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Course: MATH 621, Spring 2012
School: George Mason
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621, MATH Algebra I Assignment sheet 4 Spring 2012 - George Mason University. Professor: Geir Agnarsson (geir@math.gmu.edu) Homework to be handed in: (1): A group G is called solvable if there is a normal chain, or sequence G = G0 G1 Gn = {e}, with each Gi+1 normal in Gi and each quotient Gi /Gi+1 abelian. If G is a solvable group, show that any subgroup of G is solvable and that any homomorphic image of G is...

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621, MATH Algebra I Assignment sheet 4 Spring 2012 - George Mason University. Professor: Geir Agnarsson (geir@math.gmu.edu) Homework to be handed in: (1): A group G is called solvable if there is a normal chain, or sequence G = G0 G1 Gn = {e}, with each Gi+1 normal in Gi and each quotient Gi /Gi+1 abelian. If G is a solvable group, show that any subgroup of G is solvable and that any homomorphic image of G is solvable. (2): Let G be a group with a subgroup H . Show that for any g G the set gHg1 is also a subgroup of G and that H gHg1 . Show further |H if | = k and = G has only one subgroup of order k then H is normal in G. Other exercises to be discussed: (3): Let I be a set of indices and let Hi be a normal subgroup of Gi for each i I . Show that iI Hi is a normal subgroup of iI Gi and that Hi Gi iI = (Gi /Hi ) . iI iI (4): Let N be a normal subgroup of G of nite index [G : N ]. Let H be a nite subgroup of G such that [G : N ] and |H | are relatively prime integers. Show that H is a subgroup of N . Deadline: Monday, February 27., in the start of the class. Geir Agnarsson February 21, 2012 1
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George Mason - MATH - 621
MATH 621, Algebra IAssignment sheet 5Spring 2012 - George Mason University.Professor: Geir Agnarsson (geir@math.gmu.edu)Homework to be handed in:(1):Show that a group G on four elements, |G| = 4 is abelian. Let N4 be thesubset of the symmetric grou
George Mason - MATH - 621
MATH 621, Algebra IAssignment sheet 6Spring 2012 - George Mason University.Professor: Geir Agnarsson (geir@math.gmu.edu)Homework to be handed in:(1):Let G be an abelian group with |G| = pn for some n N where p is a primenumber. Show that G has a co
George Mason - MATH - 621
MATH 621, Algebra IAssignment sheet 7Spring 2012 - George Mason University.Professor: Geir Agnarsson (geir@math.gmu.edu)Homework to be handed in:(1):Let G be a group with |G| = pn m where p is a prime, gcd(p, m) = 1 andp &gt; m 1. Show that G has a un
George Mason - MATH - 621
MATH 621, Algebra IAssignment sheet 8Spring 2012 - George Mason University.Professor: Geir Agnarsson (geir@math.gmu.edu)Homework to be handed in:(1):Let G and H be nite groups of relatively prime orders. Show thatAut(G H ) Aut(G) Aut(H ).=Here Au
George Mason - MATH - 621
MATH 621, Algebra IAssignment sheet 9Spring 2012 - George Mason University.Professor: Geir Agnarsson (geir@math.gmu.edu)Homework to be handed in:(1):Let R = cfw_m + n 2 : m, n Z show that R is a subring of R. Showfurther that R (Z2 , +, ) where Z2
George Mason - MATH - 621
MATH 621, Algebra IAssignment sheet 10Spring 2012 - George Mason University.Professor: Geir Agnarsson (geir@math.gmu.edu)Homework to be handed in:(1):Let R be an integral domain. If a R is a unit and b R, show there isa unique automorphism of R[X ]
George Mason - MATH - 621
MATH 621, Algebra IAssignment sheet 11Spring 2012 - George Mason University.Professor: Geir Agnarsson (geir@math.gmu.edu)Homework to be handed in:(1):For a ring R, the least n N such that n = 1 + + 1 (n times) = 0 iscalled the characteristic of R a
George Mason - MATH - 621
MATH 621 Algebra I Lecture NotesGeir AgnarssonApril 19, 201211.1GroupsSome preliminariesThese lecture notes will follow the class text [1] fairly closely. They are, inpart, based on lectures of the graduate algebra class MATH 250A given bythe lat
