
	Where results make sense

Topic: Kernel of a homomorphism

Related Topics

Associative rings

Fundamental theorem on homomorphisms

Direct summand

Lie algebra

Real number

Arithmetic

	Homomorphism - Wikipedia, the free encyclopedia
		Homomorphism is one of the fundamental concepts in abstract algebra.
		In abstract algebra, a homomorphism is a map from one algebraic structure to another of the same type that preserves all the relevant structure.
		In that case the image of X in Y under the homomorphism f is necessarily isomorphic to X/~; this fact is one of the isomorphism theorems.
	en.wikipedia.org /wiki/Homomorphism (632 words)

	Kernel (algebra) - Wikipedia, the free encyclopedia
		Note that the definition of kernel here (as in the monoid example) doesn't depend on the algebraic structure; it is a purely set-theoretic concept.
		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).
		The categorical generalisation of the kernel as a congruence relation is the kernel pair.
	en.wikipedia.org /wiki/Kernel_(algebra) (1892 words)

	kernel (mathematics) (Site not responding. Last check: 2007-11-07)
		Unrelated to this, if f is any function in any context, then the kernel of f is a certain equivalence relation on the domain of f which is defined in terms of f.
		But in the case of Mal'cev algebras, it can be replaced by a simpler definition; the kernel of a homomorphism f is the preimage under f of the zero element of the codomain.
		Finally, for this last notion of kernel is generalised in a certain sense in category theory; the kernel of a morphism f is the difference kernel of f and the corresponding zero morphism (if this exists).
	www.yourencyclopedia.net /Kernel_(mathematics) (242 words)

	Category theory - LearnThis.Info Enclyclopedia (Site not responding. Last check: 2007-11-07)
		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.
		If all morphisms have a kernel and a cokernel, and all epimorphism are cokernels and all monomorphisms are kernels, then we speak of an abelian category.
	encyclopedia.learnthis.info /c/ca/category_theory.html (3218 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		A homomorphism f from a group G to a group Gbar is a mapping from G into Gbar that preserves the group operation; that is, f(ab) = f(a)f(b) for all a,b in G. Page 194 Definition: Kernel of a Homomorphism.
		The kernel of a homomorphism f from a group G to a group with identity e is the set {x in G
		Let f be a homomorphism from a group G to a group Gbar and let H be a subgroup of G. Then (1) f(H) is a subgroup (2) The image of a cyclic group is cyclic.
	orion.math.iastate.edu /hentzel/class.301.03/Dec.01 (327 words)

	Element Operations
		Given a homomorphism a belonging to a submodule of Hom(M, N), and a homomorphism b belonging to a submodule of Hom(N, P), return the composition of the homomorphisms a and b as an element of Hom(M, P).
		The kernel of the homomorphism a belonging to the module Hom(M, N), returned as a submodule of M. Note that if the domain and codomain of a are matrix modules themselves, the kernel will be with respect to the appropriate action (right or left).
		The row nullspace of the homomorphism a belonging to the module Hom(M, N), returned as a submodule of M. This is equivalent to the kernel of the transpose of a.
	www.math.wisc.edu /help/magma/text518.html (821 words)

	Homomorphism (Site not responding. Last check: 2007-11-07)
		Some take it to mean any kind of structure preserving map (such as continuous maps in topology), or even a more abstract kind of map-what we term a morphism-used in category theory.
		In this setting, a homomorphism φ : A → B is a map between two algebraic structures of the same type such that
		Any homomorphism f : X → Y defines an equivalence relation ~ on X by a ~ b iff f(a) = f(b).
	www.yotor.com /wiki/en/ho/Homomorphism.htm (475 words)

