[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 = x2y2 iff xy = yx. (Hint: Use the definition g2 = gg). ii) Using part (i) prove that if g2 = 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, 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 g2 = u. How does it apply to the relation in part (i)? In the case of part (i) would this be it? x o y = x2y2. Then x o u = x = u o x --> x2 * u = x = u * x2? (In which case 1 would be the identity element. Correct?). Any help/clarification is appreciated. Thankyou.