- #1
Felix542
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Homework Statement
Let D4 = { (1)(2)(3)(4) , (13)(24) , (1234) , (1432) , (14)(23) , (12)(34) , (13), (24) }
and N=<(13)(24)> which is a normal subgroup of d4 .
List the elements of d4/N .
Homework Equations
The Attempt at a Solution
I computed the left and right cosets to prove that N is a normal subgroup of D4 ( this was a previous part to the question )
The left cosets looked something like ;
N (1)(2)(3)(4) = {((1)(2)(3)(4) , (13)(24)}
N (1234) = {(1432),(1234)}
N (13)(24) = {((1)(2)(3)(4) , (13)(24)}
N (1432) = {(1432),(1234)}
N (14)(23) = {(14)(23) , (12)(34)}
N(12)(34) = {(14)(23) , (12)(34)}
N(13) = {(24,13)}
N(24 ) = {( (24),(13)}
And the right cosets were equal i.e N(1234)=(1234)N . To compute the quotient group d4/N , I know there will be four elements one will naturally be N , but the other three I'm not too sure about . From the above cosets I noticed that say N(14)(23) and N(12)(34) give the same set , but which would I choose to be in d4/N ? This problem is again for , N(24) and N(13) .
Hopefully this makes sense , thank you for any help :) .