# Dihedral Group of Order 8

wubie
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.

Hurkyl
Staff Emeritus
Gold Member
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.

wubie
Thanks Hurkyl. I still have some questions regarding this dihedral group.

Part of the question states:

Let g = xyi for some integer i.

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:

For each a which is an element of G there exists a-1 which is an element of G such that

a o a-1 = a-1 o a = u

If so, where are the inverses of the elements y, y2, y3, xy, xyy2, xy3 in the group D8?

Hurkyl
Staff Emeritus
Gold Member
Why not -4 <= i <= 4 since i can be any integer?

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.

If so, where are the inverses of the elements y, y2, y3, xy, xyy2, xy3 in the group D8?

There are only 64 different ways to multiply 2 elements in D8. Exhaust! More pragmatically, you can use the fact I mentioned above, coupled with the fact that (xy)-1 = y-1x-1 to compute inverses.

wubie
Thanks alot Hurkyl. That was very helpful to me. I really appreciate it.

Cheers.