What Are the Cosets in Q/Z(Q)?

[missavvy]

Homework Statement

Find the cosets in Q/Z(Q)

Homework Equations

The Attempt at a Solution

So Z(Q) is the centre of Q..
Then Z(Q) is normal in Q.

I don't get what the cosets would be without any given elements of Q or Z(Q)..
But I'm assuming since it is the centre of Q there is some trick?

Thanks.

 

[rasmhop]

What is Q here? The quaternion group? I will assume it is.

What is the center of Q? Does i commute with every element? does j? does k? does -1? etc.? Once you have determined the center you should be able to find representatives for every coset in Q/Z(Q).

 

[losiu99]

Hmm, is the operation simply addition? If so, Q is commutative, and Z(Q)=Q.

 

[missavvy]

I get:
{1,-1} = Z(Q)= Z(Q)(-1)
{i, -i} = Z(Q)i = Z(Q)(-i)
{j, -j} = Z(Q)j = Z(Q)(-j)
{k, -k} = Z(Q)k = Z(Q)(-k)

is this correct?

 

[losiu99]

Yes, this is correct. Sorry for the first answer, I thought Q ment rationals...