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Topic: Coset

	Coset - Wikipedia, the free encyclopedia
		A coset representative is a representative in the equivalence class sense.
		All left cosets and all right cosets have the same number of elements (or cardinality in the case of an infinite H).
		Furthermore, the number of left cosets is equal to the number of right cosets and is known as the index of H in G, written as [G : H].
	en.wikipedia.org /wiki/Coset (929 words)

	Coset code generator for computer memory protection - Patent 4569052
		A coset is a set of elements in a group formed by applying the group operation between a fixed element of the group and each of the elements of a subgroup.
		For cosets of weight two, since no two cosets have a word in common and all their members have even weight, the minimum distance between these codes is at least two.
		The resulting coset is of odd weight, and its syndrome cannot be the same as the syndrome of any weight one coset, whose syndromes are precisely the columns of the correction (parity-check) matrix.
	www.freepatentsonline.com /4569052.html (6353 words)

	Coset Spaces and Tables
		Given a coset table T for a subgroup H of G, construct the permutation group image of G given by its action on the cosets of H, using the columns of T. This is the second return value of CosetTableToRepresentation(G, T).
		The indexed coset space for G corresponding to the permutation representation f of G, where f is a homomorphism of G onto a transitive permutation group.
		Right coset of the subgroup H of the group G, where g is an element of G (as an element of the right coset of H).
	www.math.ufl.edu /help/magma/text231.html (3245 words)

	PlanetMath: double coset
		In contrast to the situation with ordinary cosets, the
		-double cosets need not all be of the same cardinality.
		This is version 5 of double coset, born on 2006-10-01, modified 2006-10-06.
	planetmath.org /encyclopedia/DoubleCoset.html (101 words)

	Coset Spaces and Tables
		Given a coset table T for a subgroup H of G, construct the permutation group image of G given by its action on the cosets of H, using the columns of T. This is the second return value of CosetTableToRepresentation
		When printing the coset table, the action of the generators and of the non-trivial inverses of generators on the enumerated transversal is shown in table form.
		Using the coset table, we now construct the permutation representation of G on the cosets of S in G. We assign the representation (a homomorphism), the image (a permutation group of degree [G:S] = 10) and the kernel of the permutation representation (a subgroup of G).
	www.math.wayne.edu /answers/magma2.10/htmlhelp/text408.htm (2805 words)

	Coset enumeration - Wikipedia, the free encyclopedia
		In mathematics, coset enumeration is the problem of counting the cosets of a subgroup H of a group G given in terms of a presentation.
		As a by-product, one obtains a permutation representation for G on the cosets of H.
		Coset enumeration is usually considered to be one of the fundamental problems in computational group theory.
	en.wikipedia.org /wiki/Coset_enumeration (295 words)

	Common Systems of Coset Representatives - Ashay Dharwadker
		Thus G is the disjoint union of the left cosets of H.
		A set consisting of exactly one representative of each left coset from the set of all left cosets of H in G is called a system of representatives for the left cosets of H in G.
		A set that is simultaneously a system of representatives for the left cosets of H in G and a system of representatives for the right cosets of H in G is called a common system of representatives for the left and right cosets of H in G.
	www.geocities.com /dharwadker/coset.html (2321 words)

	Subgroups and Cosets
		All the elements in a coset are distinct.
		If z is in the coset of y is in the coset of x, z is in the coset of x (transitivity).
		Each member of the right coset is a member of the left coset, and h is normal.
	www.mathreference.com /grp,sub.html (791 words)

	PlanetMath: coset
		Accordingly, the collection of left cosets (or right cosets) partitions the group
		This is version 5 of coset, born on 2002-01-05, modified 2002-11-04.
		Object id is 1306, canonical name is Coset.
	planetmath.org /encyclopedia/Coset.html (65 words)

	Cosets
		Cosets of this group are {Y, b} and {Z, a}.
		Subgroups whose right cosets are also left cosets are very important in group theory.
		The operation of "multiplication on the right by a group element" which transforms subsets into subsets (and subgroups into right cosets) will also transforms any right coset of a subgroup H into a right coset of H (This could be a different right coset than we started with or the same one).
	members.tripod.com /~dogschool/cosets.html (1008 words)

