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Topic: Coset enumeration

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	Interactive Coset Enumeration
		This function creates a coset enumeration process for enumerating the cosets of the subgroup H of the finitely presented group G. Note that no actual coset enumeration is started for the created coset enumeration process.
		This means that a coset enumeration process P for the cosets of H in G is transformed into a coset enumeration process for the cosets of < H, w > in G, where < H, w > denotes the subgroup of G generated by H and w.
		Change enumeration parameters of the coset enumeration process P. The set of parameters accepted by this function is the same as for the function CosetEnumerationProcess; see there for a description.
	www.umich.edu /~gpcc/scs/magma/text471.htm (5151 words)

	Coset Spaces and Tables
		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).
		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.umich.edu /~gpcc/scs/magma/text300.htm (2820 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.
		The indexed coset space obtained from the coset space V by forcing cosets i and j to be equal.
	www.math.ufl.edu /help/magma/text231.html (3245 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.
		In Mathematical Introduction the calculation performed by the double coset enumerator, and the meaning of the input is described more DCE Words and DCE Presentations describe how the input is organized as Examples of Double Coset Enumeration.
	www.mcs.kent.edu /system/documentation/gap/CHAP061.htm (4131 words)

	Small Group Identification
		Hence for performance reasons, a coset limit of 100.o is imposed, where o is the maximal order of groups in the database, unless the order of G is known to be less or equal to o.
		If the coset enumeration for G fails with the coset limit 100.o, this can be seen as a reasonable indication that G is probably too large to be contained in the database of small groups.
		To deal with cases where the coset enumeration fails although G is known or suspected to be small enough, it is recommended to attempt to compute the order of G using the function Order before the actual group identification.
	www.math.wayne.edu /answers/magma2.10/htmlhelp/text411.htm (533 words)

	[ACE] 3 Some Basics
		The state of an enumeration at any time is stored in a 2-dimensional array known as a coset table whose rows are indexed by coset numbers and whose columns are indexed by the group generators and their inverses.
		The key to performance of coset enumeration procedures is good selection of the next coset number to be defined.
		standard coset table whose columns correspond, in order, to the already-described alphabet, of generators and their inverses, has an important property: a scan of the body of the table row by row from left to right, encounters new coset numbers in numeric order.
	www-groups.dcs.st-and.ac.uk /gap/Manuals/pkg/ace/htm/CHAP003.htm (2767 words)

	Subgroups
		The subgroup N is obtained by computing the coset table of the trivial subgroup in the group defined by the relations of G together with relators corresponding to the words generating H. For a sample application of this function, see Example H22E17.
		Try to enumerate the cosets of the trivial subgroup in G. Check the subgroups of known or easily computable index in G. If we can compute the order of such a subgroup or prove that it is infinite, we're done.
		The default behaviour for such implicitly called coset enumerations is the same as the one for coset enumerations invoked explicitly, e.g.
	www.math.niu.edu /help/math/magmahelp/text300.html (2825 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)

	Coset Enumeration (Site not responding. Last check: 2007-10-12)
		ACE is a complete rewrite of the premier coset enumerator TC/(A)CE by Havas, a programme with a 30 year pedigree.
		The manual might also be helpful to users of P(E)ACE, since their coset enumerators were derived from ACE's.
		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)

	[ref] 45 Finitely Presented Groups
		If the enumeration does not finish with this number of cosets, an error is raised and the user is asked whether she wants to continue.
		A coset enumeration will not finish if the subgroup does not have finite index, and even if it has it may take many more intermediate cosets than the actual index of the subgroup is. To avoid a coset enumeration ``running away'' therefore
		Using variations of coset enumeration it is possible to compute the abelian invariants of a subgroup of a finitely presented group without computing a complete presentation for the subgroup in the first place.
	www.msri.org /about/computing/docs/gap/htm/ref/CHAP045.htm (5184 words)

