(See Exercise 48 in Chapter 2 for the definition of multiplication.) Show that H is an Abelian group of order 9. Is H isomorphic to Z Groups 176 or to Z, Z,? 39. Let G = (3"6"Im,n EZ} under multiplication. Prove that G is isomor- phic to ZZ. Does your proof remain valid if G = {3"9 | m, n E Z}? 40. Let (a, a,...,a) E G, G, GGive a necessary and sufficient condition for I(a, a,..., an 41. Prove that D, D DZ 42. Determine the number of cyclic subgroups of order 15 in Zon 36 Provide a generator for each of the subgroups of order 15. 43. List the elements in the groups U,(35) and U,(35). 44. Prove or disprove that U(40) Z, is isomorphic to U(72) Z 45. Prove or disprove that C has a subgroup isomorphic to Z, Z. 46. Let G be a group isomorphic to Z Z ..Z. 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 8EG} (see Exercise 17, Supplementary Exercises for Chapters 1-4). If H and K are Abelian, show that (H K = H" K". 49. Express Aut(U(25)) in the form Z 50. Determine Aut(Z, Z). 51. Suppose that n, n elements of order 2 does Z Z, Z, have ? How many there if we drop the requirement that n, n2,...,n, must be even? 52. Is Z10 Z12Z Z0ZZ2? 53. Is Z10 Z12Z Z1sZ,Z2? 54. Find an isomorphism from Z2 to Z Z. 55. How many isomorphisms are there from Z, to Z, Z? nk " {g" n are positive even integers. How many 56. Suppose that is an isomorphism from Z3 Z to Z,s and 57. If is an isomorphism from Z, Z, to Z, what is (2, 0)? What d2, 3) 2. Find the element in Z Z, that maps to 1. are the possibilities for (1, 0)? Give reasons for your answer. 58. Prove that ZZ, has exactly six subgroups of order 5. 59. Let (a, b) belong to Z Z. Prove that I(a, bl divides lem(m, n). 60. Let G lax+ bx + cla, b, c Z). Add elements of G as you would polynomials with integer coefficients, except use modulo 3 addition. Prove that G is isomorphic to Z, Z, Z Generalize.