Lagrange's theorem (group theory)

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	PlanetMath: group
		See Also: subgroup, cyclic group, simple group, symmetric group, free group, ring, field, group homomorphism, Lagrange's theorem, identity element, proper subgroup, groupoid, fundamental group, topological group (obsolete), Lie group, Proof: The orbit of any element of a group is a subgroup, locally cyclic group, existence of Hilbert class field, abelian group,
		(Group theory and generalizations :: General reference works)
		Groups often arise as the symmetry groups of other mathematical objects; the study of such situations uses group actions.
	planetmath.org /encyclopedia/Group.html (311 words)

	PlanetMath: group
		See Also: subgroup, cyclic group, symmetric group, free group, ring, field, group homomorphism, Lagrange's theorem, identity element, proper subgroup, groupoid, fundamental group, topological group, Lie group, Proof: The orbit of any element of a group is a subgroup, locally cyclic group, existence of Hilbert class field, abelian group,
		(Group theory and generalizations :: General reference works)
		Groups often arise as the symmetry groups of other mathematical objects; the study of such situations uses group actions.
	planetmath.org /encyclopedia/Group.html (283 words)

	PlanetMath: group
		See Also: subgroup, cyclic group, simple group, symmetric group, free group, ring, field, group homomorphism, Lagrange's theorem, identity element, proper subgroup, groupoid, fundamental group, topological group, Lie group, Proof: The orbit of any element of a group is a subgroup, locally cyclic group, existence of Hilbert class field, abelian group,
		(Group theory and generalizations :: General reference works)
		Groups often arise as the symmetry groups of other mathematical objects; the study of such situations uses group actions.
	planetmath.org /encyclopedia/Group.html (277 words)

	PlanetMath: group
		See Also: subgroup, cyclic group, simple group, symmetric group, free group, ring, field, group homomorphism, Lagrange's theorem, identity element, proper subgroup, groupoid, fundamental group, topological group, Lie group, Proof: The orbit of any element of a group is a subgroup, locally cyclic group, existence of Hilbert class field, abelian group,
		(Group theory and generalizations :: General reference works)
		Groups often arise as the symmetry groups of other mathematical objects; the study of such situations uses group actions.
	planetmath.org /encyclopedia/Group.html (277 words)

	Sylow theorem
		In mathematics, especially group theory, the Sylow theorems, named after Ludwig Sylow, form a partial converse to Lagrange's theorem, which states that if H is a subgroup of a finite group G, then the order of H divides the order of G.
		The group G acts on itself or on the set of its p-subgroups in various ways, and each such action can be exploited to prove one of the Sylow theorems.
		The proofs of the Sylow theorems exploit the notion of group action in various creative ways.
	www.toshare.info /en/Sylow.htm (1053 words)

	Advanced Algebra
		In particular, you should be familiar with the following concepts and theorems in group theory: group, subgroup, order of an element, cyclic group, Lagrange's theorem, homomorphism, normal subgroup, factor group, homomorphism and isomorphism theorems, symmetric and alternating groups, direct product.
		Text: Concerning group theory we shall use D. Robinson, A Course in the Theory of Groups and depending on the pace of the course we might cover also parts of M. Isaacs, Character Theory of Finite Groups.
		Two sample theorems (the second of which requires representation theory):
	www.stolaf.edu /people/budapest/WebPages/course_AAL.html (279 words)

	RUMEC APOS Theory Glossary
		This is followed by a proof of Lagrange's theorem in a group discussion format.
		In abstract algebra a student has a trans level of development for a group schema when the student is guided by the definition or relevant theorems in determining whether a certain subset of an abstract group is also a group under the same binary operation.
		Understanding a coset in group theory only as a set of calculations that are actually performed to obtain a definite set is an action conception.
	www.cs.gsu.edu /~rumec/Papers/glossary.html (279 words)

	Cosets and Lagrange's Theorem
		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.
		This establishes the following extremely important theorem in the theory of finite groups.
	www.ew.usna.edu /~wdj/tonybook/gpthry/node22.html (643 words)

