Abstract Algebra Theory and Applications - Computer Science ...

86 CHAPTER 4 PERMUTATION GROUPS(a) (1345)(234)(b) (12)(1253)(c) (143)(23)(24)(d) (1423)(34)(56)(1324)(e) (1254)(13)(25)(f) (1254)(13)(25) 2(g) (1254) −1 (123)(45)(1254) (h) (1254) 2 (123)(45)(i) (123)(45)(1254) −2(j) (1254) 100(k) |(1254)|(l) |(1254) 2 |(m) (12) −1(n) (12537) −1(o) [(12)(34)(12)(47)] −1 (p) [(1235)(467)] −13. Express the following permutations as products of transpositions and identifythem as even or odd.(a) (14356)(c) (1426)(142)(e) (17254)(1423)(154632)4. Find (a 1 , a 2 , . . . , a n ) −1 .(b) (156)(234)(d) (142637)5. List all of the subgroups of S 4 . Find each of the following sets.(a) {σ ∈ S 4 : σ(1) = 3}(b) {σ ∈ S 4 : σ(2) = 2}(c) {σ ∈ S 4 : σ(1) = 3 and σ(2) = 2}Are any of these sets subgroups of S 4 ?6. Find all of the subgroups in A 4 . What is the order of each subgroup?7. Find all possible orders of elements in S 7 and A 7 .8. Show that A 10 contains an element of order 15.9. Does A 8 contain an element of order 26?10. Find an element of largest order in S n for n = 3, . . . , 10.11. What are the possible cycle structures of elements of A 5 ? What about A 6 ?12. Let σ ∈ S n have order n. Show that for all integers i and j, σ i = σ j if andonly if i ≡ j (mod n).13. Let σ = σ 1 · · · σ m ∈ S n be the product of disjoint cycles. Prove that the orderof σ is the least common multiple of the lengths of the cycles σ 1 , . . . , σ m .14. Using cycle notation, list the elements in D 5 . What are r and s? Write everyelement as a product of r and s.