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Algebra2003Jan
Course: MATH 699, Spring 2011School: University of North...
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University of North Carolina School of the Arts - MATH - 699
University of North Carolina School of the Arts - MATH - 699
University of North Carolina School of the Arts - MATH - 699
University of North Carolina School of the Arts - MATH - 699
University of North Carolina School of the Arts - MATH - 699
University of North Carolina School of the Arts - MATH - 699
University of North Carolina School of the Arts - MATH - 699
Ph. D. Algebra Preliminary Exam Monday, August 20, 2007Show the work you do to obtain an answer. Give reasons for your answers.There are 10 questions. You should answer each question.1. (10 points) Let G be a group of order p2 for some prime number p.
University of North Carolina School of the Arts - MATH - 699
Ph. D. Algebra Preliminary Exam Wednesday, January 10, 2007Show the work you do to obtain an answer. Give reasons for your answers.There are 10 questions. Do all parts of all questions.Notations: C (x) is the centralizer of a group element x, , is the
University of North Carolina School of the Arts - MATH - 699
Algebra Preliminary Examination, August 19, 2008Print name:Show your work, give reasons for your answers, provide all necessary proofsand counterexamples. There are 10 problems on 15 pages. Each problem is worth 10points for a total of 100 points. Ple
University of North Carolina School of the Arts - MATH - 699
Algebra Preliminary Examination, January 10, 2008Print name:Show your work, give reasons for your answers, provide all necessary proofs andcounterexamples. There are 10 problems on 18 pages worth 10 points each for the total of 100points. Check that y
University of North Carolina School of the Arts - MATH - 699
Ph. D. Algebra Preliminary Exam Tuesday, August 25, 2009Show the work you do to obtain an answer. Give reasons for your answers.There are 10 questions. Do all parts of all questions. Each question is worth10 points. When a question has two parts (a) an
University of North Carolina School of the Arts - MATH - 699
ALGEBRA PHD PRELIMINARY EXAM, 9 JANUARY 20091. (10 points) Let G be a group of order 132 = 22 3 11. Prove that G is not simple.2. (10 points) Let H and K be normal subgroups of a group G , and assume that G = HK .Prove that there is an isomorphismG /(
University of North Carolina School of the Arts - MATH - 699
Ph. D. Algebra Preliminary Exam August 23, 2010Show the work you do to obtain an answer. Give reasons for your answers.There are 10 questions. Do all parts of all questions. Each question is worth 10points. Q is the eld of rational numbers. C is the el
University of North Carolina School of the Arts - MATH - 699
NAME:Algebra Ph.D. Preliminary ExamJanuary 12, 20101. Recall that a subgroup H of a group G is called characteristic if (H ) H for every automorphism of G.(a) Prove that characteristic subgroups are always normal.(b) Let P be a p-Sylow subgroup of a
University of North Carolina School of the Arts - MATH - 699
University of North Carolina School of the Arts - MATH - 699
University of North Carolina School of the Arts - MATH - 699
University of North Carolina School of the Arts - MATH - 699
University of North Carolina School of the Arts - MATH - 699
University of North Carolina School of the Arts - MATH - 699
Syllabus for Preliminary Exam for 631-632 Algebra I, IILinear algebra: vector spaces, linear transformations, eigenvectors anddiagonalization, Jordan canonical form, bilinear forms and inner product spaces,normal operators.Groups: cosets, quotient gro
University of North Carolina School of the Arts - MATH - 600
University of North Carolina School of the Arts - MATH - 600
University of North Carolina School of the Arts - MATH - 600
University of North Carolina School of the Arts - MATH - 600
University of North Carolina School of the Arts - MATH - 600
University of North Carolina School of the Arts - MATH - 600
University of North Carolina School of the Arts - MATH - 600
University of North Carolina School of the Arts - MATH - 600
University of North Carolina School of the Arts - MATH - 600
University of North Carolina School of the Arts - MATH - 600
University of North Carolina School of the Arts - MATH - 600
University of North Carolina School of the Arts - MATH - 600
University of North Carolina School of the Arts - MATH - 600
University of North Carolina School of the Arts - MATH - 600
University of North Carolina School of the Arts - MATH - 600
University of North Carolina School of the Arts - MATH - 600
University of North Carolina School of the Arts - MATH - 600
Preliminary Exam Jan 20071. Let X be a metric space and let Aj be subsets of X , j = 1, 2, . . . . For each of the following statements, prove it or give a counterexample (the means limit points):(i) (A1 A2 ) A1 A2(ii)j =1Aj Ajj =12. Prove that th
University of North Carolina School of the Arts - MATH - 600
Analysis Preliminary ExamAugust, 20081. Let f : R2 R be given by the formulax2 yx2 + y 2f (x, y ) =0if (x, y ) = (0, 0)if (x, y ) = (0, 0).(a) Show that f is continuous at (0, 0).(b) Prove that the rst order partial derivatives of f at (0, 0) ex
