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37. 38. 39. . Explain why SB contains subgroups isomorphic to Z1 41. 42. Show that the mapping (110.1 + bi) = a " bi is an automorphism of the group of complex numbers under addition. Show that (p pre- serves complex multiplication as wellthat is, (say) = gb(x)d1(y) for all x and yin C. (This exercise is referred to in Chapter 15.) Let G 3 {a + bVEI a, b are rational} Ht: if] 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.) and a, b are rational}. Prove that Z under addition is not isomorphic to Q under addition. 5, U(16), and DE. Let C be the complex numbers and _ [a crib] M F { b a Prove that C and M are isomorphic under addition and that (3* and M'", the nonzero elements of M, are isomorphic under multiplication. Let R\" = {0:11, (12,. . . , an) la}. E R}. Show that the mapping :(a1, a2, . . . , an) 4 (mal, \"oz, . . . , no\") is an automorphism of the group R\" under componentwise addition. This automorphism is n I _ r! I cab ER}