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Math Help - Cosets and Normal Subgroups

  1. #1
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    Cosets and Normal Subgroups

    The group D_8 has two generators x, y with the relations x^4 = e, y^2 = e, xy = yx^3. Let H be the cyclic group generated by x. Write down the four elements in H. Write down the four elements in the left coset yH and the four elements in the right coset Hy. Show that yH = Hy.

    This is what I have so far.

    H =  <x>  = \{e, x, x^2, x^3\}
    yH = \{ye, yx, yx^2, yx^3\} = \{y, yx, yx^2, yx^3\}
    Hy = \{ey, xy, x^2y, x^3y\} = \{y, x^2y, x^3y, yx^3\} (because xy = yx^3)

    Now I am supposed to show that yH = Hy and I cannot seem to manipulate the sets so they are equal. Thanks in advance for any help.
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  2. #2
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    Ok, let's take the Hy and see how we could get it to look like yH. So the y's are the same, no issue there. Try this for the x^{2}y in Hy:

    x^{2}y=x(xy)=x(yx^{3})=(xy)x^{3}=yx^{3}x^{3}=yx^{6  }=yx^{4}x^{2}=yex^{2}=yx^{2}, which is in yH.

    I think you'll find that the other elements behave in a similar fashion. Can you finish from here?
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  3. #3
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    Thank you so much for your help. I knew the answer was staring at me in the face. Thank you for pointing it out to me.
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  4. #4
    A Plied Mathematician
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    You're very welcome. Have a good one!
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