39. Let G = {3"6" I m, n E Z) under multiplication. Prove that G is isomor- phic to Z Z. Does your proof remain valid if G = {39" \\ m, n E Z}? 40. Let (a , a,, . . ., a) EG OG, O . . . OG . Give a necessary and sufficient condition for I(a, , a2, . . ., a,)| = co. 41. Prove that D3 D DA * D12 D Zz. 42. Determine the number of cyclic subgroups of order 15 in Z,, D Z36. Provide a generator for each of the subgroups of order 15. 43. List the elements in the groups Us(35) and U,(35). 44. Prove or disprove that U(40) Zo is isomorphic to U(72) D Z. 45. Prove or disprove that C* has a subgroup isomorphic to Zz D Zz. 46. Let G be a group isomorphic to Z,,, O Z,, O . . . OZ . Let x be the product of all elements in G. Describe all possibilities for x. 47. If a group has exactly 24 elements of order 6, how many cyclic subgroups of order 6 does it have? 48. For any Abelian group G and any positive integer n, let G" = {g" | g E G} (see Exercise 17, Supplementary Exercises for Chapters 1-4). If H and K are Abelian, show that (HOOK)" = H" ( K. 49. Express Aut(U(25)) in the form Z, OZ, 50. Determine Aut(Zz (D Z2). 51. Suppose that n, n2, . . . , n, are positive even integers. How many elements of order 2 does Z,, O Z,, OD . . . OZ, have ? How many are there if we drop the requirement that n , n,, .. ., n, must be even? 52. Is Z10 D Z 12 0 76 - 260 D Z D Zz ? 53. Is Z10 O Z 12 0 76 = Z15 DZ DZ 12? 54. Find an isomorphism from Z12 to Z, D Z3. 55. How many isomorphisms are there from Z12 to Z, D Z,? 56. Suppose that o is an isomorphism from Z, O Z, to Z15 and (2, 3) = 2. Find the element in Z, O Z, that maps to 1. 57. If d is an isomorphism from Z, OZ, to Z2, what is d(2, 0)? What are the possibilities for d(1, 0)? Give reasons for your answer. 58. Prove that Z Z, has exactly six subgroups of order 5. 59. Let (a, b) belong to Z O Z,. Prove that I(a, b)I divides Icm(m, n). 60. Let G = (ax2 + bx + cla, b, c E Z, }. Add elements of G as you would polynomials with integer coefficients, except use modulo 3