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# Topic: Group homomorphisms

###### In the News (Thu 21 Jun 18)

 Group homomorphism - Wikipedia, the free encyclopedia For example, a homomorphism of topological groups is often required to be continuous. If the homomorphism h is a bijection, then one can show that its inverse is also a group homomorphism, and h is called a group isomorphism; in this case, the groups G and H are called isomorphic: they differ only in the notation of their elements and are identical for all practical purposes. For example, the endomorphism ring of the abelian group consisting of the direct sum of two copies of Z/2Z (the Klein four-group) is isomorphic to the ring of 2-by-2 matrices with entries in Z/2Z. en.wikipedia.org /wiki/Group_homomorphism   (838 words)

 Abelian group - Wikipedia, the free encyclopedia   (Site not responding. Last check: 2007-10-22) In mathematics, an abelian group, also called a commutative group, is a group (G, *) such that a * b = b * a for all a and b in G. If f, g : G → H are two group homomorphisms between abelian groups, then their sum f + g, defined by (f + g)(x) = f(x) + g(x), is again a homomorphism. The abelian group, together with group homomorphisms, form a category, the prototype of an abelian category. www.newlenox.us /project/wikipedia/index.php/Abelian_group   (824 words)

 Homomorphisms and Isomorphisms Conversely, let f be a homomorphism from the group g onto the set h, which possesses a preexisting operator *. 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)

 Category theory - LearnThis.Info Enclyclopedia   (Site not responding. Last check: 2007-10-22) In the example of groups, these are the group homomorphisms. Algebra of continuous functions: a contravariant functor from the category of topological spaces (with continuous maps as morphisms) to the category of real associative algebras is given by assigning to every topological space X the algebra C(X) of all real-valued continuous functions on that space. Homomorphism groups: to every pair A, B of abelian groups and can assign the abelian group Hom(A,B) consisting of all group homomorphisms from A to B. encyclopedia.learnthis.info /c/ca/category_theory.html   (3218 words)

 Homomorphisms   (Site not responding. Last check: 2007-10-22) Classification of Homomorphisms to Oriented Cycles and of... Fibonacci sequences and homomorphisms of free submonoid for unimodal maps... Reduction of abstract homomorphisms of lattices mod p and rigidity, by Chandrash... www.scienceoxygen.com /math/263.html   (183 words)

 GAP Manual: 7.105. Group Homomorphisms   (Site not responding. Last check: 2007-10-22) A group homomorphism phi is a mapping that maps each element of a group G, called the source of phi, to an element of another group H, called the range of phi, such that for each pair x, y inG we have (xy)^phi= x^phiy^phi. Examples of group homomorphisms are the natural homomorphism of a group into a factor group (see NaturalHomomorphism) and the homomorphism of a group into a symmetric group defined by an operation (see OperationHomomorphism). More general, since group homomorphisms are just a special case of mappings all functions described in chapter Mappings are also applicable, e.g., the function to compute the image of an element under a group homomorphism (see Image). www.math.uiuc.edu /Software/GAP-Manual/Group_Homomorphisms.html   (187 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. Any subgroup of the symmetric group Sym(S) on a set S is called a permutation group or group of permutations. www.math.niu.edu /~beachy/aaol/groups.html   (1115 words)

 [ref] 38 Group Homomorphisms Group homomorphisms are mappings, so all the operations and properties for mappings described in chapter Mappings are applicable to them. To compute the kernel of a homomorphism (unless the mapping is known to be injective) requires the capability to compute a presentation of the image and to evaluate the relators of this presentation in preimages of the presentations generators. The different representations of group homomorphisms are used to indicate from what type of group to what type of group they map and thus determine which methods are used to compute images and preimages. www.math.sunysb.edu /~sorin/online-docs/gap4r3/htm/ref/CHAP038.htm   (2933 words)

 Groups A group  is the least structure in which we can define an internal operation which generalizes the multiplication and division on nonzero real numbers. A vector space, or linear space, is the nicest way in which a group (with a commutative operation +) can be combined with a field so as to preserve all three identity elements; a module  is similar, but uses a ring instead of a field for its scalars. is a group homomorphism from the additive group of integers www.ma.umist.ac.uk /kd/knots/node5.html   (1003 words)