	Subgroups
		Given a homomorphism f from G onto a transitive subgroup of Sym(n), construct the subgroup of G that is the normal closure of the subgroup K of G which affords this permutation representation.
		Given a subgroup H of the fp-group G, this function attempts to determine the index of H in G by enumerating the cosets of H using the Todd-Coxeter procedure.
		Given a finite fp-group G, this function attempts to determine the order of G by enumerating the cosets of the trivial subgroup in G using the Todd-Coxeter procedure.
	www.math.wisc.edu /help/magma/text240.html (1741 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 a closed coset table is needed and has not been computed, a coset enumeration will be invoked.
		If the coset enumeration does not produce a closed coset table, a runtime error is reported.
		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.niu.edu /help/math/magmahelp/text301.html (3419 words)

	[ACE] 4 Options for ACE
		One of the reasons for the non-orthogonality of options is to protect the user from obtaining invalid enumerations from bad combinations of options; another reason is that commonly one may specify a strategy option and override some of that strategy's defaults; the general rule is that the later option prevails.
		R style; see Section Enumeration Style) definitions, rows of the coset table are scanned for holes after its coset number has been applied to all relators, and definitions are made to fill any holes encountered.
		Compaction may be performed multiple times during an enumeration, and the table that results from an enumeration may or may not be compact, depending on whether or not there have been any coincidences since the last compaction (or from the start of the enumeration, if there have been no compactions).
	www.maths.uwa.edu.au /~gregg/ACE/htm/CHAP004.htm (6073 words)

	PEACE 1.000: Proof Extraction after Coset Enumeration (ResearchIndex)
		0.5: Scalable Parallel Coset Enumeration: Bulk Definition and..
		1 Theorem proving in coset enumeration (context) - Lockwood - 1991
		Parallel Coset Enumeration Using Threads - Havas, Ramsay
	citeseer.ist.psu.edu /ramsay00peace.html (267 words)

	GAP Manual: 23.4. CosetTableFpGroup (Site not responding. Last check: 2007-10-12)
		Basically a coset table is the permutation representation of the finitely presented group on the cosets of a subgroup (which need not be faithful if the subgroup has a nontrivial core).
		The coset table is standardized, i.e., the cosets are sorted with respect to the smallest word that lies in each coset.
		The coset table is computed by a method called coset enumeration.
	www.math.uiuc.edu /Software/GAP-Manual/CosetTableFpGroup.html (214 words)

	[No title] (Site not responding. Last check: 2007-10-12)
		One strategy for drawing the limit set of such a group $G$ is to draw all the circles which are equivalent under the group to the limit circle $C$ of the fuchsian subgroup $H$.
		This is equivalent to enumerating all the left cosets $aH$ of $H$ in $G$.
		Then the left cosets are enumerated at the same time with representatives $a^{-1}$, and we may plot the circles $a^{-1}(C)$, thus filling out the limit set.
	www.newton.cam.ac.uk /programmes/SKG/poster/wright.html (254 words)

	Ian Redfern - maths (Site not responding. Last check: 2007-10-12)
		It includes the proof that a quasiconvex subgroup of a hyperbolic group is strongly geodesically coset automatic and therefore Short-Lex coset automatic, together with an algorithm (implemented in about 15,000 lines of C++, available on request) that will determine the word acceptor and multipliers (and hence a coset enumeration) for any coset automatic coset space.
		In general, if it's possible to enumerate the cosets of a finitely generated subgroup of a finitely presented group, this algorithm will find you a way to do it.
		Using the enumeration automata above, John Parker, Greg Mcshane, André Rocha and I were able to enumerate cosets in many hyperbolic groups, producing some fascinating orbit plots.
	www.redferni.uklinux.net /maths.html (283 words)

	[ACE] 1 The ACE Package
		The coset enumeration was then successful, allowing the computation of what turned out to be a trivial coset table.
		otal number of 3127 coset numbers needed to be defined before the final collapse to 1 coset number.
		Package users is the change of the default standard for the numbering of cosets in a coset table (see Section Coset Table Standardisation Schemes).
	www.math.rwth-aachen.de /~Greg.Gamble/gap4r3/pkg/ace/htm/CHAP001.htm (3435 words)

	GAP Manual: 23.10. Subgroup Presentations
		After having constructed this set of primary subgroup generators, say, the coset table is extended to an augmented coset table which describes the action of the group generators on coset representatives, i.e., on elements instead of cosets.
		For this purpose, suitable words in the (primary) subgroup generators have to be associated to the coset table entries.
		CosetTableFpGroup), but as the product of a coset representative by a group generator or its inverse need not be a coset representative itself, the Modified Todd-Coxeter has to store a kind of correction element for each coset table entry.
	www.math.uiuc.edu /Software/GAP-Manual/Subgroup_Presentations.html (1382 words)

