# Dihedral Group of Order 8

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.

## Answers and Replies

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.

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?

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.

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

Cheers.