Question: MAT301 ASSIGNMENT 6 DUE DATE: MARCH 1, 2017, AT THE BEGINNING OF YOUR TUTORIAL NOTE: Zn is a group under addition modulo n, and U

MAT301 ASSIGNMENT 6 DUE DATE: MARCH 1, 2017, AT THE BEGINNING OF YOUR TUTORIAL NOTE: Zn is a group under addition modulo n, and U (n) is a group under multiplication modulo n. Question 1. Suppose : G G0 is a group homomorphism. Prove that ghg 1 ker for all g G and h ker . Question 2. (a) Define : Z12 Z10 via (n) = 3n. Prove that is not a group homomorphism. (b) Suppose that : Z50 Z50 is an isomorphism with (7) = 9. Determine a formula for (x). Question 3. Let G be a group and define : G G via (g) = g 1 . (a) Prove that is a bijection. (b) Prove that is an isomorphism if and only if G is Abelain. 1

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