Algebra2009aug - Ph D Algebra Preliminary Exam Tuesday Show...

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Ph. D. Algebra Preliminary Exam Tuesday, August 25, 2009 Show 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 10 points. When a question has two parts (a) and (b) each part is worth 5 points. 1. Let V be a real vector space with subspaces A , B , and X with A B . Prove that if A + X = B + X and A X = B X , then A = B . 2. (a) Let G be a finite Abelian group and let p be a prime number that di- vides the order of G . Without using the fundamental theorem of finite Abelian groups, prove that G contains an element of order p . (b) Let G = { 1 = g 1 , g 2 , ..., g n } be a finite Abelian group. If g 1 g 2 ...g n 6 = 1, prove that the order of G must be even. 3. Prove that no group of order 48 is simple. 4. (a) Let E be a Euclidean space - that is a finite dimensional vector space over R , the real numbers, with a positive definite, symmetric, inner product denoted by ( , ). Let E * = Hom R ( E, R ) be the dual space. Prove that the map φ : E E * , given by φ ( v ) = ρ v where ρ v ( w ) = ( w, v
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