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Topic: Dihedral group

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	Dihedral group - Wikipedia, the free encyclopedia
		The cycle graphs of dihedral groups consist of an n-element cycle and n 2-element cycles.
		is the identity, and we have a finite dihedral group.
		It can also be visualized as the automorphism group of the graph consisting of a path infinite to both sides, and is isomorphic to one of the (classes of) discrete symmetry groups in one dimension: that of repetitive patterns which also have mirror image symmetry.
	en.wikipedia.org /wiki/Dihedral_group (1178 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		The quaternion group is isomorphic to the group generated by a, b with presentation a^2 = b^2 = aba^(-1)b^(-1) and a^4 = 1.
		The dihedral group of order 2n is isomorphic to the group generated by a, b with presentation a^2 = b^n = 1 and a b a = b^(-1).
		This is the group with three generators a, b, and c and relations c = a^2 = b^n, ca = ac, bc = cb, ab = ba*c, and c^2 = 1.
	www.win.tue.nl /~amc/oz/om/cds/groupname1.html (236 words)

	Weyl group - Wikipedia, the free encyclopedia
		In mathematics, in particular the theory of Lie algebras, the Weyl group of a root system Φ is the subgroup of the isometry group of the root system generated by reflections through the hyperplanes orthogonal to the roots.
		The Weyl group is generated by reflections through the lines bisecting pairs of opposite sides of the hexagon; it is the dihedral group of order 6.
		The Weyl group of a semi-simple Lie group, a semi-simple Lie algebra, a semi-simple linear algebraic group, etc. is the Weyl group of the root system of that group or algebra.
	en.wikipedia.org /wiki/Weyl_group (555 words)

	Encyclopedia: Dihedral group
		Group theory is that branch of mathematics concerned with the study of groups.
		groups consisting of rotations (about the origin) and reflections (across axes through the origin) of the plane, the group operation being composition of these reflections and rotations.
		is not In mathematics, an abelian group is a commutative group, i.e.
	www.nationmaster.com /encyclopedia/Dihedral-group (2672 words)

	PlanetMath: dihedral group (Site not responding. Last check: 2007-11-07)
		The polynomials left invariant by all the group transformations form an algebra.
		Showing that these two invariants polynomially generate the full algebra of invariants is somewhat trickier, and is best done as an application of Chevalley's theorem regarding the invariants of a finite reflection group.
		This is version 9 of dihedral group, born on 2002-02-18, modified 2004-02-15.
	planetmath.org /encyclopedia/DihedralGroup.html (222 words)

	Dihedral (Site not responding. Last check: 2007-11-07)
		Dihedral is the upward angle of an aircraft's wings from root to tip, as viewed from directly in front of or behind the aircraft.
		The Dihedral group, D n, is the symmetry group of a regular n sided polygon.
		Dihedral is the angle between the wings axis and the lateral axis.
	bigletterlist.net /w/d/Dihedral.htm (506 words)

	Dihedral and General Linear Groups (Site not responding. Last check: 2007-11-07)
		For n = 1, the dihedral group is Z
		The group consists of vertical flips cross horizontal flips.
		The special linear group is the kernel of the homomorphism implemented by the determinant, namely the n×n matrices with determinant 1.
	www.mathreference.com /grp,dih.html (310 words)

	Group Library (Site not responding. Last check: 2007-11-07)
		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/groups.html (1656 words)

	Crystallographic Point Group (Site not responding. Last check: 2007-11-07)
		In crystallography, a crystallographic point group or crystal class is a set of symmetry operations that leave a point fixed, like rotations or reflections, which leave the crystal unchanged.
		The point group of a crystal, among other things, determines the symmetry of the crystal's optical properties.
		(for dihedral, or two-sided) indicates that the group has an n-fold rotation axis plus a two-fold axis perpendicular to that axis.
	www.wikiverse.org /crystallographic-point-group (294 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.
		The group of rigid motions of a regular n-gon is called the nth dihedral group, denoted by D
	www.math.niu.edu /~beachy/aaol/groups.html (1115 words)

