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Math Help - Homomorphic images of the dihedral group D4 of order 8

  1. #1
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    Homomorphic images of the dihedral group D4 of order 8

    Find all homomorphic images of the dihedral group D4 of order 8, and give a homomorphism which produces each image. Give a familiar group that is isomorphic to each of the images.
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    Re: Homomorphic images of the dihedral group D4 of order 8

    hint: by the first isomorphism theorem, any homomorphic image of D4 is isomorphic to a quotient group of D4.

    so, first, find all the normal subgroups of D4. if memory serves me right, there are only 4 of them.
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    Re: Homomorphic images of the dihedral group D4 of order 8

    [r = rotation, s = reflection]

    {1, r^2}
    {1, r, r^2, r^3}
    {1, r^2, s, sr^2}
    {1, r^2, sr, sr^3}

    i think its right, but i don't know what to do after this
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  4. #4
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    Re: Homomorphic images of the dihedral group D4 of order 8

    call the subgroups H,K,M and N (we should also probably include the trivial normal subgroups {1} and D4).

    find D4/D4 ≅ {1}, D4/H, D4/K, D4/M, D4/N and D4/{1} = D4.

    here's what you know: K,L, and M are of order 4, so D4/K, D4/M and D4/N are of order 2. that makes describing D4/K, D4/M and D4/N easy.

    so the only really "interesting example" is D4/H, which is of order 4. there are only 2 group types of order 4.
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    Re: Homomorphic images of the dihedral group D4 of order 8

    What is the homomorphic image then?
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    Re: Homomorphic images of the dihedral group D4 of order 8

    calculate D4/H. i'll get you started:

    H = {1, r^2}. so here are the 4 cosets:

    H
    Hr = {r,r^3}
    Hs = {s, r^2s} = {s,sr^2}
    Hsr = {sr, r^2sr} = {sr,sr^3}

    so D4/H = {H,Hr,Hs,Hsr}.

    now...make a multiplication table. is D4/H abelian? is it cyclic? what are the orders of each coset?
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    Re: Homomorphic images of the dihedral group D4 of order 8

    i think it is both abelian and cyclic and orders are 1, and 3?
    I'm confused
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    Re: Homomorphic images of the dihedral group D4 of order 8

    yes, you are. how can any group of order 4 possibly have an element of order 3?????

    if you think it is cyclic, prove it. which element has order 4?
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    Re: Homomorphic images of the dihedral group D4 of order 8

    According to Lagrange's theorem every element must have an order that divides 4.
    So, if we take any non-identity element, the order will be be 2 or 4.
    If there is one of order 4, then the group is cyclic.

    And all cyclic groups are abelian
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  10. #10
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    Re: Homomorphic images of the dihedral group D4 of order 8

    all of this is true, but you have yet to convince me that you have calculated ANY of the orders of {H,Hr,Hs,Hsr}.

    you seem to be convinced this group has an element of order 4. i ask you: which one?
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    Re: Homomorphic images of the dihedral group D4 of order 8

    r r^3
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  12. #12
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    Re: Homomorphic images of the dihedral group D4 of order 8

    so you claim that Hr = {r,r^3} has order 4. hmm.

    (Hr)(Hr) = Hr^2 = {r^2,r^4} = {r^2,1} = H.

    it appears to me that Hr has order TWO.
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