity element alone (the identity element meets the axiomatic requirements for being a one-element group).
		The set of all transforms of a by elements of the group is the set of conjugates of a and is a conjugate set (or class) of elements of the group.
		The cosets of any normal subgroup H of a group G form a group under complex multiplication and this group is called the quotient group (or factor group) of G by H and is denoted by G/H. The normal subgroup H plays the role of the identity in the quotient group.
	pages.prodigy.net /jmiller.cb/c301.htm (1967 words)

	ABSTRACT ALGEBRA ON LINE: Groups
		A group G is said to be a finite group if the set G has a finite number of elements.
		Let G be a group, and let H be a subset of G. Then H is called a subgroup of G if H is itself a group, under the operation induced by G. Proposition.
		Any subgroup of the symmetric group Sym(S) on a set S is called a permutation group or group of permutations.
	www.math.niu.edu /~beachy/aaol/groups.html (1115 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.
		The Weyl group of SU(N) is the permutation group S_N with N! elements.
		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)

	Women's Surgery Group
		Because of this, the Writing Group for the WHI concluded that EPT should not be prescribed for the purpose of preventing cardiovascular disease, stroke, colon cancer or fractures.
		To avoid cyclic bleeding and to improve compliance, the continuous and combined EPT preparations were introduced with prolonged exposure to progestin.
		Several studies have demonstrated that the cyclic administration of progestin (two weeks every 2-3 months) may be as protective against endometrial cancer as the monthly administration.
	www.womenssurgerygroup.com /conditions/Menopause/overview.asp (3564 words)

	Homework Assignments
		It may be helpful to consult the Group Library page, and to make either a multiplication table or a Cayley diagram for each group by hand.
		Give an example of a group G and a normal subgroup H such that H is neither trivial nor all of G.
		Find a group in the Group Library that has need of an object of symmetry (or need of a better one).
	www.platosheaven.com /groupexplorer_v1/documentation/assignments.html (3060 words)

	3-D Crystals VII
		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).
		Said differently, the group of all symmetry transformations of the trigonal bipyramid is isomorph to the group of all symmetry transformations of the regular hexagon.
		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)

	Length Functions of a Finitely Generated Group
		It is clear that not every function satisfying (D1)-(D3) can be a length function of the cyclic group in a finitely presented group: the cardinality of the set of O-equivalence classes of functions satisfying (D1)-(D3) is the continuum, and the set of embeddings of the infinite cyclic group into finitely presented groups is countable.
		Although Theorem 9 shows that all ``reasonable" functions are length functions of a given finitely generated recursively presented group inside finitely presented groups, it does not give a characterization of these functions.
		By the proper choice of a universal group H it is not difficult to sharpen the formulation of Theorems 9 and 11.
	math.vanderbilt.edu /~msapir/Talk1/node5.html (924 words)

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