George Mason - MATH - 621
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George Mason - MATH - 621
MATH 621, Algebra ISelected solutions for HW assignment 1Spring 2012 - George Mason University.Professor: Geir Agnarsson (geir@math.gmu.edu)(1):Let f : A B and g : B C be functions.(i) Assume that both f and g are injective, we show that g f is also
George Mason - MATH - 621
MATH 621, Algebra ISelected solutions for HW assignment 2Spring 2012 - George Mason University.Professor: Geir Agnarsson (geir@math.gmu.edu)(1): Let G be a group and S G a subset. Clearly C (S ) G, so to showthat C (S ) = cfw_g G : gs = sg for every
George Mason - MATH - 621
MATH 621, Algebra ISelected solutions for HW assignment 3Spring 2012 - George Mason University.Professor: Geir Agnarsson (geir@math.gmu.edu)(1): Assume that G is a group with a subgroup H of index two in G, so[G : H ] = 2. Let g G. If g H then clearl
George Mason - MATH - 621
MATH 621, Algebra ISelected solutions for HW assignment 4Spring 2012 - George Mason University.Professor: Geir Agnarsson (geir@math.gmu.edu)(1):Assume G is a solvable group, so G has a normal chainG = G0G1Gn = cfw_e,(1)where the quotient Gi /Gi1
George Mason - MATH - 621
MATH 621, Algebra ISelected solutions for HW assignment 5Spring 2012 - George Mason University.Professor: Geir Agnarsson (geir@math.gmu.edu)(1):Let G be a group on four elements. For each element a G the cyclicsubgroup a is a subgroup of G, and henc
George Mason - MATH - 621
MATH 621, Algebra ISelected solutions for HW assignment 6Spring 2012 - George Mason University.Professor: Geir Agnarsson (geir@math.gmu.edu)(1):Let G be an abelian group with |G| = pn for some n N where p is a primenumber. If n = 1 there is nothing
George Mason - MATH - 621
MATH 621, Algebra ISelected solutions for HW assignment 7Spring 2012 - George Mason University.Professor: Geir Agnarsson (geir@math.gmu.edu)(1):Let G be a group with |G| = pn m where p is a prime, gcd(p, m) = 1 andp &gt; m 1. If G has N p-Sylow subgrou
George Mason - MATH - 621
MATH 621, Algebra ISelected solutions for HW assignment 8Spring 2012 - George Mason University.Professor: Geir Agnarsson (geir@math.gmu.edu)(1):Let G and H be nite groups of relatively prime orders. We want to showthat Aut(G H ) Aut(G) Aut(H ), wher
George Mason - MATH - 621
MATH 621, Algebra ISelected solutions for HW assignment 9Spring 2012 - George Mason University.Professor: Geir Agnarsson (geir@math.gmu.edu)(1):Using our more recent notation from class we haveR = cfw_m + n 2 : m, n Z = Z[ 2],all polynomial express
George Mason - MATH - 108
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George Mason - MATH - 108
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George Mason - MATH - 108
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George Mason - MATH - 108
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George Mason - MATH - 108
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George Mason - MATH - 108
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George Mason - MATH - 108
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George Mason - MATH - 108
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George Mason - MATH - 108
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George Mason - MATH - 108
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George Mason - MATH - 108
MATH 108, Introductory Calculus with BusinessApplicationsReview sheet for rst midterm(1) For the quadratic function f (x) = x2 + 6x 4 nd x and y intercepts.Sketch the graph. Find the vertex and the extreme value. Does it have amaximum or a minimum va
George Mason - MATH - 108
MATH 108, Introductory Calculus with BusinessApplicationsReview sheet for second midterm(1) Show that f (x) = |x 3| + 5 is not dierentiable at x = 3. Is fdierentiable everywhere else? Explain.(2) Is f (x) = x2/3 dierentiable at x = 0?(3) Use the pro
