
	Where results make sense

Topic: Fundamental theorem on homomorphisms

Related Topics

Elementary group theory

Group theory

Jennifer Hale

Obscuris vera involvens

Kernel of a homomorphism

	Fundamental theorem on homomorphisms - Wikipedia, the free encyclopedia
		In abstract algebra, the fundamental theorem on homomorphisms, also known as the fundamental homomorphism theorem, relates the structure of two objects between which a homomorphism is given, and of the kernel and image of the homomorphism.
		The homomorphism theorem is used to prove the isomorphism theorems.
		Given two groups G and H and a group homomorphism f : G→H, let K be a normal subgroup in G and φ the natural surjective homomorphism G→G/K.
	en.wikipedia.org /wiki/Fundamental_theorem_on_homomorphisms (163 words)

	Lie algebra
		The composition of such homomorphisms is again a homomorphisms, and the Lie algebras over the field F, together with these morphisms, form a category.
		If such a homomorphism is bijective, it is called an isomorphism, and the two Lie algebras g and h are called isomorphic.
		The ideals are precisely the kernels of homomorphisms, and the fundamental theorem on homomorphisms is valid for Lie algebras.
	www.ebroadcast.com.au /lookup/encyclopedia/li/Lie_algebra.html (976 words)

	Factor group
		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.
		This correspondence holds for normal subgroups of G and G/N as well, and is formalized in the lattice theorem[?].
		Several important properties of factor groups are recorded in the fundamental theorem on homomorphisms and the isomorphism theorems.
	www.ebroadcast.com.au /lookup/encyclopedia/fa/Factor_group.html (553 words)

	Fundamental theorem on homomorphisms: Definition and Links by Encyclopedian.com (via CobWeb/3.1 planetlab2.cs.unc.edu) (Site not responding. Last check: 2007-10-11)
		For some algebraic structures the fundamental theorem on homomorphisms relates the structure of two objects between which a homomorphism is given, and of the kernel and image of the homomorphism.
		Let G and H be groups; let f : G->H be a group homomorphism; let K be the kernel of f; let φ be the natural surjective homomorphism G->G/K.
		Similar theorems are valid for vector spaces, modules, and rings.
	www.encyclopedian.com.cob-web.org:8888 /fu/Fundamental-theorem-on-homomorphism.html (176 words)

	Algebraic Topology: Homotopy
		Then the fundamental group of X is generated by (the images of) the fundamental groups of A and B.
		Theorem The group operation on X induces a group operation on P(X;x) that coincides with the old group operation, and P(X;x) is commutative.
		The fundamental group of SO(2,R), the group of orthogonal transformations of determinant 1 of the real plane, is isomorphic to Z, the additive group of the integers.
	www.win.tue.nl /~aeb/at/algtop-3.html (2011 words)

	Fundamental theorem - Wikipedia, the free encyclopedia
		The names are mostly traditional; so that for example the fundamental theorem of arithmetic is basic to what would now be called number theory.
		Theorems may be called fundamental because they are results from which further, more complicated theorems follow, without reaching back to axioms.
		The mathematical literature will sometimes refer to the fundamental lemma of a field; this is often, but not always, the same as the fundamental theorem of that field.
	en.wikipedia.org /wiki/Fundamental_theorem (161 words)

	GW Bulletin
		The impossibility theorems of Balinsky and Young and of Arrow.
		Divisibility of integers, prime numbers, greatest common divisor, the Euclidean algorithm, congruence, the Chinese remainder theorem, number theoretic functions, M bius inversion, Euler's phi function, primitive roots and indices, and applications to cryptography and primality testing.
		Fundamental concepts, techniques, and results of graph theory, including applications to operations research, computer science, chemistry, and the social sciences.
	www.gwu.edu /~bulletin/ugrad/math.html (1186 words)

	Course Information
		This course is intended for education majors and is designed to provide a mathematical treatment of the fundamental concepts of arithmetic, algebra, and number theory as they relate to the elementary school mathematics curriculum.
		Topics studied are limits and continuity; derivatives and antiderivatives of the algebraic and trigonometric functions; the definite integral; and the fundamental theorem of the calculus.
		Topics include probablility definitions and theorems; discrete and continuous random variables including the binomial, hypergeometric, Poisson and normal random variables; and the applications of probability to such statistical topics as sampling distrubutions, point estimation, confidence intervals, and/or tests of hypothesis.
	www.math.geneseo.edu /dept/courses.html (1314 words)

	MIT OpenCourseWare \| Mathematics \| 18.904 Seminar in Topology, Fall 2005 \| Calendar
		Use the computations of the fundamental group of the torus and projective plane to compute the fundamental groups of the rest of the closed surfaces.
		Explain the action of the fundamental group on the fibers of a covering space, and identify the "isotropy" of this action.
		Show that the fundamental group of a graph is free, and if there is time, discuss the number of generators.
	ocw.mit.edu /OcwWeb/Mathematics/18-904Fall-2005/Calendar/index.htm (1895 words)

