homomorphic images

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December 2nd 2006, 07:09 PM
nanotek887

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.
December 3rd 2006, 06:37 AM
ThePerfectHacker

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

Those all the possible homomorphic images are subgroups of Q.

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

You can also look heir.
December 3rd 2006, 08:49 AM
nanotek887

So those are the homomorphic images of D4?

What's Q then?
December 3rd 2006, 08:52 AM
ThePerfectHacker

Quote:

Originally Posted by nanotek887

So those are the homomorphic images of D4?

Yes, any homomorphic image must be one of those.

Quote:

What's Q then?

That is the Octions,
$i^2=j^2=k^2=ijk=-1$
They are a subgroup of the multiplicative group of the Quaternions.
December 3rd 2006, 09:36 AM
nanotek887

But are the homomorphic images of Q the subgroups of anything else like the one before?
December 3rd 2006, 11:12 AM
ThePerfectHacker

Similarly the homomorphic images of Q are the subgroups of D4

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