prelim1 - disjoint cycles and factorization into a product...

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Spring 2010 MATH 4320: Prelim 1 Instructor: Yuri Berest Problem 1. (15 points) a . Find all integer solutions to the congruence 72 x 36 (mod 376) . b . Find the smallest positive integer which leaves remainders 1 , 3 , 4 after dividing by 9 , 7 , 5 respectively. Problem 2. (30 points) a . Prove that if ( a, b ) = 1 , then ( ab, c ) = ( a, c )( b, c ) for all a, b, c Z . b . Prove that if ( a, b ) = 1 , the equation ( a + bx, c ) = 1 is solvable in integers for any c Z . Problem 3. (20 points) a . Let f : X Y be a map between two finite sets of the same size. Prove that f is injective if and only if f is surjective. b . Let X = { 0 , 1 , 2 , . . . , 9 , 10 } . Define a map σ : X X by the rule: σ ( n ) = the remainder after dividing 4 n 2 - 3 n 7 by 11 . Show that σ is a permutation of X . Find its complete factorization into a product of
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Unformatted text preview: disjoint cycles and factorization into a product of transpositions. Compute sign ( σ ) and σ-1 . Problem 4. (35 points) For a permutation σ ∈ S n , define | σ | to be the least integer r > 0 such that σ r = (1). ( | σ | is called the order of σ in S n .) a . If σ = σ 1 σ 2 . . . σ k is a product of disjoint cycles, show that | σ | = lcm {| σ 1 | , | σ 2 | , . . . , | σ k |} . b * . Suppose that σ ∈ S n has k 1 cycles of length 1, k 2 cycles of length 2, k 3 cycles of length 3, . . . , k r cycles of length r , so that n = k 1 + 2 k 2 + 3 k 3 + . . . + rk r . Find | σ | ....
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This note was uploaded on 06/07/2010 for the course MATH 201 taught by Professor Crissinger during the Spring '08 term at University of Delaware.

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