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Math Help - homomorphic images

  1. #1
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    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.
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  2. #2
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    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.
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  3. #3
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    So those are the homomorphic images of D4?

    What's Q then?
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  4. #4
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    Quote Originally Posted by nanotek887 View Post
    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,
    i^2=j^2=k^2=ijk=-1
    They are a subgroup of the multiplicative group of the Quaternions.
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  5. #5
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    But are the homomorphic images of Q the subgroups of anything else like the one before?
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  6. #6
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    Similarly the homomorphic images of Q are the subgroups of D4
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