Explain

40. Let (a , a,, . . ., a) EG OG, O . . . O G. Give a necessary and sufficient condition for I(a] , a,, . . . , a )| = co. 41. Prove that D, D DA * DO7 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 Z, D Zz. 46. Let G be a group isomorphic to Z,, OZ,, 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,, DZ,. 50. Determine Aut(Z, D Zz). 51. Suppose that n , n,, . .., n, are positive even integers. How many elements of order 2 does Z,,, O Z,, O . . . OOZ, have ? How many are there if we drop the requirement that n , n,, . . ., n, must be even? 52. Is ZOZ12 DZ6 = Zoo DZ DZ? 53. Is Z10 O Z12 0 76 ~ ZIs OZ, OOZ12? 54. Find an isomorphism from Z, to Z, D Z3