MT1 - x 3(mod 10 and x 7(mod 15(3 Prove that every...

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Math 113 (Bernd Sturmfels), Midterm Exam # 1 Thursday, July 13, 9:00 a.m.–10:00 a.m. Please start by writing your name and your student ID on the cover of your blue book. This exam is closed book . Do not use any notes, calculators, cell phones etc. Show all your work, and write full sentences if time permits. Each problem is worth 20 points, for a total of 100 points. (1) Let a, b be two integers and let p be a prime number. Prove that ( a + b ) p a p + b p (mod p) . (2) Determine the set of all integers x which satisfy the three congruences x 1 (mod 6)
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Unformatted text preview: , x ≡ 3 (mod 10) and x ≡ 7 (mod 15) . (3) Prove that every subgroup of an abelian group is normal. (4) Consider the product of cyclic groups G = Z / 3 Z × Z / 4 Z × Z / 5 Z . (a) Show that the group G is cyclic. (b) How many elements in G are generators of G ? (c) How many elements in G have order 10? (5) The cycles σ = (123) and τ = (124) are in the symmetric group S 4 . (a) Compute the two products στ and τσ in S 4 . (b) Write both στ and τσ as products of disjoint cycles....
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This note was uploaded on 02/03/2011 for the course MATH 113 taught by Professor Ogus during the Spring '08 term at Berkeley.

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