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

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.