	[ITC] 4 The Tables
		Remember that coset numbers denote cosets defined by representatives given by words in the generators, but until a coset enumeration is finished, different coset numbers may represent the same coset (by different words).
		An entry j in row i and column k means that the coset with number i multiplied by the generator or inverse generator heading column k yields the coset with number j.
		The coset representative displayed is the one that is obtained by retracing in the List (not the Table) of Definitions from this coset number to the coset number 1.
	www-groups.dcs.st-and.ac.uk /~gap/Manuals/pkg/itc/htm/CHAP004.htm (2901 words)

	GAP Manual: 61 The Double Coset Enumerator
		Double Coset Enumeration (DCE) can be seen either as a space- (and time-) saving variant of ordinary Coset Enumeration (the Todd-Coxeter procedure), as a way of constructing finite quotients of HNN-extensions of known groups or as a way of constructing groups given by symmetric presentations in a sense defined by Robert Curtis.
		A double coset enumeration works with a finitely-presented group G, a finitely generated subgroup H (given by generators) and a finite subgroup K, given explicitly, usually as a permutation group.
		Coset Enumeration can be considered as a means of constructing a permutation representation of a finitely-presented group.
	www.math.jussieu.fr /~jmichel/htm/CHAP061.htm (4100 words)

	Coset - ExampleProblems.com
		Any two left cosets are either identical or disjoint.
		The left cosets form a partition of G: every element of G belongs to one and only one left coset.
		Furthermore, the number of left cosets is equal to the number of right cosets and is known as the index of H in G, written as [G : H].
	www.exampleproblems.com /wiki/index.php/Coset (366 words)

	Factor (Quotient) Groups
		Normal subgroups are subgroups for which the partition of G is the same for right cosets as for left cosets.
		It must be shown that the coset of a'b' is the same as the coset of ab.
		The group identity for the cosets is the normal subgroup N, which is the coset of the identity element e of N. Thus the identity element e can be chosen as the representive of the coset N. The inverse for the coset aN is the coset of a
	www.sjsu.edu /faculty/watkins/factorgroup.htm (684 words)

	Construction of Incidence and Coset Geometries
		Construct the coset geometry CG with set of types I, obtained from the group G and the set S of subgroups of G. The sets S and I must have same cardinality.
		If S and I are indexed sets, then the cosets of the subgroup S[i] are elements of type I[i], for all i in {0,..., n - 1} where n is the cardinality of S (and I).
		Construct the coset geometry CG obtained from the group G and the set S of subgroups of G. If G is a permutation group and S is a set of subgroups of G, the corresponding coset geometry, obtained by using Tits' algorithm (see above) is constructed.
	www.math.wayne.edu /answers/magma2.10/htmlhelp/text1315.htm (732 words)

	Section (ii) A Novice's Inquiry on the Concept of Equivalence Class and of Coset: Bestowing Meaning Through Ambivalent ...
		C6 is a nearly platonistic enquiry on the nature of cosets as objects, as entities.
		Finally Camille ceases the effort to interpret further the notion of coset once she acquires an image of cosets that is satisfying and clear to her.
		That Camille is content with what she has acquired can be assumed on the basis of the evidence, given during observation, that this student does not bring a conversation to an end until she acquires a satisfactory (to her) understanding.
	www.uea.ac.uk /~m011/thesis/chapter9/9ii.htm (1364 words)

	Double cosets (Site not responding. Last check: 2007-11-03)
		Thus the double coset decomposition of a group may be viewed as a generalization of the coset (right or left) decomposition of a group.
		However, the reader should be careful not to generalize all facts related to coset decompositions to the case of double coset decompositions.
		For example, we saw that any two cosets of a finite group have the same number of elements.
	web.usna.navy.mil /~wdj/tonybook/gpthry/node44.html (336 words)

	Coset Enumeration (Site not responding. Last check: 2007-11-03)
		ACE is a complete rewrite of the premier coset enumerator TC/(A)CE by Havas, a programme with a 30 year pedigree.
		It could also be used as a (reasonably) clean starting point for people wishing to investigate deduction handling (the most time-consuming part of an enumeration), or gap-filling strategies.
		Implicit in a coset enumeration is a sequence of coset definitions which is sufficient (along with the presentation) to recreate the coset table.
	www.itee.uq.edu.au /~cram/ce.html (383 words)

	Heuristic Example
		Nevertheless coset enumeration is often effective in verifying that a particular presentation defines a finite group or more generally in showing that a finitely generated subgroup is of finite index.
		When it succeeds, coset enumeration produces a coset table whose rows correspond to cosets and columns to generators.
		We try to enumerate the cosets of the identity subgroup < 1 > from the presentation G = < a, b \| ab = ba >.
	personal.stevens.edu /~rgilman/ccny/cosets.htm (811 words)

