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39. 40. 41. 42. 43. 45. 46. (1995 Putnam Competition) Let S be a set of real numbers that is closed under multiplication. Let T and U be disjoint subsets of S whose union is S. Given that the product of any three (not neces- sarily distinct) elements of T is in T and that the product of any three elements of U is in U, show that at least one of the two sub sets T and U is closed under multiplication. If p is an odd prime, prove that there is no group that has exactly p elements of order p. Give an example of a group G with infinitely many distinct sub groups H1,H2,H3, . . . such that H1 C H2 C H3. . . . Suppose a and b are group elements and b i e. If a'lba = b2 and Ial : 3, nd Ibl. Whatis lbl, if lal : 5? What can you say about Ibl in the case where Ial = it? Let a and I) belong to a group G. Show that there is an element g in G such that g1 abg : be. . Suppose G is a group and x3y3 : y3x3 for every x and y in G. Let H = {x E G | lxl is relatively prime to 3}. Prove that elements of H commute with each other and that H is a subgroup of G. Is your argument valid if 3 is replaced by an arbitrary positive integer :1? Explain why or why not. Let G be a finite group and let S be a subset of G that contains more than half of the elements of G. Show that every element of G can be expressed in the form 5132 where 31 and 32 belong to S. Let G be a group and let f be a function from G to some set. Show thatH = {g E G |f(xg) =f(x) for allx E G} is asubgroup of G. In the case that G is the group of real numbers under addition and