Please solve this question

47. Suppose that g and h induce the same inner automorphism of a group G. Prove that h-'g 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 2 3), prove that do = de implies that a = B. (Here, a is the inner automorphism of S,, induces by a.) 50. Prove or disprove that the mapping d from Q, the positive rational numbers under multiplication, to itself given by p(x) = x is an automorphism. 51. Suppose the o and y are isomorphisms of some group G to the same group. Prove that H = (g E G Ip(8) = y(8) } 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 o is an automorphism of G. Prove that H = (x E Glo(x) = x 1} is a subgroup of G. 54. Let be an automorphism of D. What are the possibilities for p(R45)? 55. Let d be an automorphism of C", the group of nonzero complex numbers under multipleation. Determine o(-1). Determine the possibilities for d(i). 56. Let G = 10, +2, +4, +6, . . .} and H = (0, +3, +6, 19, . . . ). 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. 58. Let o be an automorphism of D, such that d(H) = D. Find b( V). 59. Suppose that o is an automorphism of D4 such that o( Rgo) = R270 and o(V) = V. Determine 4(D) and o(H). 60. In Aut(Z.), let a, denote the automorphism that sends 1 to i where god(i, 9) = 1. Write as and a as permutations of {0, 1, . . ., 8} in disjoint cycle form. [For example, a2 = (0)(124875)(36).] 61. Write the permutation corresponding to Roo in the left regular repre- sentation of D, in cycle form