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Homework Help: Isomorphism question with D4

  1. May 23, 2010 #1
    1. The problem statement, all variables and given/known data
    Let G=<x, y|x^4=y^4=e, xyxy^–1=e>. Show that |G|≤16. Assuming |G|=16, show G/<y^2> is isomorphic to D4.

    2. Relevant equations

    3. The attempt at a solution
    Here is what I have:
    since xyxy^-1=e, we know that yxy^-1=x^-1=x^3, so we know that x is a conjugate and partions G. So G= <x> union y<x> and |G|<= 16.
    Lets assume that |G|=16.
    So |G|/|<y^2>|=8, thus |G|/|<y^2>|>= |D4|.
    D4=<a,b|a^4=b^2=(ab)^2=e, or ab=ba^3>.
    Let w=x<y^2> , z=y<y^2> and q=e<y^2> (where q is are identity element in G/<y^2>.
    Need to show that G/<y^2>= <w,z|w^4=z^2=(wz)^2=e>.
    1. Since x and y are generates in G, then w and z are generates in G/<y^2>. I am not sure if this is right, could someone explain this to me? (I need to show that w and z generate G/<y^2>).
    2. w^4=e since (x<y^2>)^4=x^4<y^2>=e<y^2>=q.
    3. z^2=e since (y<y^2>)^2=y^2<y^2>=e<y^2>=q.
    4. (wz)^2=(wz)(wz)=e. Since wz=(z^-1)(w^-1), then wz=xy<y^2> and z^-1w^-1=y^3x^3<y^2>=yx^3<y^2>. Thus xy<y^2>=yx^3<y^2>.
    Therefore, G/<y^2> is isomorphic to D4.
  2. jcsd
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