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

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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 = (123) and = (124) are in the symmetric group S 4 . (a) Compute the two products and in S 4 . (b) Write both and as products of disjoint cycles....
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