Express the following permutations in cycle notation Explain

Express the following permutations in cycle notation. Explain your solution. pi pi where pi is the permutation from part (b). pi^-1 where n is the permutation from part (b).

Solution

a) (1 2 4) (3 6 5)

    since 1 goes to 2, 2 goes to 4 n four goes to 1 completes one cycle. And 3 goes to 6, 6 goes to 5 and 5 goes to 3 completes another cycle.

b)(1 2 3 4 5 6)

    since 1 --> 2 --> 3-->4-->5 -->6-->1

c)pi * pi = (1 2 3 4 5 6) (1 2 3 4 5 6) = (1 3 5 6) (2 4)

     1 ----------------> 2 (for 1st permutation)

the 2 is send to 2nd permutatn....there 2 goes to 3. So, (1, 3.....)

     3----(1st pi)------->4---(2nd pi)------>5      so, (1,3,5.....)

     5----(1st pi)-------->6 ---(2nd pi)------>1(cycle completes)     So, (1,3,5,6)

now taking the next smallest number...its 2.

   2----(1st pi)----->3---(2nd pi)----->4   so, (2,4)

s, the product is (1 3 5 6) (2 4)

d) (6 5 4 3 2 1)