 Descriptions of the labs in Exploring Abstract Algebra with Mathematica   (Site not responding. Last check: 2007-10-22) Determining the symmetry group of a given figure -- The focus of this lab is to determine the symmetry group of a figure chosen randomly from a list of regular polygons and "cyclic" objects. Subversively grouping our elements -- This lab explores the notion of a subgroup, including looking at the subgroups of Z_n and U_n, calculating the probability that a random subset of Z_n is a subgroup and determining what elements in a subset are necessary so that the closure yields the whole group. Normality and Factor groups -- A normal group is defined and explored and then used to define and explore factor groups. www.central.edu /eaam/EAAMInfo/LabDescriptions.asp   (713 words)

 Functor   (Site not responding. Last check: 2007-10-22) Functors were first considered in algebraic topology, where algebraic objects (like the fundamental group) are associated to topological spaces, and algebraic homomorphisms are associated to continuous maps. Group actions/representations: Every group G can be considered as a category (or groupoid) with a single object. Forgetful functors: The functor U : Grp → Set which maps a group to its underlying set and a group homomorphism to its underlying function of sets is a functor. www.yotor.com /wiki/en/fu/Functor.htm   (1530 words)

 Functor Article, Functor Information   (Site not responding. Last check: 2007-10-22) Functors were first considered in algebraic topology, wherealgebraic objects (like the fundamental group) are associated to topological spaces, and algebraic homomorphisms are associated to continuous maps. Algebra of continuous functions: a contravariant functor from the category of topological spaces (with continuous maps as morphisms) to the category of real associative algebras is given by assigning to every topological space X the algebraC(X) of all real-valued continuous functions on that space. Forgetful functors: The functor U : Grp → Set whichmaps a group to its underlying set and a group homomorphism to its underlying function of sets is a functor.Functors like these, which "forget" some structure, are termed forgetful functors. www.anoca.org /functors/category/functor.html   (1494 words)

 Normal subgroup In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element x in N and each g in G, the element g A normal subgroup can also be defined by: A subgroup N of a group G is a normal subgroup if N is a union of conjugacy classes of G. Normal subgroups of G are precisely the kernelss of group homomorphisms f : G → H. www.brainyencyclopedia.com /encyclopedia/n/no/normal_subgroup.html   (215 words)

 GAP Manual: 7.108 Mapping Functions for Group Homomorphisms   (Site not responding. Last check: 2007-10-22) For group homomorphisms with equal sources and ranges only the images of the smallest irredundant generating system are compared. The image of a subgroup under a group homomorphism is computed by computing the images of a set of generators of the subgroup, and the result is the subgroup generated by those images. The preimages of a subgroup under a group homomorphism are computed by computing representatives of the preimages of all the generators of the subgroup, adding the generators of the kernel of www-groups.dcs.st-and.ac.uk /gap/Gap3/Manual3/C007S108.htm   (289 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)

 Group homomorphism   (Site not responding. Last check: 2007-10-22) Given two groups (G, *) and (H, ·), a group homomorphism from (G, *) to (H, ·) is a function h : G The exponential map yields a group homorphism from the group of real numbers R with addition to the group of non-zero real numbers R This map is surjective and has the kernel { 2πki : k in Z }, as can be seen from Euler's formula. www.sciencedaily.com /encyclopedia/group_homomorphism   (880 words)

 Kernel (algebra) : QuicklyFind Info   (Site not responding. Last check: 2007-10-22) Let G and H be groups and let f be a group homomorphism from G to H. The first isomorphism theorem for groups states that this quotient group is naturally isomorphic to the image of f (which is a subgroup of H). The notion of ideal generalises to any Mal'cev algebra (as subspace in the case of vector spaces, normal subgroup in the case of groups, two-sided ring ideal in the case of rings, and submodule in the case of modules). www.quicklyfind.com /info/Kernel_(algebra).htm   (1938 words)

 Category theory - FreeEncyclopedia   (Site not responding. Last check: 2007-10-22) Very commonly, certain "natural constructions", such as the fundamental group, can be expressed as functors. There is also the category of groups, with group homomorphisms as morphisms. One may check that the map from the category of Hausdorff topological spaces with a distinguished point to the category of groups is functorial: a topological (homo/iso)morphism will naturally correspond to a group (homo/iso)morphism. openproxy.ath.cx /ca/Category_theory.html   (2075 words)

 Modern Algebra Lecture Notes, 10/30/02   (Site not responding. Last check: 2007-10-22) Definition: Let phi be a homomorphism from the group G to the group G'. Important fact: The kernel of the group homomorphism phi is a subgroup of G, by the previous theorem. Theorem: Let phi be a homomorphism from the group G to the group G', and let H be the kernel of phi. www.assumption.edu /Alfano/MAT351-FA02/Notes/103002.html   (272 words)

