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Topic: Dihedral group of order 6

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	Dihedral group - Wikipedia, the free encyclopedia
		In mathematics, the dihedral group of order 2n is the abstract group of which one representation is the symmetry group in 2D of a regular polygon with n sides.
		is the identity, and we have a finite dihedral group of order 2n.
		As subgroups of the isometry group of Z they are different: the reflections in one subgroup all have a fixed point, the mirrors are at the integers, while none in the other subgroup has, the mirrors are in between (the translations of both are the same: by even numbers).
	en.wikipedia.org /wiki/Dihedral_group (1864 words)

	Examples of groups - Wikipedia, the free encyclopedia
		This is an Abelian group and our first (nondiscrete) example of a Lie group: a group whose underlying set is a manifold.
		Groups are very important to describe the symmetry of objects, be they geometrical (like a tetrahedron) or algebraic (like a set of equations).
		Free groups are important in algebraic topology; the free group in two generators is also used for a proof of the Banach–Tarski paradox.
	en.wikipedia.org /wiki/Examples_of_groups (1636 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)

	[No title]
		The mapping class group, g, is defined to be the group of path components o* f the group* of orientation preserving homeomorphisms of the oriented closed surface S* *g of genus g.
		Unlike the case of finite groups,* * the p-period of a p-periodic infinite group, p a fixed prime, may be arbitrarily large.
		Groups acting on spherical spaces We call a (not necessarily compact) manifold X a spherical space, if X is h* *omotopy equivalent to a sphere.
	www.math.purdue.edu /research/atopology/Glover-Mislin-Xia/cohmcg.txt (2116 words)

	Groups of small order.
		Order 1 and all prime orders (1 group: 1 abelian, 0 nonabelian)
		All groups of prime order p are isomorphic to C_p, the cyclic group of order p.
		There are the three of order 2 generated by (1 2), (1 3) and (2 3), and the one of order 3 generated by (1 2 3).
	www.math.usf.edu /~eclark/algctlg/small_groups.html (1543 words)

	[No title]
		are both dihedral of order 6, and G is isomorphic to the free produc* t of H and K amalgamating L = H \ K. We study K0(kG), the Grothendieck group* of isomorphism classes of finitely generated projective kG-modules, and in particu* *lar the dependence of K0(kG) on the choice of field k.
		The proof Throughout this section, G stands for the group presented in equation 1.1, a* *nd H, K are the subgroups generated by {a; c} and {b; c} respectively.
		The groups H and K are both isomorphic to the dihedral group of order 6, so have 3 isomorphism classes of irreducible representations over Q. Let ff * *(resp.
	www.math.purdue.edu /research/atopology/Leary/ijltork.txt (1170 words)

	[No title]
		Library module found as /share/procedures/lamp 0 6 1 7 1 10 1 12 1 14 1 16 4 18 1 20 1 22 1 26 1 30 2 32 1 34 1 42 1 60 1 120 1 Library module found as /usr/local/cayley/caylibs/gps100/g84n7 Group of order 84: number 7.
		Library module found as /share/procedures/lamp 0 6 1 10 1 12 2 14 1 16 1 18 1 20 3 22 1 24 1 26 1 27 1 28 1 30 2 40 1 60 1 72 1 Library module found as /usr/local/cayley/caylibs/gps100/g96n87 Group of order 96: number 87.
		Library module found as /share/procedures/lamp 0 6 1 7 1 9 2 10 1 11 1 13 2 14 1 15 1 16 1 18 2 20 2 22 1 23 1 29 1 30 1 40 1 Library module found as /usr/local/cayley/caylibs/gps100/g96n101 Group of order 96: number 101.
	www.rose-hulman.edu /Users/reu/reu92/reu92jf/adalin-3.log (21521 words)

	Abstract
		Figure 11, we see that Maschke associated the dihedral groups with the rotations of a dihedron (a solid determined by a regular n-gon on the equator of a sphere, with an additional vertex at each pole).
		Figure 2, for the dihedral group determined by the triangle in this case, is easily explained in terms of a cyclic element of order n and any reflection.
		The truncated cube (Figure 6) is generated by “a rotation of period 3 about the diameter passing through the middle points of the two faces a, and a rotation of period 2 about the diameter bisecting the edges 4-5 and 3-6.” These become permutations a = (1,5,4)(2,6,3) and b = (1,2)(3,6)(4,5).
	www.lcsc.edu /csteenbe/abstract.htm (3076 words)

	[No title]
		Library module found as /usr/local/cayley/caylibs/gps100/g8n1 Group of order 8: number 1.
		Library module found as /usr/local/cayley/caylibs/gps100/g12n1 Group of order 12: number 1.
		Library module found as /usr/local/cayley/caylibs/gps100/g12n2 Group of order 12: number 2.
	www.rose-hulman.edu /Users/reu/reu92/reu92ds/head4 (9793 words)

	Symmetry Detection Software (Site not responding. Last check: 2007-11-01)
		Running time depends primarily on the group order, not the number of vertices.
		If it hasn't crossed your mind yet, there is straightforward way to compute the symmetry group by choosing n independant vectors (n is the dimension) and running through all transformations defined by taking those n vectors to n other vectors, then checking if this transformation is orthogonal and fixes the set of points.
		The group is the dihedral group of order 6.
	www.gang.umass.edu /library/symmetry (441 words)

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