Proving g2 = u in 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]

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]

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 y^m = 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 a lot Hurkyl. That was very helpful to me. I really appreciate it.

Cheers.