 [tut] 5 Groups and Homomorphisms Permutation groups are so easy to input because their elements, i.e., permutations, are so easy to type: they are entered and displayed in disjoint cycle notation. Groups and the functions for groups are treated in Chapter Groups. Group homomorphisms are the subject of Chapter Group Homomorphisms. www.mathematik.uni-kassel.de /gap4/tut/CHAP005.htm   (4806 words)

 ABSTRACT ALGEBRA ON LINE: Groups (part 2) G. Definition If N is a normal subgroup of G, then the group of left cosets of N in G is called the factor group of G determined by N. It will be denoted by G/N. Example 3.8.5. Since any vector space is an abelian group under vector addition, any linear transformation between vector spaces is a group homomorphism. (Homomorphisms defined on cyclic groups) Let C be a cyclic group, denoted multiplicatively, with generator a. www.math.niu.edu /~beachy/aaol/groups2.html   (602 words)

 5.5 Gauge Group Hom(, R+) The gauge group is, therefore, a rather universal object. as a kernel of the homomorphism of groups, moreover it is also a vector subspace. Thus (5.19) shows that the elements of the gauge group are in fact linear transformations and not merely group homomorphisms. www.immt.pwr.wroc.pl /kniga/node22.html   (354 words)

 [No title] RATIONAL ISOMORPHISMS OF p-COMPACT GROUPS JESPER MICHAEL MOLLER Abstract.A rational isomorphism is a p-compact group homomorphism in- ducing an isomorphism on rational cohomology. It is shown that rational isomorphisms of p-compact groups restrict to admissible rational isomorphisms of the maximal tori and the* * clas- sification of rational isomorphisms between connected p-compact groups is reduced to the simply connected case. As the center [18, 8] of a connected p-compact group is a Weyl group invariant subgroup of the maximal torus this theorem implies [Corollary 3.1] th* *at the center is a functor on the category of connected p-compact groups with fini* *te covering homomorphisms as morphisms. hopf.math.purdue.edu /Moller/rateq.txt   (4216 words)

 American Mathematical Monthly, The: Which functor is the projective line?   (Site not responding. Last check: 2007-10-22) Also, there are two homomorphisms per subgroup rather than one because, once we fix the kernel H in G, it still remains to be decided which coset of H gets sent to 1 in Z/3 and which coset goes to 2. Furthermore, if we replace the groups Z/2 and Z/3 with a classical group C such as C = GL^sub n^C or C = U(n), the problem of determining the group homomorphisms from arbitrary groups G to C encompasses essentially all of representation theory! Thus, understanding a group explicitly via an analysis of group homomorphisms into it can be exceedingly complicated. www.findarticles.com /p/articles/mi_qa3742/is_200308/ai_n9300781   (1294 words)

 Math 423, Fall, 2002   (Site not responding. Last check: 2007-10-22) is a homomorphism of the underlying group structure, and is smooth. Aut(V) is a representation of the Lie group G, where V is a vector space. This is actually an important point, since for Lie algebras and their associated Lie groups, understanding homomorphisms of the groups will be reduced to understanding homomorphisms of their Lie algebras - which are linear maps. www.lehigh.edu /dlj0/yesterday/courses/423f02-lect13-redux.html   (514 words)

 Homology   (Site not responding. Last check: 2007-10-22) We talk about ascending and descending chains of groups etc. Well this is a different kind of chain, and perhaps that is why it is sometimes called a chain complex. The internal homomorphisms are called boundary operators, a term that is associated with algebraic topology. Given a ring r, the groups in a chain could be replaced with modules, withmodule homomorphisms carrying each into the next. www.mathreference.com /mod-hom,intro.html   (359 words)

 [No title] d) The image of a group homomorphism is isomorphic to a factor group. If f:G -> H and g:K-> K are group homomorphisms, prove that their composition is a group homomorphism. Determine all homomorphisms from Z4 to Z2 (Z2. www.ilstu.edu /~lmiones/336E3S02.DOC   (152 words)

 CVM 1.1 (VW): Group Homomorphisms When we set out to construct recipes for functions with negating symmetries, we started with the group G, the actual symmetries of a pattern, and extended it to E Group homomorphisms are useful for discussing the alternative, starting with E, the group of extended symmetries, and reducing it to G. From this we see that G is normal in E, as the kernel of P. www.maa.org /cvm/1998/01/vw/article/algebra/gh.html   (254 words)

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