	Springer Online Reference Works
		On the basis of this theorem, A.I. Mal'tsev created a method of proof of local theorems in algebra (see Mal'tsev local theorems).
		Central in the theory of ultraproducts is the theorem of J.
		The Gödel compactness theorem and the Löwenheim–Skolem theorem are in the Russian literature sometimes known as the Gödel–Mal'tsev theorem and the Löwenheim–Skolem–Mal'tsev theorem, respectively.
	eom.springer.de /m/m064390.htm (1345 words)

	Quotient group (Site not responding. Last check: 2007-10-11)
		If G is the group of invertible 3×3 matrices and N is the subgroup of 3×3 real 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 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.
		Several important properties of quotient groups are in the fundamental theorem on homomorphisms and the isomorphism theorems.
	www.freeglossary.com /Factor_group (1002 words)

	IMACS (Site not responding. Last check: 2007-10-11)
		Groups and subgroups; symmetric groups; cycle notation; Lagrange's Theorem; permutation representations of groups; generators and generated subgroups; the Sylow Theorems; a classification of all finite Abelian groups; normal subgroups; quotient groups; the Isomorphism Theorems; simple groups; solvable groups; the Jordan-Hlder Theorem.
		Differentiability; the Linear Approximation Theorem; properties of derivatives; the calculus within its historical context; the Mean Value Theorem for Derivatives; curve sketching; the chain rule; parametric representation of relations; various forms of l'Hpital's rule; Cauchy's Mean Value Theorem; implicit differentiation; antiderivatives.
		The axiom of choice; the Hausdorff Chain Theorem; Zorn's lemma; the Well-Ordering Theorem; the principle of transfinite induction; Bourbaki's Theorem; transfinite recursion; ordinal numbers; cardinal numbers; a discussion of the continuum hypothesis; the Fundamental Theorem of Cardinal Arithmetic.
	imacs.org /IMACSWeb/default.aspx?page=Mathematics (1391 words)

	Supplementary ProofPower Examples
		This is followed by the definitions of the concepts of group, homomorphism between groups, subgroup, normal subgroup, kernel of a homomorphism, congruence modulo a subgroup, coset of a subgroup, and quotient group.
		The theorems include: more facts about finiteness and the size of finite sets; algebraic properties of indexed sums; induction over finitely-supported functions; the inclusion/exclusion principle; the binomial coefficients and their basic properties, including the formula for the number of combinations and the binomial theorem; Bertrand's ballot problem.
		The theorem and its proof are presented as a Z specification and a series of Z conjectures all of which have been mechanically verified using the ProofPower system.
	www.lemma-one.com /ProofPower/examples/examples.html (964 words)

	The Van Kampen Theorem
		We have sketched a proof in the tutorials of the fact that the fundamental group of a wedge of two pointed spaces is the free product of the respective fundamental groups.
		The next step is to compute the homomorphism induced by inclusion of the circle as the boundary of the respective punctured surface.
		Later on we will consider the classification theorem for closed surfaces and will in particular be able to present a calculation using the Van Kampen theorem of all possible fundamental groups of closed surfaces.
	www.maths.abdn.ac.uk /~ran/mx4509/mx4509-notes/node22.html (1284 words)

	Centre College Course Offerings: Mathematics
		The majority of the course focuses on the definition of the integral, the Fundamental Theorem of Calculus, and applications of the integral.
		An introduction to single variable calculus reviewing the real number line, inequalities and absolute value, and discussing functions and graphing, limits, continuity, the derivative, rules of differentiation, the Mean Value Theorem, applications of the derivative, antiderivatives, Riemann sums and the definite integral, the Fundamental Theorem of Calculus, and applications of the integral.
		MAT 331 Abstract Algebra-II A continuation of MAT 330, in which key properties from the integers and the real numbers are used as models for the algebraic structures known as rings and fields.
	www.centre.edu /web/academic/coursecatalog/matcc.html (1547 words)

	Fundamental theorem on homomorphisms... (via CobWeb/3.1 planetlab2.cs.unc.edu) (Site not responding. Last check: 2007-10-11)
		In abstract algebra, for a number of algebraic structures, the fundamental theorem on homomorphisms relates the structure of two objects between which a homomorphism is given, and of the kernel of a homomorphism and image of the homomorphism.
		The homorphism theorem is used to prove the isomorphism theorem s.
		Similar theorems are valid for monoid s, vector space s, module (mathematics), and ring (mathematics).
	vvikipedia.biz.cob-web.org:8888 /8998 (185 words)

