- #1

**[SOLVED] Proof Using Def. of Groups**

Hello,

This is my question:

Let G be a group.

i) Let x and y be elements of G. Prove that (xy)

^{2}= x

^{2}y

^{2}iff xy = yx. (Hint: Use the definition g

^{2}= gg).

ii) Using part (i) prove that if g

^{2}= u (the unit element) for all g which is an element of G, then G is abelian.

Now I BELIEVE that I have properly proved the part (i) of the question. But I am not sure how to proceed with part (ii). In fact, the second part question makes me wonder if I did part (i) correctly.

I know that the definition of abelian is:

For every x and y which are elements of G, a group G with the property x o y = y o x is called abelian ( or commutative). To rephrase, I would think this is the same as F(y,x) = F(x,y).

Now I am not sure what the definition would be in context of the question. Is the question saying,

Proposition: If g^{2}= u then xy = yx?

Is that the proposition that I am supposed to prove? And if that is the case, I am still not sure how to use the nfo g

^{2}= u. How does it apply to the relation in part (i)?

In the case of part (i) would this be it?

x o y = x

^{2}y

^{2}. Then

x o u = x = u o x --> x

^{2}* u = x = u * x

^{2}? (In which case 1 would be the identity element. Correct?).

Any help/clarification is appreciated. Thankyou.