Solve

43. Let G be a group and let g E G. If z E Z(G), show that the inner automorphism induced by g is the same as the inner automorphism induced by zg (that is, that the mappings , and o, are equal). 44. Show that the mapping a - log , a is an isomorphism from R+ under multiplication to R under addition. 45. Suppose that g and h induce the same inner automorphism of a group G. Prove that h- g E Z(G). 46. Combine the results of Exercises 43 and 45 into a single "if and only if" theorem. 47. If x and y are elements in S (n 2 3), prove that o = p implies x = y. (Here,