	Learn more about List of small groups in the online encyclopedia. (Site not responding. Last check: 2007-11-07)
		For each order, all groups of that order up to group isomorphism are listed.
		: the symmetric group of degree n, containing the n permutations of n elements.
		The group theoretical computer algebra system GAP (available for free at http://www.gap-system.org/) contains the "Small Groups library": it provides access to descriptions of the groups of "small" order.
	www.onlineencyclopedia.org /l/li/list_of_small_groups.html (306 words)

	An interesting group: Dihedral group
		This is a group whose elements are symmetries of geometric objects.
		A symmetry (or an element of the group) is any rigid motion of a copy of the original n-gon, which move the n-gon in any fashion in 3-place and then placing the copied one back to the original one, so that it exactly covers it.
		The argument is sketched as follow: if 1 is sent to i, then 2, which is adjacent to 1, must be sent to i-1 or i+1.
	www.math.ucla.edu /~malmlui/groupex1.htm (175 words)

	Frieze Groups and Other Things (Site not responding. Last check: 2007-11-07)
		There are only two possible angles for reflections in a frieze group, parallel to the translation on the center line of the pattern or perpendicular to the translation.
		A permutation group is all the possible arrangments of a set of objects.
		Dihedral groups are all realizable in the plane.
	jwilson.coe.uga.edu /EMT668/EMAT6680.F99/McCallum/WALLPA~1/FRIEZE~1.HTM (555 words)

	AN INTERESTING DIHEDRAL GROUP, ITS AUTOMORPHISMS, AND TRANSFORMATION (Site not responding. Last check: 2007-11-07)
		Specifically, a certain group of operations (mathematically a "dihedral group") acting on an octatonic pitch-class collection describes significant events in the work.
		Moreover, the operations of the octatonic dihedral group are used to interpret pitch-class sets in the manner of Klumpenhouwer Networks.
		By invoking the automorphisms of the dihedral group, a network of "K-networks" demonstrates recursive structure at a deeper level, illustrating how the structure of the cadential sonorities is manifest in the structure of phrases progressing toward those sonorities themselves.
	www.societymusictheory.org /html/events/abstracts/smt-97.abstracts/gollin.html (140 words)

	Read This: Adventures in Group Theory
		Adventures in Group Theory is a tour through the algebra of several "permutation puzzles." Although the main focus is on the Rubik's Cube, several other puzzles are explored to a lesser degree.
		The quaternions, finite cyclic groups, dihedral groups, and symmetric groups are presented.
		For example, the structure of this group, the center of the Rubik's Cube Group, the structure of some of the subgroups of the Rubik's Cube Group, including an embedding of the quaternions into the group, and an example of two elements which generate the whole Rubik's Cube group.
	www.maa.org /reviews/joynergroups.html (960 words)

	MODELING PERMUTATION GROUP
		Chicago, IL 60625 1-312-588-2370 Objectives: Throughout this unit the students are: to construct equilateral triangles on cardboard; to develop the transformation concept with a one to one correspondence; to generate the rotation and reflection table; and to deduce from the model the abstract axioms of a group.
		verify the model with its laws as a group C General Plan With the large equilateral triangle on the overhead demonstrate the concepts of rotation and reflection.
		Again affirm that the patterns created by the transformations of the equilateral triangle form a set of elements which is well structured set of axioms with the the binary operations of the transformation necessary for a group D. Overhead Transparencies 1.
	www.iit.edu /%7Esmile/ma8604.html (529 words)

	Creation of Finite Soluble Groups
		The simplest method of producing a pc-presentation for a group is to use one of the built-in construction functions.
		Construct the abelian group defined by the sequence Q = [n_1,..., n_r] of positive integers as a pc-group.
		A map f from the free group of rank n to G is returned as well.
	magma.maths.usyd.edu.au /magma/htmlhelp/text313.htm (1566 words)

