Thread: 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

Please help

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

Take an element not found in this set, say and this gives us .

Take an element not found in any of these sets, say and this gives us .

Can you continue?

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???