Let f and g be permutations on the set 123456789 defined as...

Let f and g be permutations on the set {1,2,3,4,5,6,7,8,9}, defined as follows f= ( 1 2 3 4 5 6 7 8 9 2 8 9 6 5 7 4 1 3 ) g= ( 1 2 3 4 5 6 7 8 9 3 9 6 2 4 5 7 1 8 ) Write each of the following permutations as a product of disjoint cycles, separated by commas (e.g. (1,2),(3,4,5),… ). Do not include 1-cycles (e.g. (2) ) in your answer. (a) f g= _________. (b) f^(-1)=________. (c) f g f^(?1)=_________.

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00:01 In this question we are given with f is equal to 6 4, 2, 5 and 7, 2, 5, 3, 4, 8.

00:18 So first we have to find the product of this disjoint cycles.

00:22 So first let us write this as 1, 2, 3, 4, 5, 6, 7, 8.

00:31 6 goes to 4, 4 goes to 2, 2 goes to 5 and again 5 goes to 6.

00:49 So we have here 1, 3, 7 and 8.

00:55 Similarly for this we get 1, 2, 3, 4, 5, 6, 7, 8.

01:03 So in the same mapping we get 1 5 4 8 3 2 7 sorry here we get 6 2 and 7 so simplifying this in the same mapping process we have mapped this upon multiplying this disjoint cycles we get 1 2 3 4 5 6 7 8 so we get 162, 8 3, 4, 5, 7.

01:55 So from this we can write f as 264, 8, 7 5, 3.

02:07 And g is equal to 48, 3 71, 265.

02:17 So let us solve the first subdivision.

02:24 The first subdivision is to find f multiplied by g.

02:31 So we have f and g here.

02:33 So f is 1, 2, 3, 4, 5, 6, 7, 8, sorry 1, 6, 2, 8, 3, 4, 5, 5, 6, 7, 7, 7, 1, 6, 6, 8, 3, 3, 4, 5, 7, 7, 7, 1, 6, 6, 7, 7, 1, 6, 5, 7, 7.

03:01 1, 2, 3, 4, 5, 6, 7, 8.

03:07 So here we have the map.

03:09 2, 6, 7, 8, 4, 5, 1, 3, which is equals to.

03:24 So, product of disjoint cycle is 1, 2, 3, 4, 5, 6, 7, 8...