	Amazon.com: "coset enumeration": Key Phrase page (Site not responding. Last check: 2007-10-12)
		It is possible to extract formal proofs from the internal working of coset enumerations.
		Proof: Coset enumeration over H = (XI, X2i x3) reveals that this subgroup has index 600 in G. By Theorem 1 and Lemma...
		Using coset enumeration with a group given in...
	www.amazon.com /phrase/coset-enumeration (581 words)

	LMS JCM (4) 74-134 (Site not responding. Last check: 2007-10-12)
		Abstract: The authors study a new method for coset enumeration in finitely presented groups.
		The method is compared to well-known methods for Todd–Coxeter enumeration, using examples from the literature where studies of these methods are reported.
		New insights into coset enumeration were gained using three different kinds of orderings, combined with new frameworks and strategies implemented in M
	www.lms.ac.uk /jcm/4/lms2000-010 (126 words)

	4. The Todd-Coxeter Coset Enumeration Procedure
		is finite the procedure halts and produces a set of coset representatives and a coset table with entries
		The coset representative of the word aba can be deduced by either tracing the coset table:
		which is in fact the minimal representative of this coset with respect to the chosen ordering.
	www.mathematik.uni-kl.de /~zca/Reports_on_ca/23/paper_html/node4.html (275 words)

	[No title]
		Computations with semigroups in which the multiplication and comparison of elements can be described using automata is currently implied as with "coset enumeration" but is not done explicitly.
		Knowing what type of algorithm to use for a given semigroup especially if little structural information is known about the semigroup is not easily determined mechanically e.g.
		what kind of enumeration strategy is optimal or feasible for a particular semigroup.
	faculty.evansville.edu /rm43/compalg/standsumary.html (550 words)

	GAPCosetEnumLesson (Site not responding. Last check: 2007-10-12)
		Exercise: Find out (as closely as you can in a reasonable time) how many cosets GAP had to create to get the coset table for the group neweight.
		Determine the order in which definitions of cosets are made for eight.
		Now determine how many cosets GAP uses before it finishes enumerating the cosets of one.
	www.math.umn.edu /~webb/GAPfiles/GAPCosetEnumLesson.html (231 words)

	Proving a Group Trivial Made Easy: A Case Study in Coset Enumeration - Havas, Ramsay (ResearchIndex)
		Abstract: : Coset enumeration, based on the methods described by Todd and Coxeter, is one of the basic tools for investigating finitely presented groups.
		The process is not well understood, and various pathological presentations of, for example, the trivial group have been suggested as challenge problems.
		Proving a group trivial made easy: a case study in coset enumeration.
	citeseer.ist.psu.edu /289949.html (592 words)

	Atlas: Developments in coset enumeration by George Havas (Site not responding. Last check: 2007-10-12)
		Coset enumeration is one of the fundamental tools for investigating finitely presented groups.
		We describe recent developments in computer coset enumeration: the advanced coset enumerator, ACE; the parallel advanced coset enumerator, PACE; and proof extraction after coset enumeration, PEACE.
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafm-23.
	atlas-conferences.com /c/a/f/m/23.htm (91 words)

	[ref] 8 Options Stack (Site not responding. Last check: 2007-10-12)
		Such guidance should not change mathematically the specification of the computation to be performed, although it may change the algorithm used.
		A typical example is the selection of a strategy for the Todd-Coxeter coset enumeration procedure.
		This is not felt to be adequate because many procedure calls might cause, for example, a coset enumeration and each would need to make provision for the possibility of extra arguments.
	www.math.temple.edu /computing/gap/ref/CHAP008.htm (398 words)

	Constructing a Subgroup
		Process version of coset enumeration allowing the user complete control over its execution
		Coset enumeration is performed using George Havas's ACE version of the Todd-Coxeter procedure.
		It has the capability of enumerating up to one hundred million cosets on a sufficiently large machine.
	magma.maths.usyd.edu.au /magma/Features/node41.html (61 words)

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