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Dihedral Group of Order 8

  1. Oct 29, 2003 #1

    I am having trouble understanding groups in my group theory class. I am not confident on how to approach the following question:

    I know that y4 = u. So then,

    g = xy4 = xu = x. Then

    g2 = x2 = u

    which is what I am trying to prove.

    Now if i = 1 then,

    g = xy. Then

    g2 = xy xy = x yx y = x xy-1 y. Then

    xx y-1y = x2 y-1y = u y-1y since

    x2 = 2. Then

    u y-1y = u u = u since

    y-1y = u.

    First question: Is the work I have completed so far correct?

    Second question: Do I need to prove this in a case by case basis? That is, I would think that I would have to prove this for i = 1,2,3,4. Since I have already completed 1 and 4, I would have to do cases in which i = 2,3. Correct?

    This may seem elementry, but like I stated above, my confidence in answering such questions is not great. And my understanding of the material is very weak.

    Any comments, input, help is appreciated.

  2. jcsd
  3. Oct 29, 2003 #2


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    Yes, you do have to prove it for i = 1..4. (actually, you could do it for (i = 0..3).

    The reason is because you can use y4 = u to reduce the general case to one of these 4 selected cases.

    Your work looks correct, except for the typo that you wrote x2 = 2 instead of x2 = u.
  4. Oct 29, 2003 #3
    Thanks Hurkyl. I still have some questions regarding this dihedral group.

    Part of the question states:

    Now, why would I just assume that i = 1 to 4? Why not -4 <= i <= 4 since i can be any integer?

    Also isn't one of the properties of a group that:

    If so, where are the inverses of the elements y, y2, y3, xy, xyy2, xy3 in the group D8?
  5. Oct 29, 2003 #4


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    The same reason you don't need to worry about i > 4.

    Because you know y4 = u, we know that:

    y-1 = y-1 * u = y-1 * y4 = y3

    In general, if m = n mod 4, we can use induction to prove that ym = yn.

    There are only 64 different ways to multiply 2 elements in D8. Exhaust! :smile:

    More pragmatically, you can use the fact I mentioned above, coupled with the fact that (xy)-1 = y-1x-1 to compute inverses.
  6. Oct 29, 2003 #5
    Thanks alot Hurkyl. That was very helpful to me. I really appreciate it.

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