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43. 45. 47. 49. 50. 51. 52. 53. 54. 55. 56. 5'7. 58. 59. of permutations of its vertices (see Example 3). Prove that S" is non-Abelian for all n a 3. . Prove that A\" is non-Abelian for all n a 4. Porn: 3,1etH= {BESHIB(1)= lor2andB(2)= lor2}. Prove that H is a subgroup of Sn. Determine IHI. . Show that in 8,, the equation x2 = (1234) has no solutions but the equation 13 = (1234) has at least two. If (ab) and (ed) are distinct 2-cycles in S", prove that (ab) and (ed) commute if and only if they are disjoint. . Let a: and [3 belong to S". Prove that Ba" and a are both even or both odd. Viewing the members of D4 as a group of permutations of a square labeled 1, 2, 3, 4 as described in Example 3, which geometric sym- metries correspond to even permutations? Viewing the members of D5 as a group of permutations of a regular pentagon with consecutive vertices labeled 1, 2, 3, 4, 5, what geo- metric symmetry corresponds to the permutation (14253)? Which symmetry corresponds to the permutation (25)(34)? Let n be an odd integer greater than 1. Viewing D" as a group of permutations of a regular n-gon with consecutive vertices labeled 1, 2, . . . , n, explain why the rotation subgroup of D,I is a sub- group of A\". Let a}. or2 and :13 be 2-cycles. Prove that (11:12:13 #5 e. Generalize. Show that A 5 has 24 elements of order 5, 20 elements of order 3, and 15 elements of order 2. (This exercise is referred to in Chapter 25.) Find a cyclic subgroup of As that has order 4. Find a noncyclic sub- group of As that has order 4. Show that a permutation with odd order must be an even permutation. Compute the order of each member of A 4. What arithmetic relation- ship do these orders have with the order of A4? Show that every element in A" for n a 3 can be expressed as a 3-cycle or a product of 3-cycles. Show that for n a: 3, Z(Sn) = [3]. Verify the statement made in the discussion of the Verhoe' check digit