	The kernel of a homomorphism is the set of all elements in the domain that map to the identity element in the co-domain
		The kernel of a homomorphism is the set of all elements in the domain that map to the identity element in the co-domain.
		In other words, if F is a homomorphism from the group G to the group G’, the kernel, denoted Ker F, is the set of all elements x in F such that F(x)=e’ where e’ is the identity of G’.
		Since F is a homomorphism that is a bijection (injective and surjective), F can be classified as an isomorphism.
	students.uww.edu /muellerbt15/Kernel.htm (476 words)

	Ring homomorphisms and isomorphisms
		A ring homomorphism which is a bijection (one-one and onto) is called a ring isomorphism.
		The kernel of a (ring) homomorphism is the set of elements mapped to 0.
		Note the similarity with the corresponding result for groups: the kernel of a group homomorphism is a normal subgroup.
	www-groups.dcs.st-and.ac.uk /~john/MT4517/Lectures/L7.html (513 words)

	[12pt] (Site not responding. Last check: 2007-11-07)
		Thus either the kernel is {e} or G. In the former case, the map is one-to-one and hence an isomorphism.
		Note that K is the kernel of the map and the result follows from the first isomorphism theorem.
		The map is a surjective homomorphism with kernel equal to {-1, 1}.
	people.clarkson.edu /~dobrowb/courses/ma311f99/hw13sol.html (658 words)

	ABSTRACT ALGEBRA: OnLine Study Guide, Section 3.7
		(Homomorphisms defined on cyclic groups) Let C be a cyclic group, denoted multiplicatively, with generator a.
		There are many important examples of group homomorphisms that are not isomorphisms, and, in fact, homomorphisms provide the way to relate one group to another.
		Examples 3.7.4 and 3.7.5 are important, because they give a complete description of all group homomorphisms between two cyclic groups.
	www.math.niu.edu /~beachy/abstract_algebra/study_guide/37.html (626 words)

	Pointers in creating a homomorphism
		Finding a homomorphism between two groups is often difficult especially for a first course in group theory.
		The set of homomorphisms between groups G and H is known as Hom(G,H), or in the case H=G, End(G).
		Another application shows that there are no non trivial homomorphisms from a cyclic group of order m to one of order n whenever m and n are coprime.
	www.math.csusb.edu /notes/advanced/algebra/gp/node17.html (338 words)

	[No title]
		The kernel of a homomorphism from A to any other ring is an "ideal": a set closed under addition and also multiplication by all elements of A.
		Even better, the kernel of a homomorphism from A to a field is a "prime" ideal, meaning it's not not all of A, and whenever the product of two elements of A lies in the ideal, at least one of them must be in the ideal.
		This is the kernel of the homomorphism from Z into the rationals.
	math.ucr.edu /home/baez/twf_ascii/week199 (4337 words)

	GAP Manual: 43. Homomorphisms (Site not responding. Last check: 2007-11-07)
		A mapping map is a homomorphism if the source and the range are domains of the same category, and map respects their structure.
		The last section describes the function that computes the kernel of a homomorphism (see Kernel).
		Because homomorphisms are just a special case of mappings all operations and functions described in chapter Mappings are applicable to homomorphisms.
	www.math.uiuc.edu /Software/GAP-Manual/Homomorphisms.html (161 words)

	Elements of M_n as Homomorphisms (Site not responding. Last check: 2007-11-07)
		Given an element of M_n(S), return the kernel of the homomorphism represented by the matrix a (as an element of S^((n))).
		Given an element of M_n(S), return the row nullspace of the homomorphism represented by the matrix a (as an element of S^((n))).
		This is equal to the kernel of the transpose of a.
	www.math.niu.edu /help/math/magmahelp/text874.html (132 words)

	Rings (Site not responding. Last check: 2007-11-07)
		If the kernel of this homomorphism is nZ we say that A has characteristic n.
		A homomorphism from a ring A to a ring B is a function f: A --> B such that
		The kernel of a homomorphism of rings f: A --> B is the ideal in A consisting of those elements a in A such that f(a) = 0.
	mcraeclan.com /MathHelp/BasicAARings.htm (1016 words)

