Thread: Dihedral Group of Order 8 View Single Post
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Hello,

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

 Let D = D8 be dihedral of order 8 so D = {u,y,y2,y3,x,xy,xy2,xy3} where x2 = u, y4 = u, and yx = xy-1. Let g = xyi for some integer i. Prove that g2 = u.

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.

Thankyou.