	Mathematical Groups
		The following are the most important concepts and theorems in undergraduate group theory: equivalence classes, conjugation, Lagrange's Theorem, the Class Equation, Cauchy's Theorem.
		This chapter deals with the most important concepts and theorems in undergraduate group theory as I see them.
		A mathematical group is a nonempty set S with some simple structure added to it.
	www.ajnpx.com /html/Groups.html (294 words)

	Joseph-Louis Lagrange
		Lagrange also established the theory of differential equations, and provided many new solutions and theorems in number theory, including Wilson's theorem.
		Lagrange's classic Theorie des fonctions analytiques laid some of the foundations of group theory, anticipating Galois.
		Lagrange succeeded Euler as the director of the Berlin Academy.
	lagrange-bio.net (165 words)

	Sylow theorems - Wikipedia
		The Sylow theorems of group theory form a partial converse to the theorem of Lagrange, which states that if H is a subgroup of a finite group G, then the order of H divides the order of G.
		The Sylow theorems guarantee, for certain divisors of the order of G, the existance of corresponding subgroups:
		Let p be a prime number and write the order of G as p
	nostalgia.wikipedia.org /wiki/Sylow_theorems (229 words)

	Subsets of a Group and Lagrange's Theorem
		In order to establish Lagrange's theorem we first investigate subsets of a group and partitions of the group with respect to these subsets.
		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
	www.ew.usna.edu /~wdj/tonybook/gpthry/node17.html (62 words)

	Contents
		Subjects that will be discussed are: induced representations, Mackey's imprimitivity theorem, Heisenberg group, Stone-von Neumann theorem, highest weight theory and Borel-Weil description of representations of compact groups, induced representations of non-compact semisimple groups, Harish-Chandra modules, representations of the principal and the discrete series of SL(2, R).
		The theory then leads to the structure theory of compact Lie groups and their Lie algebras, including the description of the maximal tori, roots and root spaces, the Weyl group and Weyl's integration theorem.
		The first case is the setting of classical mechanics with a symmetry group; quantization of such a situation leads to questions of representation theory of K and spectral theory of differental operators commuting with (spectral degeneration).
	www.math.uu.nl /people/kolk/SpringSchool2004/contents.html (62 words)

	Errata for Adventures in Group Theory
		In the form given in the book, Lagrange's theorem (for permutation groups) was probably known to Galois and appeared in a work of Serret and of Jordan.
		It is worth remarking that, contrary to what might be suggested by the name of the result, Lagrange's theorem, in the forms stated here, is not due to Lagrange.
		Generally, if the table says "semi-direct product of A with B (with B normal)", it should read "semi-direct product of A with B (with A normal)".
	web.usna.navy.mil /~wdj/cubebook_errata.html (62 words)

	Errata for Adventures in Group Theory
		In the form given in the book, Lagrange's theorem (for permutation groups) was probably known to Galois and appeared in a work of Serret and of Jordan.
		It is worth remarking that, contrary to what might be suggested by the name of the result, Lagrange's theorem, in the forms stated here, is not due to Lagrange.
		Page 117, lines 2-3: Not a typo but there is no need to assume that G is a permutation group to define a Cayley graph.
	web.usna.navy.mil /~wdj/cubebook_errata.html (62 words)

	RUMEC APOS Theory Glossary
		This is followed by a proof of Lagrange's theorem in a group discussion format.
		In order to understand the proof of Lagrange's Theorem in abstract algebra, one must have an object conception of the set of equivalence classes.
		Understanding a coset in group theory only as a set of calculations that are actually performed to obtain a definite set is an action conception.
	www.cs.gsu.edu /~rumec/Papers/glossary.html (3991 words)

