Abelian group (Commutative group) Prove that if in a group $(ab)^2= a^2 b^2$ then the group is commutative.  I am having a hard time doing this. Here is what I have so far: Proof:  $a^2 b^2= a^1 a^1 b^1 b^1$ =$aa^{-1}bb$ =ebb Hence,$aa^{-1}=e$  I am stuck, I do not know if this is the right process in proving this

$aabb=(ab)^2=abab$ implies that $a^{-1}aabbb^{-1} = a^{-1}ababb^{-1}$. So, $eabe = ebae$. Hence $ab=ba$.