	The Catholic University of America - Mathematics Department
		Riemann-Stieltjes integral; equicontinuous families of functions and Arzela-Ascoli theorem; Tietze's extension theorem; Baire category theorem; differentiation and integration of a function of several variables; fixed point theorem; implicit function theorem; inverse function theorem; existence and uniqueness theorems for ordinary differential equations.
		Mean value theorem for Banach space-valued piecewise differentiable functions, differentiable and Lipschitzian operators, construction of epsilon-approximate solution to a differential equation, existence and uniqueness of local solutions, extensions to a maximal solution.
		Linear mappings, Banach's homomorphism theorem, uniform boundedness and the Banach-Steinhaus theorem, duality, dual systems and weak topologies; strong dual, bi-dual, and reflexive spaces; theorems of Grothendieck, weak compactness, open mappings, and closed graph theorems; linear manifolds and applications.
	math.cua.edu /courses.cfm (2191 words)

	MATHEMATICS
		Culminates in the theorems of Green and Stokes, along with the Divergence Theorem.
		Homomorphisms, normal subgroups, quotient groups, and the fundamental isomorphism theorems.
		First quarter of a three-quarter sequence covering Weierstrass preparation theorem and its immediate consequences, analytic continuation, domains of holomorphy, pseudoconvexity, Cartan-Oka theory of coherence, embedding theorems; the CR equations, CR manifolds, connections with algebraic geometry.
	www.washington.edu /students/crscat/math.html (5102 words)

	Fundamental Theorem of Isomorphism
		The Fundamental Theorem of Homomorphisms (also known as the First Isomorphism Theorem) states that
		The Fundamental theorem provides a different way to think about the problem.
		Then if H contains a subgroup isomorphic to that factor group there is a homomorphism from G to H.
	www.math.csusb.edu /notes/advanced/algebra/gp/node20.html (271 words)

	Van Wyk's 431 Sections
		Making the set of cosets of a normal subgroup into a group, the "fundamental homomorphism theorem", automorphisms, inner automorphisms, conjugate elements, conjugate subgroups.
		Ring homomorphisms (again) and what they preserve, kernels of such, ideals, quotient rings, the "fundamental homomorphism theorem" for rings.
		Extension fields, Kronecker's Theorem, algebraic and transcendental elements over a field (special case: algebraic and transcendental numbers), monic polynomials, irreducible (monic) polynomial for an element over a field, simple extensions.
	www.math.jmu.edu /~vanwyk/courses/431/hw.html (864 words)

	Mathematics Descriptions (Site not responding. Last check: 2007-10-11)
		Emphasis will be placed on structure theorems such as the fundamental theorem of group homomorphisms and will use the sophistication developed in the prerequisites.
		The standard topics in logic from the statement calculus (primitive statements, compound statements, truths tables, valid statements, rules of inference, and proof) and from the restricted predicate calculus (quantifiers, variables, rules of inference, and proofs using quantified statements) are developed.
		Applications of logic to abstract deductive systems, theorems in geometry, number theory, and algebra are given.
	www.ship.edu /~mathcs/matdes.html (1309 words)

	American International College
		This course covers an in-depth analysis of the fundamental properties of the real number system, including the completeness property, sequences, limits and continuity, differentiation through the Mean Value Theorem, and the Riemann integral.
		The material on groups includes subgroups, group homomorphisms and factor groups as well as the fundamental group homomorphism theorem.
		Fermat’s Little Theorem and Euler’s Theorem are used to simplify congruences in the integers.
	www.aic.edu /pages/544.html (1456 words)

	Course Descriptions
		Limits and continuity of functions; the concept of derivative; calculating derivatives; applications of derivatives such as optimization and related rates; integration through the Fundamental Theorem.
		an exploration of fundamental concepts involving natural numbers, integers, rational numbers, real numbers, and complex numbers, and their operations.
		Topics covered include combinatorics, basic axioms and theorems, random variables and probability distributions, expectation, moment, moment generating functions, and functions of random variables.
	cas.bellarmine.edu /mathematics/CourseDescriptions.asp (1738 words)

	Solvable Groups, Double Cosets and Isomorphism Theorems
		Here, for the most part, the theorems established will be introductory, and will subsequently be used to establish much deeper theorems.
		So, for example, what is called the second isomorphism theorem here may be called the third isomorphism theorem in another text.
		Also some authors include what we have called the Fundamental Homomorphism Theorem (Theorem 7.1.8) as one of the isomorphism theorems.
	web.usna.navy.mil /~wdj/tonybook/gpthry/node41.html (308 words)

	UIC Graduate College -- Courses: Mathematics (Site not responding. Last check: 2007-10-11)
		Properties of Cartesian n-space the derivative, inverse and implicit function theorems, extrema, line integrals, vector calculus theorems, change of variables, differential forms, generalized Stokes's theorem.
		The fundamental group and its applications, covering spaces, classification of compact surfaces, introduction to homology, development of singular homology theory, applications of homology.
		Cohomology theory, universal coefficient theorems, cohomology products and their applications, orientation and duality for manifolds, homotopy groups and fibrations, the Hurewicz theorem, selected topics.
	www.uic.edu /depts/grad/courses/math.shtml (2065 words)

Try your search on: Qwika (all wikis)