Prove that in a group ab2 a2b2 if and only if ab ba Prove

Prove that in a group, (ab)^2 = a^2b^2 if and only if ab = ba. Prove that in a group, (ab)^-2 = b^-2a^-2 if and only if ab = ba Solve, the equation axb

Solution

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 .