- #1

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

Let D = D_{8}be dihedral of order 8 so

D = {u,y,y^{2},y^{3},x,xy,xy^{2},xy^{3}}

where x^{2}= u, y^{4}= u, and yx = xy^{-1}.

Let g = xy^{i}for some integer i. Prove that g^{2}= u.

I know that y

^{4}= u. So then,

g = xy

^{4}= xu = x. Then

g

^{2}= x

^{2}= u

which is what I am trying to prove.

Now if i = 1 then,

g = xy. Then

g

^{2}= xy xy = x yx y = x xy

^{-1}y. Then

xx y

^{-1}y = x

^{2}y

^{-1}y = u y

^{-1}y since

x

^{2}= 2. Then

u y

^{-1}y = u u = u since

y

^{-1}y = 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.