Suppose that g and h induce the same inner automorphism of a group G. Prove that h-lg E Z(G). 48. Combine the results of Exercises 45 and 47 into a single "if and only if theorem. 49. If a and B are elements in S, (n 3), prove that d be implies that a = B. (Here, %a is the inner automorphism of S, induces by a.) 50. Prove or disprove that the mapping from @*, the positive rational numbers under multiplication, to itself given by b(x) = ? is an automorphism. 51. Suppose the b and y are isomorphisms of some group G to the same group. Prove that H = (g E Gl *(g) = y(g)} is a subgroup of G. 52. Let G be a group. Complete the following statement: (Inn(G)| =1 if and only if 53. Suppose that G is an Abelian group and p is an automorphism of G. Prove that H = {x E G| b(x) =x -1} is a subgroup of G. 54. Let $ be an automorphism of Dg. What are the possibilities for *(R45)? 55. Let & be an automorphism of C*, the group of nonzero complex numbers under multiplcation. Determine b(-1). Determine the possibilities for d(i). 56. Let G = (0, 2, 4, +6, .} and H = (0, 3, 6, 9, ..l. Prove that G and H are isomorphic groups under addition by defin- ing a mapping that has the required properties. Does your isomor- phism preserve multiplication? Generalize to the case when G (m) and H = (n), where m and n are integers. 57. Give three examples of groups of order 120, no two of which are isomophic. Explain why they are not isomorphic