prove that if (ab)^2 = a^2b^2 in a group G, then ab = ba?

Answer 1

assume (ab)^2 = a^2 b^2
but (ab)^2 = abab
so we have that abab = a^2b^2
multiply from the left by a^{-1} and from the right by b^{-1} both sides of the equality,
then ba = ab which means that G is abelian .

Answer 2

(ab)^2 = (ab)(ab) = a(ba)b = a^2b^2 Now, let's bring in the inverses:

a^-1 a (ba) b b^-1 = ba = a^-1 a^2 b^2 b^-1 = ab QED