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Topic: Theorem of Lagrange

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Coset

Conjugacy class

	Joseph Louis Lagrange Summary
		Lagrange did not explicitly recognize groups, but he obtained implicitly some of the simpler properties, including the theorem known after him, which states that the order of a subgroup is a divisor of the order of the group.
		Lagrange did not regard the principle as an axiom but rather as a general expression of the law of equilibrium deduced from the laws of the lever and the composition of forces or, alternatively, from the properties of strings and pulleys.
		Lagrange proved to be a mathematical prodigy; he was teaching geometry at the Royal Artillery School in Turin at the age of eighteen, and he established the Turin Academy of Sciences in 1758.
	www.bookrags.com /Joseph_Louis_Lagrange (7560 words)

	Lagrange's theorem (group theory) Summary
		Lagrange's theorem is one of the fundamental theorems of finite group theory.
		Lagrange, who was prolific in his career studying mathematics and physics, had a special talent for number theory and developed several theories dealing with numbers and their relations.
		Lagrange's theorem, in the mathematics of group theory, states that if G is a finite group and H is a subgroup of G, then the order (that is, the number of elements) of H divides the order of G.
	www.bookrags.com /Lagrange's_theorem_(group_theory) (623 words)

	Joseph Louis Lagrange
		Lagrange did not seek fame and saw no reason to come to Prussia while Euler, who he had much respect for, already had the position of director of mathematics at the Berlin Academy.
		Lagrange's work, which included subjects such as the calculus of variations, probabilities, the principle of least action, kinetic energy, and propagation of sound, appears in the first three volumes, published in 1759, '62, and '66.
		In 1764, Lagrange won the prize competition from the Académie des Sciences in Paris on the subject of the libration, or "wobble", of the moon.
	numericalmethods.eng.usf.edu /anecdotes/lagrange.html (971 words)

	PlanetMath: calculus of subgroup orders
		The theorem of Lagrange is the first of many basic theorems on the calculus of indices of subgroups and it can be stated as follows:
		As the proof of (2) can be written with bijections instead of specific integer values, the proof of (1) is immediate from the usual proof of the Lagrange's theorem.
		This is to allow for infinite indices, as we can multiply cardinal numbers, but we may not always be able to make sense of cardinal number division.
	planetmath.org /encyclopedia/TheoremOfLagrange.html (485 words)

	PlanetMath: Lagrange's theorem
		See Also: group, proof of Fermat's little theorem using Lagrange's theorem, proof of Euler-Fermat theorem using Lagrange's theorem
		proof of the converse of Lagrange's theorem for finite cyclic groups
		This is version 2 of Lagrange's theorem, born on 2002-01-23, modified 2002-02-02.
	planetmath.org /encyclopedia/LagrangesTheorem.html (68 words)

	Lagrange inversion theorem - Wikipedia, the free encyclopedia
		In mathematical analysis, the Lagrange inversion theorem, also known as the Lagrange-Bürmann formula, gives the Taylor series expansion of the inverse function of an analytic function.
		The theorem was proved by Lagrange and generalized by Bürmann, both in the late 18th century.
		There is a straightforward derivation using complex analysis and contour integration (the complex formal power series version is clearly a consequence of knowing the formula for polynomials, so the theory of analytic functions may be applied).
	en.wikipedia.org /wiki/Lagrange_inversion_theorem (457 words)

	The Cellular Automata pages
		Lagrange’s Four-square Theorem states that any positive integer n can be written as the sum of four squares.
		Bezout’s theorem is simple enough when you state it: Two polynomial curves of degree m and n respectively intersect in nm points.
		Of course, Godel’s Theorem is never proved but there’s a nice demonstration of a non-computable function (marred by typographical accidents) and an important link to a translation of Godel’s original paper.
	www.math.uwaterloo.ca /navigation/ideas/reviews/theoremmonth.shtml (359 words)

	The Nichols-Zoeller Theorem (Site not responding. Last check: 2007-11-03)
		Students of Group Theory are familiar with Lagrange's Theorem, which states that the order of a finite group is divisible by the orders of each of its subgroups.
		The most important consequence of what has become known as the Nichols-Zoeller Theorem is the analogue for Hopf algebras: every finite-dimensional Hopf algebra is free as module over each of its Hopf subalgbras.
		Lagrange's Theorem is a special case, since group algebras are Hopf algebras.
	www.math.fsu.edu /~nichols/hp/nicholszoellertheorem.html (187 words)

	Cosets and Lagrange's Theorem (Site not responding. Last check: 2007-11-03)
		This establishes the following extremely important theorem in the theory of finite groups.
		Theorem 4.3.1 (Lagrange's Theorem) The order of a subgroup of a finite group is a divisor of the order of the group.
		We end this section with an application of Lagrange's theorem, in particular of the first corollary of this theorem, to number theory.
	web.usna.navy.mil /~wdj/tonybook/gpthry/node22.html (643 words)

