# Thread: right and left cosets

1. ## right and left cosets

I am suppose to find the right and left cosets of <(1 2 3),2> in S_3 (+) Z_3

(+) is the External Direct Product sign

I know that 0 1 2 make up Z_3
and I think S_3 is 1 2 3

Then how do I start to solve to find the left and right cosets of <(1 2 3),2> in S_3 (+) Z_3

2. Originally Posted by mandy123
I am suppose to find the right and left cosets of <(1 2 3),2> in S_3 (+) Z_3
We have $H=\left< ((123), 2) \right> = \{ (\text{id},0), ((123),2), ((132),1) \}$

Take an element not found in this set, say $((12),1)$ and this gives us $((12),1)H = \{ ((12),1), ((23), 0), ((13), 2) \}$.

Take an element not found in any of these sets, say $((12),2)$ and this gives us $((12),2)H = \{ ((12),2), ((23),1), ((13),0) \}$.

Can you continue?

3. ## is this right

So this is how i finished it

((12),0)H={((12),0), ((23),2), ((13),1)}

((123),1)H={((123),1), ((132),0), ((123),2)}

((123),0)H={((123),0), ((132),2), ((123),1)}

((132),1)H= {((132),1), ((123), 0), ((123),1)

And so the left cosets would be

((12),1), ((23), 0), ((13), 2), ((12),2), ((23),1), ((13),0), ((12),0), ((23),2), ((13),1)

And the right cosets are

((123),1), ((132),0), ((123),2), ((123),0), ((132),2), ((132),1)

would this be right???