Algebra2008Jan - Algebra Preliminary Examination, January...

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Algebra Preliminary Examination, January 10, 2008 Print name: Show your work, give reasons for your answers, provide all necessary proofs and counterexamples. There are 10 problems on 18 pages worth 10 points each for the total of 100 points. Check that you have a complete exam, print your name on each page. Problem 1 Problem 6 Problem 2 Problem 7 Problem 3 Problem 8 Problem 4 Problem 9 Problem 5 Problem 10 Total Do NOT write on this page 1
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Print name: 1. Let G,H be cyclic groups generated by elements x,y of finite orders m,n , respectively. (a) (7 points) Determine the necessary and sufficient condition on m,n so that sending x i to y i , for all i Z , is a well-defined homomoprhism of groups. 2
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Print name: 1. (continued) (b) (3 points) Describe all homomorphisms of the cyclic group of order 6 into the cyclic group of order 24. 3
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2. Given a subgroup K of a group G , the set S of left cosets of K in G is a left G -set by means of g · xK = gxK , for all g,x G . If H is another subgroup of G , then S is a left H -set by restriction. Recall that the set HxK = { y G | y = hxk for some h H,k K } is called a double coset . For any set X , | X | denotes the cardinality of X . (a) (3 points) Prove that the orbit of the element xK of the H -set S is the set of left cosets of K in G contained in the double coset HxK and compute the stabilizer of
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Algebra2008Jan - Algebra Preliminary Examination, January...

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