	Constrained Optimization
		The latter result in the theorem says that the Hessian of the Lagrangian evaluated at its stationary points is non-negative definite with respect to all vectors orthogonal to the gradient of the constraint.
		If it is not real, we cannot use the theorem that permits computation of stationary points by computing the gradient with respect to z alone.
		To prove the theorem, not only does the gradient with respect to x need to be considered, but also with respect to the vector s of slack variables.
	cnx.org /content/m11223/latest (1485 words)

	Fermat's Last Theorem: Johann Dirichlet
		The purpose of this blog is to present the story behind Fermat's Last Theorem and Wiles' proof in a way accessible to the mathematical amateur.
		Johann Peter Gustav Lejeune Dirichlet was born on February 13, 1805 in Duren which at the time was part of Napoleon's empire.
		Despite his fame from his result with Fermat's Last Theorem, his appointment at the University of Breslau created a great controversy among the German math professors.
	fermatslasttheorem.blogspot.com /2005/10/johann-dirichlet.html (667 words)

	AMLEV Chapter 2
		When proving Lagrange‑Dirichlet theorem these peculiarities of a solid body (in our case, levitator) were considered: a real‑conservative system comprising bodies of different configurations was substituted by a system of mass points.
		Therefore in order to employ Lagrange‑Dirichlet theorem for creating a stable suspension of real bodies of definite shapes, the equivalent force must be expanded into components in such a way to satisfy all necessary equations for static body equilibrium.
		According to this theorem a conservative system has a stable equilibrium if its potential energy has a local (i.e., not coinciding with boundaries of magnetized bodies) minimum.
	www.amlevtrans.com /AMLEV-Publish/AMLEV-Publish-Chapter-2.htm (11849 words)

	OpenCollege e-Learning Content Library (Site not responding. Last check: 2007-11-03)
		Theorem 4 (The test for parallelism of a line to a plane)
		Theorem 1 (Test for perpendicularity of a st. line to a plane)
		Theorem 8 (The test for perpendicularity of a plane to a plane)
	www.opencollege.com /simsim/php/CatalogManager.php?reset=1 (217 words)

	Lagrange Polynomials
		The above algorithm is sufficient for understanding and/or constructing the Lagrange polynomial.
		Investigate the error for the Lagrange polynomial approximations of degree n = 2, 3, 4, and 5 in Example 2.
		Summary of the maximum error and the error bounds over the interval [0, 1] that were found in Example 3.
	math.fullerton.edu /mathews/n2003/LagrangePolyMod.html (225 words)

	Theorem Environments
		In LaTeX, one can create `environments' for statements of theorems, lemmas, propositions, corollaries, etc., and also for proofs, definitions, examples and remarks.
		The theorem will be labelled and numbered by LaTeX, and the statement of the theorem with be automatically italicised.
		We conclude that the subgroup~$H$ and the left coset $xH$ both have $m$ elements, as required.\qed \end{proof} \begin{theorem} \emph{(Lagrange's Theorem)} \label{Lagrange} Let $G$ be a finite group, and let $H$ be a subgroup of $G$.
	www.maths.tcd.ie /~dwilkins/LaTeXPrimer/Theorems.html (688 words)

	Introduction to Remainder & Factor Theorem
		Factor theorem is just a special case of remainder theorem.
		It is used to solve equations of higher power, find the remainder when one polynomial is divided by another and factorise equations.
		And then, one of the way to solve such equations is by applying the concept of remainder and factor theorem.
	library.thinkquest.org /C0110248/algebra/remfactintro.htm (86 words)

	Existence: Lagrange's four square theorem
		Fermat conjectured that every integer could be written as a sum of at most four squares.
		Lagrange managed to prove this conjecture to hold for all positive integers.
		Before we finalize our statement of Lagrange's Theorem, we make a simple observation.
	www.math.ohio-state.edu /~econrad/Jacobi/sumofsq/node1.html (197 words)

	[No title] (Site not responding. Last check: 2007-11-03)
		It is a famous theorem of Lagrange that every positive integer is a sum of 4 perfect squares (some of which may be 0), but more subtle is the case of 3 squares because not every positive integer can be expressed in this form.
		It is a hard theorem of Legendre that classifies exactly which integers are a sum of 3 perfect squares.
		Often in number theory the problem of studying integer solutions to an equation lies much deeper than the study of rational solutions, as the case of rational solutions is often more susceptible to geometric methods.
	www.math.columbia.edu /~ums/abstracts/Fal06Conrad.htm (172 words)

