Question

The following questions pertain to permutations in S8 (a) Decompose the permutation (1 2 3 4 5 6 7 %) into a product of disjoint 13 6 4 1 8 2 5 7 cycles. = (b) Decompose the permutation T= (1,4, 3) (5,7,6,8) into a product of transpositions. (c) Determine whether o and T are even or odd permutations. (d) Compute the product OT.

Answer 1

The following Questions pertain to permutations in S8. into product of [a] De compose the permutation 1 2 3.4 5 6 7 8 1 8 2 57 disjoint cycles. Any:- We start with our first element. 3 6 4 = (134) (26)(587) o(1) = 3 0 (3) = 4, 014) = 1 5 (2) = 6 o(6)=2 015) = 8 518)=7, 017) - 5.10 / (b.] Decompose the permutation 7= (1,4,3) (5,7,6,8) into a product of transpositions. Any: Tromsposition is cycle length of two. first we break the permutation into disfaint cycle. then we pair every element with first element of the cycle in reverse seder. t is altody ento desjaint cycles. T = ( 1,4,3) (5,7,6,8) Y = (1,3) (1,4) (5,8) (516) (5,7)

[c. Determine whether o and T are even or add Permutations Any:s Even permutation can be obtained as the composition of an even number of transpositions. While an odd permutation can be abtained by an odd number of transpositions (134) (26) (587) Im composition of transposition os (14) (13) (26) 157) (58) : Total number of transposition = 5. which is odd number. hence o is odd Permentation Now T = (1,3) (1,4) (5,8) (5, 6) (57) Total number of transposition - 5 Which is odd hence T is add permutation. mumber. [d] compute the product ot. Any First decompose & and X into disjoint cycle. then Multiply it. TF (1.34)26) (5,87) (1,43) (5,7,6,8)

o T = 1 2 3 4 5 6 78 3 6 4 18 257 7 4 3 517 6 8 2 4 3 1 7 6 8 5 2 o T first Row of and second row of T should be some. 2. 1 T To = 2 3 4 5 6 7 8 Mla 4 1 8 7.5 6 1 2 3 4 5 6 7 8 36 4 1 8 2 57 o T = 3 2 4 1 8 7 56 3 64 1 8 2 5 7 disjoint Cycle OT = (3) (267) (4) (1) (8) (5) = (267) ot