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	Quotient group Information
		In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is intuitively a group that "collapses" the normal subgroup N to the identity element.
		A quotient group of a group G is a partition of G which is itself a group under this operation.
		The quotient group G / G is isomorphic to the trivial group (the group with one element), and G / {e} is isomorphic to G.
	www.bookrags.com /wiki/Quotient_group (1300 words)

	Quotient group (Site not responding. Last check: 2007-10-20)
		In mathematics given a group G and a normal subgroup N of G the quotient group or factor group of G over N is a group that intuitively "collapses" normal subgroup N to the identity element.
		Several important properties of quotient groups are in the fundamental theorem on homomorphisms and the isomorphism theorems.
		Cohomology of Quotients in Symplectic and Algebraic Geometry.
	www.freeglossary.com /Factor_group (1002 words)

	math lessons - Quotient group (Site not responding. Last check: 2007-10-20)
		The quotient group is written G/N and is usually spoken in English as G mod N (mod is short for modulo).
		Trivially, G/G is isomorphic to the trivial group (the group with one element), and G/{e} is isomorphic to G.
		Several important properties of quotient groups are recorded in the fundamental theorem on homomorphisms and the isomorphism theorems.
	www.mathdaily.com /lessons/Factor_group (790 words)

	PlanetMath: abelian group
		In fact, it is often more natural to treat abelian groups as modules rather than as groups, and for this reason they are commonly written using additive notation.
		Theorem 2 Quotient groups of abelian groups are also abelian.
		This is version 21 of abelian group, born on 2003-10-15, modified 2006-12-12.
	planetmath.org /encyclopedia/AbelianGroup2.html (198 words)

	PlanetMath: quotient group
		See Also: group, normal subgroup, ring, field, subgroup, equivalence relation, a subgroup of index 2 is normal
		proof that a subgroup of a group defines an equivalence relation on the group
		This is version 9 of quotient group, born on 2001-12-21, modified 2007-01-04.
	planetmath.org /encyclopedia/FactorGroup.html (101 words)

	Quotient - Wikipedia, the free encyclopedia
		In mathematics, a quotient is the end result of a division problem.
		For example, in the problem 6 ÷ 3, the quotient would be 2, while 6 would be called the dividend, and 3 the divisor.
		Quotients also come up in certain tests, like the IQ test, which stands for intelligence quotient.
	en.wikipedia.org /wiki/Quotient (226 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 (5201 words)

	Quotient group - Wikipedia, the free encyclopedia
		The quotient group is written G/N and is usually spoken in English as G mod N (mod is short for modulo).
		A quotient group of a group G is a partition of G which is itself a group under this operation.
		The quotient group G / G is isomorphic to the trivial group (the group with one element), and G / {e} is isomorphic to G.
	en.wikipedia.org /wiki/Quotient_group (1387 words)

	Group Theory at the Library of Math (Free Online Mathematics) (Site not responding. Last check: 2007-10-20)
		In this topic, many examples are given to explain the importance of permutation groups when the underlying set is a finite set of counting numbers; and the matrix form and cycle notation of such permutations are detailed so as to fully explore the groups of permutations of finite sets of counting numbers (called symmetric groups).
		Basically, the center of a group is the collection of elements in the group that commute with all elements in the group and the centralizer of a given element in the group is the collection of all elements in the group that commute with that given element.
		Finally, homomorphisms, kernels, normal subgroups, and quotient groups are defined; and lastly, using a quotient group, it is shown that normal subgroups are kernels of homomorphisms.
	libraryofmath.com /Group_Theory.html (1788 words)

	Construction of an FP-Group
		The group G is defined by means of a presentation which consists of the relations for F (if any), together with the additional relations defined by the list R. The expression defining F may be either simply the name of a previously constructed group, or an expression defining an fp-group.
		Given a subgroup H of the group G, construct the quotient of G by the normal closure N of H. The quotient is formed by taking the presentation for G and including the generating words of H as additional relators.
		Groups that satisfy certain properties, such as being abelian or polycyclic, are known to possess presentations with respect to which the word problem is soluble.
	www.umich.edu /~gpcc/scs/magma/text295.htm (2326 words)