	Evolution of a Computer Application
		Here, for example, are the two non-abelian groups of order 8. One of them actually is the dihedral group, and the other is the group of quaternionic units.
		It should be easy to spot group 13 as the dihedral group - it has lots of reflections (elements of order 2) and only two rotations (elements of order 4). However both groups are non-abelian and have elements of order 2 and n/2.
		Now, if one frequently needs to determine whether a group is dihedral, there is a rather interesting further possibility: Generate results by a method that might be slow -- and then use the stored results rather than re-calculating.
	www.joma.org /images/upload_library/4/vol3/evolution/wavrik6e.html (1199 words)

	Maybe this Explains the Economic Cycle... best Dihedral Group D6 (Site not responding. Last check: 2007-11-07)
		The symmetric group S3 is also the dihedral group D6, and so is presented by two involutions with...
		The symmetry group of a regular hexagon is a group of order 12, the Dihedral group D6.
		Two views of a 12-crossing knot whose symmetry group is the dihedral group D6 of order 12...
	ascot.pl /th/Fourier3/Dihedral-Group-D6.htm (643 words)

	Directions for Use -- The Abstract Algebra Helper
		Generator button requires the group to be selected (the radio button is filled), the "n" value for the group to be entered, and an element of the group to be entered.
		Inverse button requires the group to be selected, the "n" value for the group to be entered, and an element of the group to be entered.
		Subgroup button requires the group to be selected, and the "n" value for the group to be entered.
	www.stolaf.edu /people/dietz/AlgHelper/directions.html (598 words)

	Mattresses, Contra Dancing and Quilts
		This group is useful for describing the flipping of a square bed, so we don't want to rule it out.
		To see that they are the same group, imagine that the contra dancers have to dance holding a mattress by the corners.
		Both groups, by the way, are called dihedral groups, because they are the symmetry groups of two-sided polygons, as a pentagon cut out of paper.
	jimvb.home.mindspring.com /Matcontra.htm (2042 words)

	Carboxylate Binding Modes in Zinc Proteins: A Theoretical Study -- Ryde 77 (5): 2777 -- Biophysical Journal
		molecule (the unligated atom) or a tyrosine hydroxyl group (the
		group is protonated and therefore is not negatively charged.
		molecules, an aspartate carboxylate group, and a hydroxide ion.
	www.biophysj.org /cgi/content/full/77/5/2777 (5473 words)

	Standard Groups and Extensions
		The effect of these functions is to construct the group on some standard set of generators.
		The group category of the result may be specified as an argument to the function.
		We define G to be the symmetric group of degree 4 and H to be the dihedral group of order 8.
	www.math.lsu.edu /magma/text255.htm (742 words)

	PlanetMath: generalized dihedral group (Site not responding. Last check: 2007-11-07)
		Cross-references: free product, isomorphic, finite, dihedral group, cyclic, inverses, maps, order, cyclic group, semidirect product, abelian group
		This is version 1 of generalized dihedral group, born on 2004-12-13.
		(Group theory and generalizations :: Structure and classification of infinite or finite groups :: Extensions, wreath products, and other compositions)
	www.planetmath.org /encyclopedia/GeneralizedDihedralGroup.html (79 words)

	Dihedral Group Encyclopedia Article, Definition, History, Biography (Site not responding. Last check: 2007-11-07)
		Looking For dihedral group - Find dihedral group and more at Lycos Search.
		Find dihedral group - Your relevant result is a click away!
		Look for dihedral group - Find dihedral group at one of the best sites the Internet has to offer!
	www.folkartmuseum.com /encyclopedia/Dihedral_group (1323 words)

	groups
		Examples of Finite Groups: A = {1, -1, i, -i} where * is multiplication, B = {0, 1, 2, 3) where * is addition modulo 4.
		A subgroup is a group entirely inside another: {1, -1} is a subgroup of A, {0, 2) is is a subgroup of B.
		The makers of GAP have written an analysis of Rubik's Cube from a Group Theory perspective.
	www.mathpuzzle.com /groups.html (581 words)

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