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36. 37. 38. 39. . Explain why Sl3 contains subgroups isomorphic to 215, U06), and D3. 41. 42. Prove or disprove that U (20) and U(24) are isomorphic. Show that the mapping d:(a + hi) 2 a 121' is an automorphism of the group of complex numbers under addition. Show that qb pre- serves complex multiplication as wellthat is, qb(xy) : (,'b(x)(y) for all x and y in C. (This exercise is referred to in Chapter 15.) Let G = {a + (BA/El a, bare 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. Let C be the complex numbers and " ll\" "'l M P b a Prove that C and M are isomorphic under addition and that C* and M*, the nonzero elements of M, are isomorphic under multiplication. Let R\" :{(a1, (12,. .., an) I a}. E R}. Show that the mapping qb: (a1, a2, . . . , a\") > (al, -a2, . . . , -an) is an automorphism of the group R'1 under componentwise addition. This automorphism is called inversion. Describe the action of '35 geometrically. a,bER}. . Consider the following statement: The order of a subgroup divides the order of the group. Suppose you could prove this for nite permutation groups. Would the statement then be true for all nite groups? Explain. . Suppose that G is a nite Abelian group and G has no element of order 2. Show that the mapping g > g2 is an automorphism of G. Show, by example, that there is an innite Abelian group for which the mapping g > g2 is one-toone and operation-preservin g but not an automorphism