	Group theory terms
		Not all subsets of a group are subgroups.
		A collection C of elements of a group is said to generate the group (and they are called generators) if all possible combinations of multiplications of those elements with one another yields all elements of the group.
		Group theory is the study of symmetry in the abstract.
	groupexplorer.sourceforge.net /help/rf-groupterms.html (3947 words)

	Construction of an FP-Group
		The group G is defined by means of a presentation which consists of the relations for F (if any), together with the additional relations defined by the list R. The expression defining F may be either simply the name of a previously constructed group, or an expression defining an fp-group.
		Given a subgroup H of the group G, construct the quotient of G by the normal closure N of H. The quotient is formed by taking the presentation for G and including the generating words of H as additional relators.
		Groups that satisfy certain properties, such as being abelian or polycyclic, are known to possess presentations with respect to which the word problem is soluble.
	www.math.lsu.edu /magma/text423.htm (2221 words)

	Quotient Summary
		Quotient is defined in mathematics as the quantity obtained by dividing one quantity or expression by another quantity or expression.
		Quotients are most often represented by the symbols "/" (called diagonal or solidus) and "÷" (obelus) so that for numbers "a" and "b" (with a non-zero value of "b") the quotient may be expressed "a / b" or "a ÷ b".
		Therefore, the quotient "6 / 2" represents the same number as "6" (the dividend of the quotient) multiplied by the reciprocal of the divisor (i.e., 1 / 2), or written as an equation: "6 / 2 = 6 ⋅ (1 / 2)".
	www.bookrags.com /Quotient (1164 words)

	Moduloid - Abelian Unital Magma
		Naturally such visualization of quotient group is extended to that of multidimensional quotient group.
		And these quotient spaces are discretized to obtain the corresponding moduloids with the manner similar to that described in the former sections.
		In general we can not know whether the given three dimensional quotient space is the chaotic space or not from the form of its moduloid contrary to the case of two dimensional chaotic space.
	geocities.com /tontokohirorin/mathematics/moduloid/moduloid2.htm (1336 words)

	Science Fair Projects - Quotient group
		A subgroup N of a group G is normal if and only if the coset equality aN = Na holds for all a in G.
		If G is the group of invertible 3×3 real matrices, and N is the subgroup of 3×3 real matrices with determinant 1, then N is normal in G (since it is the kernel of the determinant homomorphism), and G/N is isomorphic to the multiplicative group of non-zero real numbers.
		There is a "natural" surjective group homomorphism π : G → G/N, sending each element g of G to the coset of N to which g belongs, that is: π(g) = gN.
	www.all-science-fair-projects.com /science_fair_projects_encyclopedia/Quotient_group (930 words)

	Homomorphisms and Isomorphisms
		If k is a normal subgroup, and h the resulting quotient group, let the function f take an element x in g to the coset of k represented by x.
		The image group h is indistinguishable from the quotient group g/k.
		A group homomorphism defines, and is defined by, a normal subgroup and quotient group.
	www.mathreference.com /grp,homo.html (727 words)

	Computing Quotients of Finitely Presented Groups
		Given a finitely presented group G, this function computes the elementary divisors of the derived quotient group G/G', by constructing the relation matrix for G and transforming it into Smith normal form.
		Given a subgroup H of the finitely presented group G, this function computes the elementary divisors of the derived quotient group of H. (The coset table T may be used to define H.) This is done by abelianizing the Reidemeister-Schreier presentation for H and then proceeding as above.
		We construct the class 6 quotient q of R(2, 5) and then partially construct the class 7 quotient interactively to find out how many normal words having initial segment q.1^2 need to be considered when imposing the exponent law.
	www.math.ufl.edu /help/magma/text232.html (2441 words)

	Soluble Quotients
		The soluble quotient varepsilon:F mapsur G, where F is a finitely presented group and G is a finite soluble group.
		Initialises a soluble quotient process for the epimorphism e: F mapsur G. F must be a finitely presented group and G a finite soluble group.
		Typically either a soluble group may appear such that the homomorphism is not surjective, or (the series of) the soluble group does not have the desired properties.
	www.math.niu.edu /help/math/magmahelp/text312.html (10178 words)

