Midterm - G Let n = | H | be the number of elements in H...

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Instructor: Leo Goldmakher University of Toronto at Mississauga Department of Mathematical and Computational Sciences MAT301 – Groups and Symmetries Midterm Exam (October 6, 2009) M.1 (50 points) Let Z denote the group of the integers under addition (i.e. you may assume that all the group axioms hold for Z ). (a) (20 points) Prove that if H Z then H = d Z for some integer d 0. (b) (8 points) Given any integers a and b , define the following set h a, b i = ax + by : x, y Z . Prove that h a, b i ≤ Z . (c) (12 points) If a and b are positive integers, prove that h a, b i = gcd( a, b ) Z . (d) (10 points) Given any positive integers a, b , and n , prove that ax + by = n has an integral solution ( x, y ) if and only if gcd( a, b ) n . M.2 (12 points) Given G an arbitrary group, and H a finite sub set of G which is closed under the binary operation of
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Unformatted text preview: G . Let n = | H | be the number of elements in H . Prove that H contains the identity element. M.3 (8 points) For this question, you may assume any theorems or facts from lecture. (a) (4 points) What is the last digit of 1983 21 ? Prove that your answer is correct. (b) (4 points) What are the last two digits of 1983 41 ? M.3 1 2 (0 points) What was Euler’s first name? Prove it. M.4 (10 points) Let G be an arbitrary group. (a) (5 points) Prove that for any element a ∈ G , aG = G . (b) (5 points) Suppose G is a finite group, and consider its multiplication table. Using (a) or otherwise, explain why the same element of G cannot appear twice in the same row of the table. 1...
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