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1. Let d : ZxZ - S9 be the homomorphism such that d ((1,0)) = (3, 5) (2, 4) and d ((0,1)) = (1, 7) (6, 9, 8, 7). a. Find the kernel of d. b. Compute +((-3,7)). 2. Let G = (23 X Za) / ((1, 3)). a. Find the order of G and classify G according to the Fundamental Theorem of Finitely Generated Abelian Groups. b. Find the order of the element (2,5) + ((1, 3)) in G. 3. Find the sum and the product the given polynomials f(z) = 2 + 624 - 82-7722 - 6 and g(x) = 426 - 524 + 22³ - 3x +1 in the following rings of polynomials. a. Z[z] b. 4. Let G = Zeo. a. Draw the subgroup diagram of G. b. Is G isomorphic to 2.1s X Z6? Justify your answer. Let a = 9 2 ) be a permutation in 1 2 3 4 5 6 7 8 5. 4 5 3 6 9 8 1 7 Sg. a. Express a as a product of disjoint cycles. What is the order of o? b. Express (T as a product of transpositions. Is a an even or odd permutation?

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