1. ## right/left coset calculation

The following is a worked problem:

So, could anyone please show me how they've calculated (1,2,3)H and H(1,2,3)? I mean, for example for (1,2,3)H how did they get (1,2,3)(1,2)=(1,3)? Similarly for H(1,2,3), how (1,2)(1,2,3)=(2,3)?

Thank you.

2. $$The notation (1,2,3) is a permutation function f with the property f(1) = 2, f(2) = 3, f(3) = 1.$$ And the notation (1,2) is a permutation function g with the property g(1) = 2, g(2) = 1, g(3) = 3. (Whenever an integer is omitted, that means the function fixes that integer, lke 3 in the previous example)

 So (1,2,3)(1,2) is a notation f*g (composition of f and g). Can you calculate (1,2,3)(1,2) now?

3. How about (1,2)(1,2,3)? Since there are two disjoint cycles, I said in the first cycle 1 goes to 2, and in the second cycle 2 goes to 3. Again in the first cycle 2 goes to 1, and in the second cycle 2 goes to 3. So we have:

(1,2)(1,2,3)=(3,2)

The correct answer should be (2,3) not (3,2). But I guess since this is a cycle of length two the order doesn't matter, does it? Is my method correct?

By the way I used the same method with your hint and I managed to get (1,2,3)(1,2)=(1,3).