1. ## homomorphic images

Last problem I can't figure out.

Find all the homomorphic images of the following groups, D4 and Q.

I would really really appreciate any help. Thanks in advance.

2. The homomorphic image of $\displaystyle D_4$ must be a subgroup of Q.

Those all the possible homomorphic images are subgroups of Q.

Thus,
$\displaystyle \{1\}$
$\displaystyle \{1,-1\}$
$\displaystyle \{1,-1,i,-i\}$
$\displaystyle \{1,-1,j,-j\}$
$\displaystyle \{1,-1,k,-k\}$
$\displaystyle \{1,-1,i,-i,j,-j,k,-k\}$

You can also look heir.

3. So those are the homomorphic images of D4?

What's Q then?

4. Originally Posted by nanotek887
So those are the homomorphic images of D4?
Yes, any homomorphic image must be one of those.
What's Q then?
That is the Octions,
$\displaystyle i^2=j^2=k^2=ijk=-1$
They are a subgroup of the multiplicative group of the Quaternions.

5. But are the homomorphic images of Q the subgroups of anything else like the one before?

6. Similarly the homomorphic images of Q are the subgroups of D4