	Springer Online Reference Works (Site not responding. Last check: 2007-10-20)
		is a topological group in the induced topology.
		Integration with respect to a Haar measure allows one to transfer to compact groups a significant part of the theory of representations of finite groups (for example, the orthogonality relation for characters, or for matrix entries), and also the Peter–Weyl theorem, which was first obtained for Lie groups.
		Classes of groups satisfying some finiteness property have been studied; for example, the condition of finiteness of the special rank, different variants of the maximum and minimum conditions for subgroups, etc. (cf.
	eom.springer.de /T/t093070.htm (1330 words)

	Construction of a General Permutation Group
		Given the permutation group G, construct the subgroup H of G that is the normal closure of the subgroup H generated by the elements specified by the list L, where the possibilities for L are the same as for the sub-constructor.
		We illustrate the use of the ncl-constructor by using it to construct the derived subgroup of the Hessian group H. We use the fact that the derived subgroup may be obtained as the normal closure of the subgroup generated by the commutators of the generators of H. > H := PermutationGroup< 9
		Given a normal subgroup N of the permutation group G, construct the quotient of G by N. Currently in Magma, the quotient group is constructed in its regular representation, so that the application of this operator is restricted to the case where the index of N in G is a few thousand.
	www.math.wisc.edu /help/magma/text297.html (1051 words)

	Subgroups, Quotient Groups and Extensions
		The collection of words and groups specified by the list must all belong to the group G and H will be constructed as a subgroup of G. The generators of H consist of the words specified directly by terms L[i] together with the stored generating words for any groups specified by terms of L[i].
		Construct the quotient Q of the pc-group G by the normal subgroup N, where N is the smallest normal subgroup of G containing the elements specified by the terms of the generator list L. The possible forms of a term L[i] of the generator list are the same as for the sub-constructor.
		Given a finitely presented group F, a prime p, and a positive integer c, this function constructs a consistent power-conjugate presentation for the largest p-quotient H of F having lower exponent-p class at most c.
	www.math.uiuc.edu /Software/magma/text219.html (1152 words)

	Advanced Math: Order of a quotient group.
		aka: the group of integers mod 12 cross the group of integers mod 18.
		Z12 x Z18 means the group consisting of all ordered pairs where a = 0..11 and b = 0..17, under addition.
		In that case, I think the quotient group (factor group) consists of not 32, but 12 elements.
	en.allexperts.com /q/Advanced-Math-1363/Order-quotient-group.htm (343 words)

	Abelian, Nilpotent and Soluble Quotient
		Given a subgroup H of the finitely presented group G, this function computes the elementary divisors of the derived quotient group of H. (The coset table T may be used to define H.) This is done by abelianising the Reidemeister-Schreier presentation for H and then proceeding as above.
		A nilpotent quotient algorithm constructs, from a finite presentation of a group, a polycyclic presentation of a nilpotent quotient of the finitely presented group.
		Each factor group is represented by a special form of polycyclic presentation, a nilpotent presentation, that makes use of the nilpotent structure of the factor group.
	www.umich.edu /~gpcc/scs/magma/text297.htm (2923 words)

	Normal Subgroup and Quotient Group (Site not responding. Last check: 2007-10-20)
		Subgroup and Factor Group of a Solvable Group...
		Character Table of the Symmetric Group, S4 isomorphic to the group of rotations...
		Representing subgroups of finitely presented groups by Quotient...
	www.scienceoxygen.com /math/262.html (123 words)

	GAP (nq) - Chapter 3: The Functions of the Package
		The quotient modulo the torsion subgroup is torsion-free.
		Note that the last epimorphism is a map from the group generated by a and b onto the nilpotent quotient.
		They are not generators of the group G. Also note that the left-normed commutator above is mapped to the identity as G satisfies the specified identical law.
	www-groups.dcs.st-and.ac.uk /gap/Manuals/pkg/nq/doc/chap3.html (1418 words)

	Elliptic Curves and Modular Functions
		This group has a "fundamental domain" with the property that any point in the whole plane is a transformation of a point in the fundamental domain by an element of the group.
		For instance, the set of all rotations of the plane about the origin is a group, and the orbit of any particular point in the plane is a circle whose radius is the distance of the point from the origin.
		If the surface happens to be a quotient space with respect to a symmetry group on another surface, then the space of all its meromorphic functions corresponds to a very special class of functions on the "larger" surface: the automorphic functions.
	www.mbay.net /~cgd/flt/flt05.htm (2994 words)

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