	Amazon.com: A First Course in Optimization Theory: Books: Rangarajan K. Sundaram (Site not responding. Last check: 2007-11-03)
		Lagrange multipliers and how they are used in constrained optimization problems are effectively discussed in Chapter 5.
		Whenever a theorem is stated different examples are given to emphasize the points.
		For example when stating the Lagrange Theorem and Kuhn-Tucker theorem the author points out when the theorems fail and gives detailed examples to illustrate the ideas.
	www.amazon.com /First-Course-Optimization-Theory/dp/0521497701 (1791 words)

	Representations of integers by quadratic forms by Bhargava
		One example is provided by the "Four Squares Theorem" of Lagrange, from 1770: any (positive) integer can be expressed as the sum of four squares, as in the quadratic form given above.
		But the proof of this so-called "Fifteen Theorem" was rather intricate, and as a result it was never written up or published.
		He even showed that, given any subset of integers--say, the primes--there is such a finite test to check whether a quadratic form represents that subset.
	www.ams.org /ams/bhargava.html (339 words)

	Doug, Basho, and Axel Thu
		My notebook was opened to Lagrange's Four Squares Theorem, a very nice theorem that states that any number can be written as the sum of four squares.
		Whenever I did it, it kept coming out "Akthel Thuth Theorem." I would go to bed moaning, "I know I'm gonna blow it I know I'm gonna blow it." I wanted the Lagrange Proof to be the Platonic Ideal - the quintessence of the Perfect Answer, and I was going to lisp.
		I DID get cut off while trying to invoke his theorem, and I DID have to prove the La Grange Four Squares theorem, so those parts are true...
	www.dougshaw.com /essays/thu.html (1669 words)

	Proof of Fermat's Little Theorem
		Here we are concerned with his "little" but perhaps his most used theorem which he stated in a letter to Fre'nicle on 18 October 1640:
		As usual Fermat did not provide a proof (this time saying "I would send you the demonstration, if I did not fear its being too long" [Burton80, p79]).
		If p does not divide a, then we need only multiply the congruence in Fermat's Little Theorem by a to complete the proof.
	primes.utm.edu /notes/proofs/FermatsLittleTheorem.html (260 words)

	History
		1813 - Karl Friedrich Gauss rediscovers the divergence theorem of Lagrange.
		Green's theorems are given, as well as the divergence theorem (Gauss's law), but Green doesn't know of the work of Lagrange and Gauss and only references Priestly's deduction of the inverse square law from Franklin's experimental work on the charging of hollow vessels.
		Later Stokes assigns the proof of this theorem as part of the examination for the Smith's Prize.
	maxwell.byu.edu /~spencerr/phys442/node4.html (6551 words)

	Euler's Totient Theorem (Site not responding. Last check: 2007-11-03)
		I'll see if I can come up with a convincing proof not using groups; but to be honest I'd suggest you read some basic group theory - subgroups, cosets, Lagrange's theorem - and go back and look at it again afterwards.
		The proof that primes of the form 4k+3 can be expressed as the sum of four squares uses Fermat's Little Theorem.
		A bunch of Ring Theory theorems and definitions.
	mcraefamily.com /mathhelp/BasicNumberCoprimesTotientTheoremEuler.htm (576 words)

	Subsets of a Group and Lagrange's Theorem (Site not responding. Last check: 2007-11-03)
		In this chapter, we establish one of the most important theorems in finite group theory, i.e., Lagrange's Theorem.
		This theorem gives a relationship between the order of a finite group and the order of any subgroup (in particular, if
		In order to establish Lagrange's theorem we first investigate subsets of a group and partitions of the group with respect to these subsets.
	web.usna.navy.mil /~wdj/tonybook/gpthry/node17.html (62 words)

	Cyclic subgroups, Cauchy's Theorem.
		The fact that the inverse of an element is in the set of the successive powers of an element, as we saw in the page about cyclic
		Lagrange's theorem tells us that the order of any cyclic subgroup of a group of order n, as any subgroup, will divide the
		Cauchy's theorem : In a finite group of order n will there will exist elements, and cyclic subgroups,
	hemsidor.torget.se /users/m/mauritz/math/alg/cyclsub.htm (320 words)

	Films for the Humanities and Sciences - The Counting Theorem (Site not responding. Last check: 2007-11-03)
		This program explores the orbit-stabilizer theorem in group theory, showing how this leads to the counting theorem, which is used to solve counting problems.
		This program investigates the isomorphism theorem in group theory, showing how the theorem can be used to identify quotient groups by finding suitable homomorphisms.
		By using a variety of graphics sequences, animations, and three-dimensional models, a geometric characterization of conjugacy is established.
	www.films.com /id/8607/The_Counting_Theorem.htm (199 words)

	Differentiation of functions,limits (II), maximum, minimum, inflection points.
		The theorem says that there is a suitable point R on the curve C such that the tangent line in R is parallel to PQ.
		It is not difficult to verify that the h(x) satisfies the three conditions of Rolle's theorem.
		On this function, we can use Lagrange's theorem.
	www.ping.be /~ping1339/diff.htm (2811 words)

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