George Mason - MATH - 108
Department of Mathematical SciencesGeorge Mason UniversityMath 108, Introductory Calculus with Business ApplicationsFirst midterm exam, \,cfw_arch5, 2012,Rob. Hall- 8104This midterm consists of 5 problems, each is worth 207o, This is a closedbook
George Mason - MATH - 108
EdtFHTDepartment of Mathematical SciencesGeorge Mason UniversityMath 108, Introductory Calculus with Business ApplicationsSecond midterm exam,April 9th,2012, Rob. Hall, 8104This midterm consists of 5 problems, each is worth 20%, This is a closedbo
George Mason - MATH - 108
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George Mason - CHEM - 212
GeorgeMasonUniversityGeneralChemistry212Chapter13PropertiesofMixtures:Solutions&amp;ColloidsAcknowledgementsCourseText:Chemistry:theMolecularNatureofMatterandChange,6thed,2011,MartinS.Silberberg,McGrawHill04/22/12TheChemistry211/212GeneralChemistrycou
George Mason - CHEM - 212
GeorgeMasonUniversityGeneralChemistry212Chapter14MainGroupElementPatternsAcknowledgementsCourseText:Chemistry:theMolecularNatureofMatterandChange,6thed,2011,MartinS.Silberberg,McGrawHillTheChemistry211/212GeneralChemistrycoursestaughtatGeorgeMason
George Mason - CHEM - 212
GeorgeMasonUniversityGeneralChemistry212Chapter15OrganicChemistryAcknowledgementsCourseText:Chemistry:theMolecularNatureofMatterandChange,6thed,2011,MartinS.Silberberg,McGrawHillTheChemistry211/212GeneralChemistrycoursestaughtatGeorgeMasonareinten
George Mason - CHEM - 212
GeorgeMasonUniversityGeneralChemistry212Chapter16ChemicalKineticsAcknowledgementsCourseText:Chemistry:theMolecularNatureofMatterandChange,6thed,2011,MartinS.Silberberg,McGrawHillTheChemistry211/212GeneralChemistrycoursestaughtatGeorgeMasonareinte
George Mason - CHEM - 212
GeorgeMasonUniversityGeneralChemistry212Chapter17Equilibrium:ExtentofChemicalReactionsAcknowledgementsCourseText:Chemistry:theMolecularNatureofMatterandChange,6thed,2011,MartinS.Silberberg,McGrawHill04/22/12TheChemistry211/212GeneralChemistrycour
George Mason - CHEM - 212
GeorgeMasonUniversityGeneralChemistry212Chapter18AcidBaseEquilibriaAcknowledgementsCourseText:Chemistry:theMolecularNatureofMatterandChange,6thed,2011,MartinS.Silberberg,McGrawHill04/22/12TheChemistry211/212GeneralChemistrycoursestaughtatGeorgeMa
George Mason - CHEM - 212
GeorgeMasonUniversityGeneralChemistry212Chapter19IonicEquilibriaAcknowledgementsCourseText:Chemistry:theMolecularNatureofMatterandChange,6thed,2011,MartinS.Silberberg,McGrawHill04/22/12TheChemistry211/212GeneralChemistrycoursestaughtatGeorgeMason
George Mason - CHEM - 212
GeorgeMasonUniversityGeneralChemistry212Chapter20ThermodynamicsAcknowledgementsCourseText:Chemistry:theMolecularNatureofMatterandChange,6thed,2011,MartinS.Silberberg,McGrawHill04/22/12TheChemistry211/212GeneralChemistrycoursestaughtatGeorgeMasona
George Mason - CHEM - 212
GeorgeMasonUniversityGeneralChemistry212Chapter21ElectrochemistryAcknowledgementsCourseText:Chemistry:theMolecularNatureofMatterandChange,6thed,2011,MartinS.Silberberg,McGrawHill04/22/12TheChemistry211/212GeneralChemistrycoursestaughtatGeorgeMaso
George Mason - CHEM - 212
GeorgeMasonUniversityGeneralChemistry212Chapter23TransitionElementsAcknowledgementsCourseText:Chemistry:theMolecularNatureofMatterandChange,6thed,2011,MartinS.Silberberg,McGrawHill04/22/12TheChemistry211/212GeneralChemistrycoursestaughtatGeorgeMa
George Mason - CHEM - 212
GeorgeMasonUniversityGeneralChemistry212Chapter24NuclearReactionsAcknowledgementsCourseText:Chemistry:theMolecularNatureofMatterandChange,5thed,2009,MartinS.Silberberg,McGrawHill04/22/12TheChemistry211/212GeneralChemistrycoursestaughtatGeorgeMaso
George Mason - CHEM - 212
GENERAL CHEMISTRY (Chem 212)Lecture SyllabusInstructor: Dr. James C. SchornickCourse:Chemistry 212-004 (10247)T 4:30-7:10 pmRm 253 Bull Run Hall, PWCOffice Hrs: M, T, W, R, F 9:30 am - 11:00 am Fairfax, S&amp;T IRm 318AM, R, F1:00 am - 4:00 pmFairf
