right and left cosets

mandy123 · #1 $Old$ December 15th, 2008, 08:27 AM

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

ThePerfectHacker · #2 $Old$ December 15th, 2008, 03:27 PM

Quote:

Originally Posted by mandy123 $View Post$

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?

mandy123 · #3 $Old$ December 15th, 2008, 08:08 PM

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???