University of North Carolina School of the Arts - MATH - 600
Analysis Preliminary ExamAugust 24, 20091. If F1 and F2 are closed subsets of R1 and dist(F1 , F2 ) = 0 then F1 F2 = . Prove orgive a counterexample.2. Newtons method for nding zeroes of a function f : R1 R1 is based on the recursionformulaf (xn )x
University of North Carolina School of the Arts - MATH - 600
Analysis preliminary examJan. 8, 20091. Let C be the standard Cantor set on the interval [0, 1] and let A = C c be its complement on the real line. Identify the set of all limit points A of A, explaining your answer.2. (a) Provenk=k=1n(n + 1)2(b)
University of North Carolina School of the Arts - MATH - 600
August 2010 Preliminary Exam in Analysis1. Suppose that f : R R is a function such that f (f (x) = x for all x R. Prove thatthere exists an irrational number t such that f (t) is also irrational.2. Find three subsets A, B, C of the real line R such tha
University of North Carolina School of the Arts - MATH - 600
January 2010 Preliminary Exam in Analysis.1. Let X be a connected metric space. Given two points p, q X and a numberprove that there exist an integer n> 0,0 and points a0 , a1 , . . . , an X such that a0 = p,an = q , andd(aj , aj 1 ) <for all j = 1
University of North Carolina School of the Arts - MATH - 600
University of North Carolina School of the Arts - MATH - 600
University of North Carolina School of the Arts - MATH - 600
University of North Carolina School of the Arts - MATH - 600
Topics for the Ph.D. Preliminary ExaminationANALYSISFundamentals of Analysis (MAT 601-602)The basic material is contained in Rudin's "Principles of Mathematical Analysis" 3rdEdition, Chapters 1 through 9. Important topics are:1. Properties of real an
University of North Carolina School of the Arts - MATH - 601
University of North Carolina School of the Arts - MATH - 601
University of North Carolina School of the Arts - MATH - 601
Qualifying ExaminationJanuary 10, 2008Algebra Part Please do all ve questions. Problem #5 is worth twice as much as each of the othersWe will always assume that rings have an identity element and that modulesare unitary left modules.1. Let I be an
University of North Carolina School of the Arts - MATH - 601
Algebra P art o f Qualifying Examination, A ugust 25, 2009Instructions: Do all questions, justify your answers with the necessary proofs.All rings are associative (not necessarily commutative) with identity a nd all modulesare left unitary modules. We
University of North Carolina School of the Arts - MATH - 601
Q ualifying Exam - January 2009Algebra PartInstructions: C omplete as m any q uestions a s possible. Answers should be justified withthe necessary proofs. All rings are a ssumed to be n oncom mutative unless statedotherwise. All rings have an identity
University of North Carolina School of the Arts - MATH - 601
Algebra Part of Qualifying Examination, August 23, 2010Instructions: Do all questions, justify your answers with the necessary proofs. Allrings are associative (not necessarily commutative) with identity, and all modules are leftunitary modules. We den
University of North Carolina School of the Arts - MATH - 601
(Topics for qualifying exam in algebra) 731 SYLLABUSI. Set-up (over rings with unity, including noncommutative)Modules and Homomorphism TheoremsDirect Sums and Products, Free Modules(including Universal Mapping Properties)Projective and Injective Mod
University of North Carolina School of the Arts - MATH - 601
August 2008Qualifying ExaminationAlgebra PartThere are only 6 questions. Do them all.1. Let A be a nite abelian group. Prove that A is not a projective Z-moduleand also that it is not an injective Z-module.2. Prove that Q/Z Z Q/Z = 03. Let I be an
University of North Carolina School of the Arts - MATH - 680
University of North Carolina School of the Arts - MATH - 680
Qualifying Exam in Combinatorics12 January, 20071. Suppose that G is a graph with no vertex of valence < 5, exactly 13vertices of valence 5, no vertex of valence 7, and possibly an assortmentof vertices of other valences.(a) What is the least number
University of North Carolina School of the Arts - MATH - 680
Qualifying Exam in Combinatorics19 August, 20081. Let be a 3-connected planar graph with planar dual . Prove ordisprove:If is a Cayley graph and is is vertex-transitive, then is a Cayleygraph.If you believe the statement to be true, give a proof, or
University of North Carolina School of the Arts - MATH - 680
University of North Carolina School of the Arts - MATH - 680
University of North Carolina School of the Arts - MATH - 680
Combinatorics Qualifying ExamPractice QuestionsJuly 21, 20081. (a) Dene (v , k , )-design. Derive some identities involving these three parameters.(b) How are these designs related to nite pro jective planes? Give a picture of the simplest nitepro je
University of North Carolina School of the Arts - MATH - 680
Topics for Qualifying Exam in CombinatoricsI. EnumerationA. Selections with and without repetitions (combinations &permutations)B. Partitions1. Stirling numbers of the first and second kindC. Principle of Inclusion-Exclusion1. Surjections2. Derang