George Mason - CHEM - 212
ReviewElementPropertiesCriticalAtomicProperties ElectronConfiguration AtomicSize IonizationEnergy Electronegativity03/18/101ReviewElementPropertiesElectronConfiguration (nl )Thedistributionofelectronsintheenergylevelsml = 0 ml = -1 0 +1 ml = -2 -1
George Mason - CHEM - 211
GeorgeMasonUniversityGeneralChemistry211Chapter1KeystotheStudyofChemistryAcknowledgementsCourseText:Chemistry:theMolecularNatureofMatterandChange,6thedition,2011,MartinS.Silberberg,McGrawHill04/22/12TheChemistry211/212GeneralChemistrycoursestaught
George Mason - CHEM - 211
GeorgeMasonUniversityGeneralChemistry211Chapter2TheComponentsofMatterAcknowledgementsCourseText:Chemistry:theMolecularNatureofMatterandChange,6thedition,2011,MartinS.Silberberg,McGrawHillTheChemistry211/212GeneralChemistrycoursestaughtatGeorgeMaso
George Mason - CHEM - 211
GeorgeMasonUniversityGeneralChemistry211Chapter4ThreeMajorClassesofChemicalReactionsAcknowledgementsCourseText:Chemistry:theMolecularNatureofMatterandChange,6thedition,2011,MartinS.Silberberg,McGrawHill04/22/12TheChemistry211/212GeneralChemistryco
George Mason - CHEM - 211
GeorgeMasonUniversityGeneralChemistry211Chapter5GasesandtheKineticMolecularTheoryAcknowledgementsCourseText:Chemistry:theMolecularNatureofMatterandChange,6thedition,2011,MartinS.Silberberg,McGrawHill04/22/12TheChemistry211/212GeneralChemistrycours
George Mason - CHEM - 211
GeorgeMasonUniversityGeneralChemistry211Chapter6Thermochemistry:EnergyFlowandChemicalChangeAcknowledgementsCourseText:Chemistry:theMolecularNatureofMatterandChange,6thedition,2011,MartinS.Silberberg,McGrawHill04/22/12TheChemistry211/212GeneralChem
George Mason - CHEM - 211
GeorgeMasonUniversityGeneralChemistry211Chapter7QuantumTheoryandAtomicStructureAcknowledgementsCourseText:Chemistry:theMolecularNatureofMatterandChange,6thedition,2011,MartinS.Silberberg,McGrawHillTheChemistry211/212GeneralChemistrycoursestaughtat
George Mason - CHEM - 211
GeorgeMasonUniversityGeneralChemistry211Chapter8ElectronConfigurationandChemicalPeriodicityAcknowledgementsCourseText:Chemistry:theMolecularNatureofMatterandChange,6thedition,2011,MartinS.Silberberg,McGrawHill04/22/12TheChemistry211/212GeneralChem
George Mason - CHEM - 211
GeorgeMasonUniversityGeneralChemistry211Chapter9ModelsofChemicalBondingAcknowledgementsCourseText:Chemistry:theMolecularNatureofMatterandChange,6thedition,2011,MartinS.Silberberg,McGrawHill04/22/12TheChemistry211/212GeneralChemistrycoursestaughtat
George Mason - CHEM - 211
GeorgeMasonUniversityGeneralChemistry211Chapter10TheShapes(Geometry)ofMoleculesAcknowledgementsCourseText:Chemistry:theMolecularNatureofMatterandChange,6thedition,2011,MartinS.Silberberg,McGrawHill04/22/12TheChemistry211/212GeneralChemistrycourses
George Mason - CHEM - 211
GeorgeMasonUniversityGeneralChemistry211Chapter11TheoriesofCovalentBondingAcknowledgementsCourseText:Chemistry:theMolecularNatureofMatterandChange,6thedition,2011,MartinS.Silberberg,McGrawHill04/22/12TheChemistry211/212GeneralChemistrycoursestaugh
George Mason - CHEM - 211
GeorgeMasonUniversityGeneralChemistry211Chapter12IntermolecularForces:Liquids,Solids,andPhaseChangesAcknowledgementsCourseText:Chemistry:theMolecularNatureofMatterandChange,6thedition,2011,MartinS.Silberberg,McGrawHillTheChemistry211/212GeneralChem
George Mason - CHEM - 211
GENERAL CHEMISTRY I (Chem 211)Lecture SyllabusInstructor: Dr. James C. SchornickCourse:Chemistry 211-004 T 4:30Office Hrs: M, T, W, F9:30 amR2:00 pmT2:00 amTelephone: 703-993-1091Email:jschorni@gmu.edu- 7:10 pm- 11:00 am- 4:00 pm- 4:00 pm
George Mason - CHEM - 211L
Absorption Spectroscopy Spectroscopy - Study of the Interaction ofElectromagnetic Radiation (Energy) andMatter When energy is applied to matter it can beabsorbed, emitted, cause a chemical change(reaction), or be transmitted. Electromagnetic Spectr