This is my first semester dealing with proofs and it's a little difficult for me. The question at hand is as follows:
Prove that if (ab)^2 = a^2 b^2 in a group G, then ab = ba.
What I did was this:
. . .(ab)^2 = a^2 b^2 ---> a^2 b^2 = a^2 b^2
Then I multiplied on the left by a^-1 to get:
. . .(a^-1)(a)ab^2 = (a^-1)(a)b^2 ----> ab^2 = ab^2
Then I multiplied on the left again by b^-1 to get:
. . .(b^-1)(b)ba = a(b^-1)(b)b ----> ab = ba
The problem is I don't know if I did this right or if I'm even on the right track. Am I showing what the problem is asking?