	E. Noether's Discovery of the Deep Connection Between Symmetries and Conservation Laws
		For theories whose symmetry group is an infinite continuous group, the main results of theorem II are that there are certain identities, or "dependencies" as she called them [1], between Lagrange functions of the theory and their derivatives.
		This is quite different from the results of theorem I that for each infinitesimal generator of the finite continuous group there is a quantity whose divergence vanishes when the Lagrange functions vanish (field equations are satisfied).
		Noether's theorem 11 applies in the case of general relativity and one sees that she has proved Hilbert's assertion that in this case one has `improper energy theorems', and that this is a characteristic feature of the theory.
	www.physics.ucla.edu /~cwp/articles/noether.asg/noether.html (5320 words)

	Business Fresh : Article 'Lagrange's theorem'
		Every group can, therefore, be embedded in the symmetric group on its own elements, Sym(G) — a result known as Cayley's theorem.
		An understanding of group theory is also important in the physical sciences.
		Has a proof of the Lagrange theorem, accessible to high school students.de:Waringsches Problem
	www.business-fresh.net /DisplayArticle242050.html (577 words)

	E. Noether's Discovery of the Deep Connection Between Symmetries and Conservation Laws
		For theories whose symmetry group is an infinite continuous group, the main results of theorem II are that there are certain identities, or "dependencies" as she called them [1], between Lagrange functions of the theory and their derivatives.
		The symmetry group of the theory, is a gauge group.
		It is owing to the fact that the theory is a gauge theory; i.e., that it has an infinite continuous group of symmetries of which time translations are a subgroup.
	www.physics.ucla.edu /~cwp/articles/noether.asg/noether.html (5320 words)

	PlanetMath: alternating group is a normal subgroup of the symmetric group
		Cross-references: Lagrange's theorem, first isomorphism theorem, domain, kernel, odd permutation, even permutation, epimorphism, symmetric group, normal subgroup, alternating group
		(Group theory and generalizations :: General reference works)
		This is version 2 of alternating group is a normal subgroup of the symmetric group, born on 2003-06-23, modified 2004-04-30.
	planetmath.org /encyclopedia/AlternatingGroupIsANormalSubgroupOfTheSymmetricGroup.html (108 words)

	Amazon.ca: Books: Galois Theory
		Some antecedents of Galois theory in the works of Gauss, Lagrange, Vandemonde, Newton, and even the ancient Babylonians, are explained in order to put Galois' main ideas in their historical setting.
		Other books would give a succession of "theorem-proofs" that eventually proved the Galois solvability theorem but when I was finished, I still could not intuitively understand how the solvability of Galois groups to corresponds to solvability by radicals.
		I disagree with the author that the original text of Galois is the best way to learn the theory, I found it considerably less readable than Artin's book, and it placed Galois theory too much in its historical context of solution by radicals.
	www.amazon.ca /exec/obidos/ASIN/038790980X (108 words)

	MC449 Galois Theory
		The main theorem of Galois theory is one of the most beautiful theorems in all of mathematics, and extensions and applications of Galois theory are the subject of major research activities in algebra, geometry and analysis.
		Lagrange resolvents and the proof that a radical normal extension has a solvable Galois group.
		Galois theory is one of the first examples of methods from one branch of Mathematics being applied to solve problems in an apparently completely different area.
	www.mcs.le.ac.uk /Modules/Modules01-02/node76.html (108 words)

	Modular Arithmetic, Fermat Theorem, Carmichael Numbers - Numericana
		Lagrange's Theorem (arguably the first great result of Group Theory) states that the order of any subgroup divides the order of the whole group.
		This fact may be used to prove the very important Little Theorem of Fermat presented in the next article, and it suggests a generalization due to Euler.
		Modular arithmetic: The algebra of congruences, formally introduced by Gauss.
	home.att.net /~numericana/answer/modular.htm (108 words)

	Mathematics
		Selected topics for the following fields: graph theory (planar graphs, Euler formula, connectivity), topology of surfaces (triangulation, genus), geometry of surfaces (geodesics, minimal surfaces, spherical triangles), group theory (abstract groups, Lagrange theorem, generators).
		Existence of the algebraic closure of a field, and additional topics: "constructions with straightedge and compass", the Fundamental Theorem on symmetric polynomials, norm, and trace in finite extensions, separability and trace form Kummer theory.
		Cyclotomic extensions, realization of abelian groups as Galois groups over the rational number field.
	ug.technion.ac.il /Catalog/CatalogEng/01002086.html (304 words)