	[No title] (Site not responding. Last check: 2007-11-03)
		The purpose of transmitting difference sub-coset values instead of individual pixel coset values is to minimize the number of changes in the higher bit planes (of bit-plane encoding), and hence, to achieve a lower bit rate through the bit-plane encoding in part (iv).
		This coset includes one pixel coset value (that of the initial pixel of the N pixel group) and N-1 sub-coset values for the differences.
		The first pixel belongs to sub-coset 30 of this coset, meaning that, since it is the first pixel and that there are a total of 32 sub-cosets for this position, the first pixel is one of 30,62,94...,190,222,254.
	www.stanford.edu /~ozonat/algorithm.html (862 words)

	[ITC] 3 The Basic Operations
		It should be clear from understanding the idea of CE that this new coset number may in fact (by a different word in the generators) represent a coset for which a (lower) number has already been defined, and that this fact may only become apparent much later in the CE process.
		In a first step all cosets obtained from the coset with number 1 by multiplication with generators or their inverses will be arranged in the order in which they occur in a row-wise scan of the Coset Table from left to right.
		In general not all coset numbers defined in the CE will occur in the sequence of indispensables, so we complement the sequence of indispensables by the remaining coset numbers in the order in which they occurred in the original definition sequence.
	www-groups.dcs.st-and.ac.uk /~gap/Manuals/pkg/itc/htm/CHAP003.htm (2454 words)

	Coset Geometries
		Diagram(C) computes the diagram of the coset geometry C. This algorithm is much faster than the one for incidence geometries since it uses lots of group theory machinery in the computation.
		IsGraph(C) permits to check if the coset geometry C is a graph.
		Quotient(D,K) returns the quotient of the coset geometry D by a subgroup K provided that K is a subgroup of the kernel of D. Up:
	www.umich.edu /~gpcc/scs/magma/rel/node49.htm (165 words)

	The Tate Pairing
		Elements of these cosets are not all of the same order.
		The Tate Pairing operates on a pair of points, P of prime order r (a member of G[r]) and a point Q which is a representative member of one of the cosets.
		Its value is the same irrespective of which element of a particular coset is chosen.
	www.computing.dcu.ie /~mike/tate.html (1301 words)

	Observations on Coset Enumeration
		Todd and Coxeter's method for enumerating cosets of finitely generated subgroups in finitely presented groups (abbreviated by T
		It is extended to free monoids and an algebraic characterization for the ``cosets'' enumerated in this setting is provided.
		coset enumeration, subgroup problem, prefix string rewriting, Gröbner bases in monoid and group rings.
	www.mathematik.uni-kl.de /~zca/Reports_on_ca/23/paper_html/paper.html (125 words)

	Subgroups of Finite Index
		If the coset enumeration does not produce a closed coset table, a runtime error is reported.
		Given a subgroup H of the fp-group G, this function returns a set of words in the generators of G, generating H as a subgroup of G (assuming such words are known or can be constructed).
		It should be noted that this function is evaluated by first constructing the right cosets of H in G and then computing the orbits of the cosets under the action of the generators of the subgroup K. This function requires a closed coset table for H in G. Example
	www.math.lsu.edu /magma/text427.htm (3812 words)

	atlas: atlas::weyl::Transducer Class Reference
		Reduced decomposition in W (or W_r) of minimal coset representative x.
		When x' is not equal to s, this is an equality of minimal coset representatives.
		When x'=x, the equation for minimal coset representatives is out(x,s).x = x.s.
	www-math.mit.edu /~dav/html/classatlas_1_1weyl_1_1_transducer.html (692 words)

	Construction of Incidence and Coset Geometries
		Construct the incidence geometry IG from the coset geometry C. This is done using Tits' algorithm described in the introduction of this chapter.
		The group G of the coset geometry CG is the automorphism group of D. Magma determines a chamber C of D, that is a clique of the incidence graph of D containing one element of each type.
		In order to obtain a coset geometry combinatorially isomorphic to the incidence geometry we started with, the group G must be transitive on every rank two truncation of D. If this condition is satisfied, the function returns a boolean set to the value
	www.math.niu.edu /help/math/magmahelp/text1176.html (1137 words)

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