	Math 109, Winter Quarter, 2001 (Site not responding. Last check: 2007-11-07)
		(c) Recall that the kernel of a homomorphism from group G to group H is defined to be the set of group elements in G which are mapped to the identity element of H. Find the kernel of the homomorphism f.
		Suppose that G is a cyclic group generated by an element a, and f: G --> H is a homomorphism, then f is completely determined by f(a).
		Prove that if f: G --> H is a homomorphism and A is a subgroup of G, then f(A) is a group of H. f(A) is the image of the subgroup A under the map f.
	www.math.ucsd.edu /~kbriggs/109W2001 (692 words)

	PlanetMath: kernel of a homomorphism is a congruence (Site not responding. Last check: 2007-11-07)
		PlanetMath: kernel of a homomorphism is a congruence
		"kernel of a homomorphism is a congruence" is owned by almann.
		This is version 5 of kernel of a homomorphism is a congruence, born on 2003-07-27, modified 2004-02-22.
	planetmath.org /encyclopedia/KernelOfAHomomorphismIsACongruence.html (71 words)

	8.5 The category of vector space schemes (Site not responding. Last check: 2007-11-07)
		One can easily ``relativise'' the notion of a homomorphism of modules to define the notion of a homomorphism of vector space schemes.
		The inverse image of the zero section under such a homomorphism a sub-vector space scheme of the domain of the homomorphism.
		This, defines the kernel of a homomorphism of vector space scheme.
	www.imsc.ernet.in /~kapil/crypto/notes/node43.html (457 words)

	bigint_matrix (Site not responding. Last check: 2007-11-07)
		In the first case, the class supports for example computing bases, hermite normal form, smith normal form etc., in the second case it may be used for computing the determinant, the characteristic polynomial, the image, the kernel etc. of the homomorphism.
		The last column of matrix A is a solution x of the system and the other columns form a generating system of the kernel of the homomorphism corresponding to matrix B.
		stores a basis of the kernel of the homomorphism generated by matrix B to A.
	www.math.psu.edu /local_doc/LiDIA/node77.html (2877 words)

	The Kernel Of An Homomorphism Of Harish-Chandra - Levasseur, Sta (ResearchIndex) (Site not responding. Last check: 2007-11-07)
		The Kernel Of An Homomorphism Of Harish-Chandra - Levasseur, Sta (ResearchIndex)
		Levasseur and J. Staord, The kernel of an homomorphism of Harish-Chandra, Ann.
		5 erential operators and an homomorphism of Harish-Chandra (context) - Levasseur, Sta et al.
	citeseer.ist.psu.edu /levasseur95kernel.html (610 words)

	review2
		A subgroup is normal if and only if it is the kernel of a homomorphism.
		There is a homomorphism from a group of 6 elements into a group of 12 elements.
		A homomorphism is one to one if and only if its kernel consists of just the identity.
	www2.truman.edu /~dgarth/math367/spring01/review2/review2.html (460 words)

	Math 461 Test 2 Review Information (Site not responding. Last check: 2007-11-07)
		You will not be responsible for any material in the text but not covered in class -- such as applications to periodic functions and plane isometries in 2.4 but will be responsible for the material on binary linear codes, 2.5.
		Be able to give examples of homomorphisms and their kernels.
		Kernel of a homomorphism is a normal subgroup
	www.sci.uidaho.edu /m461/m461t2rev.htm (257 words)

	Group Theory & Rubik's Cube
		A homomorphism from G into H is a map f which perserves the group operation, i.e.
		The kernel of a homomorphism G->H is the set of elements of G which are mapped to the identify of H. The kernel is always a normal subgroup of G, and its cosets form a quotient group G/(kernel) which is isomophic to H. See quotient group.
		A representation of a group G is a set of matrices M which are homomorphic to the group.
	akbar.marlboro.edu /~mahoney/courses/Spr00/rubik.html (3602 words)

Try your search on: Qwika (all wikis)