	Theorems of Abstract Algebra
		OK, the basic plan in undergraduate group theory is to lay a foundation for Lagrange's Theorem and then present a number of important partial converses to it.
		PCL3 (Cauchy's Theorem): For any (finite) group G, if p is a prime number that divides the order of G, then G has an element of order p.
		As an immediate Corollary to the last theorem, we have that G has a Sylow-p subgroup.
	www.ajnpx.com /html/Theorems.html (851 words)

	UnmathsCourses.htm
		Review of basic group theory including Lagrange's Theorem.
		Variational principles and perturbation theory: the Courant minimax theorem, Weyl's inequalities, Gershgorin's theorem, perturbations of the spectrum, vector norms and related matrix norms, the condition number of a matrix.
		Groups, subgroups, symmetric groups, cyclic groups and order of an element, isomorphisms, cosets and Lagrange's Theorem.
	www.kfupm.edu.sa /math/Ungraduate/UnmathsCourses.htm (879 words)

	MTH-2A23 : Algebra I
		Lagrange's Theorem, saying that the order of a subgroup of a finite group divides the order of the group, is proved by showing that cosets partition the group.
		One of the most useful theorems on group actions is the Orbit Stabilizer Theorem, which will be used to count orbits and patterns.
		With the theory developed so far it is possible to classify all commutative groups generated by a finite number of elements and this result is one of the principal results in the course.
	www.mth.uea.ac.uk /maths/syllabuses/0203/2A2302.html (501 words)

	Joseph-Louis Lagrange
		Lagrange's classic Theorie des fonctions analytiques laid some of the foundations of group theory, anticipating Galois.
		Lagrange also established the theory of differential equations, and provided many new solutions and theorems in number theory, including Wilson's theorem.
		Lagrange succeeded Euler as the director of the Berlin Academy.
	lagrange-bio.net (501 words)

	Department of Mathematics
		Introduction to group theory, subgroups, Lagrange's theorem, factor groups, permutation groups, group homomorphisms, isomorphism theorems, introduction to ring theory, ideals, ring homomorphisms, divisibility, polynomial rings, field of rational functions.
		Sets and counting, probability and relative frequency, conditional probability, Bayes' theorem, independence, discrete and continuous random variables, binomial, Poisson and normal distributions, functions of random variables, law of large numbers, generating functions, characteristic functions, moments, compound distributions, central limit theorems, Markov chains and their limiting probabilities.
		Projective spaces and homogeneous coordinates, subspaces, the dual space, Desargues' theorem, double ratio, collineation, projections and correlations, polarity, passage to affine and metric spaces, plane algebraic curves and their singular points, conics and cubics.
	www.boun.edu.tr /undergraduate/arts_sciences/mathematics.html (2184 words)

	Qualifying Exams: UCLA Math Graduate Handbook
		Topics- Group Theory: elementary theory dealing with properties of finite groups, Lagrange's theorem, permutation groups, abstract groups including automorphisms and homomorphisms, normal subgroups and factors groups, the homomorphism theorems, and Sylow theorems.
		Cell complexes and cellular homology, Lie brackets, Frobenius Theorem and Lie derivatives; cotangent spaces, tensors, differential forms and exterior derivative, and Stokes Theorem on manifolds; introduction to cohomology theory, and de Rham's Theorem.
		definition maxima and minima, uniform continuity, definition of derivative, the mean value theorem, Taylor expansion with remainder, Riemann integral, mean value theorem for integrals, fundamental theorem of calculus, sequences and series of functions, uniform convergence and integration, differentiation under the integral sign.
	www.math.ucla.edu /grad_programs/handbook/hbqex.html (2184 words)

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