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40. In S, find a cyclic subgroup of order 4 and a noncyclic subg 140 of order 4. 41. Suppose that B is a 10-cycle. For which integers i between 2 and 10 is Bi also a 10-cycle? 42. In S,, find elements o and B such that lal = 2, IBI = 2, and laBI = 3. 43. Find group elements a and B in S, such that lal = 3, IBI = 3, and laBl = 5. 44. Represent the symmetry group of an equilateral triangle as a group of permutations of its vertices (see Example 3). 45. Prove that S, is non-Abelian for all n 2 3. 46. Prove that A is non-Abelian for all n 2 4. 47. For n 2 3, let H = {B E S, I B(1) = 1 or 2 and B(2) = 1 or 2}. Prove that H is a subgroup of S. Determine |HI. 48. Show that in S,, the equation x2 = (1234) has no solutions but the equation x' = (1234) has at least two. 49. If (ab) and (cd) are distinct 2-cycles in S, prove that (ab) and (cd) commute if and only if they are disjoint. 50. Let a be a 2-cycle and B be a t-cycle in S. Prove that aBa is a t-cycle. 51. Use the previous exercise to prove that, if a and B belong to S, and B is the product of k-cycles of lengths n, n2, . . ., no, then aBa is the product of k-cycles of lengths n, n2, . . . nx 52. Let o and B belong to S . Prove that BaB- and a are both even or both odd. 53. What is the smallest positive integer n such that S, has an element of order greater than 2n