Show that the mapping b(a + bi) = a - bi is an automorphism of the group of complex numbers under addition. Show that b pre- serves complex multiplication as well-that is, b(xy) = b(x)$(v) for all x and y in C. (This exercise is referred to in Chapter 15.) 38. Let G= {a + bV21 a, b are rational} and 26 b are rational Show that G and H are isomorphic under addition. Prove that G and H are closed under multiplication. Does your isomorphism preserve multiplication as well as addition? (G and H are examples of rings- a topic we will take up in Part 3.) 39. Prove that Z under addition is not isomorphic to Q under addition. 40. Explain why 5, contains subgroups isomorphic to Zy, U(16), and Dg 41. Let C be the complex numbers and Prove that C and M are isomorphic under addition and that C* and M*, the nonzero elements of M, are isomorphic under multiplication. 42. Let R" = ((a,, a . . . , a,) la, E R). Show that the mapping d: (a,, , a.) -Az, -a) is an automorphism of the group R" under componentwise addition. This automorphism is called inversion. Describe the action of * geometrically. 43. Consider the following statement: The order of a subgroup divides the order of the group. Suppose you could prove this for finite permutation groups. Would the statement then be true for all finite groups? Explain. 44. Suppose that G is a finite Abelian group and G has no element