What Are the Elements of the Quotient Group D4/N?

[Felix542]

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 :) .

 

[I like Serena]

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 :) .

Welcome to PF, Felix542!

N(24) and N(13) are the same coset (why?), so you can pick either.

Btw, these are "right" cosets and not "left" cosets, so you should write (24)N instead for an element of the quotient group D4/N.

 

[Felix542]

Welcome to PF, Felix542!

N(24) and N(13) are the same coset (why?), so you can pick either.

Btw, these are "right" cosets and not "left" cosets, so you should write (24)N instead for an element of the quotient group D4/N.

Thanks for your reply ! Thanks for the welcome .
Oops yeah sorry meant right cosets sorry . Well I think they are the same because N(24 ) = {( (24),(13)}=N(13) ?

So would this be a suitable answer d4/N={N , N(1432) , N(12)(34) , 24(N)} ?

 

[I like Serena]

Thanks for your reply ! Thanks for the welcome .
Oops yeah sorry meant right cosets sorry . Well I think they are the same because N(24 ) = {( (24),(13)}=N(13) ?

So would this be a suitable answer d4/N={N , N(1432) , N(12)(34) , 24(N)} ?

Yep!

 

[Felix542]

Okay thank you :) . Sorry , just one more quick question ! Would there be other valid answers ? I'm not sure if this is correct but since say we have shown N(24 )= N(13) , we could equally have an equivalent answer for D4/N ?

 

[I like Serena]

Each element in D4/N is unique.
It's just that there is more than one way to write each element down.

You could for instance also have written {(24),(13)} to represent the element (13)N.

 

[Felix542]

So for future reference I'm looking each time for a coset representative to form the quotient group - I hope this is the right term to use here ?

 

[I like Serena]

Hmm, coset representative sounds as if you're talking about one element from the coset, like (13).
I guess you could use that to represent the coset, but I would tend to stick to (13)N or {(24),(13)}.


From wikipedia:
"Let N be a normal subgroup of a group G. We define the set G/N to be the set of all left cosets of N in G, i.e., G/N = { aN : a in G }."

and:
"gH = {gh : h an element of H } is a left coset of H in G"


I prefer to use either the notation aN, or to write out the set itself.