Question

The following questions pertain to permutations in

(a) Decompose the permutation   into a product of disjoint cycles.

(b) Decompose the permutation  into a product of transpositions.

(c) Determine whether σ and τ are even or odd permutations.

(d) Compute the product σ τ.

Answer 1

<sup>a (a) know that finite set can be every permutation of written as cycle of disjoint cycles. as product Let us show this an example. by 2 3 5 4 be Let us consider 1 2 3 1 which can be permutation 3 as expressed 1 2 3 5 4 can be Every * Recall: A transposition is cycle of order a. permutation expressed of transpositions. consider permutation 4 1 5 2 3 as product 1 2 3 4 5 Let US Then, 1 2 3 4 13:3)-611) (3) (49) (13) (3) = product of transpositions

is to А be an even permutation said permutation if it can be expressed product of number of transposition. the even as is to be o dd A can permutation said permutation if it be expressed product of of odd number of transposition. as the Let us illustrate this by some example: permutation 1 2 3 4 be Let 2 4 3 6. 5 1 2 5 4 Then, 1 2 3 2 5 6 2 3 1 5 4 3. 6 0 1 2 HCGS ) - (69) (1) (33) (4) even product of transposition so, 1 2 3 4 5 6 2 4 even permutation 3 6 5 1 Now, 2 1 3 45 (623)63) 4 2 5 23 1 - 1 4 4 1 2 1 35 53 2 product of odd transposition i an odd This permutation.

cel Let az and 002) (ar) - ocan) Cocan (acand clar) az an be two permutation. T = t(an) Then, ro (ortos) o (eca) & (ecan)) ( (az)) o(a2) of two permutation. is the product</sup>