Let a and b be elements of a group G. Prove that G is abelian if and only if (ab)^(2) = a^(2) * b^(2)
consider $\displaystyle a,b \in G$
if $\displaystyle a^2b^2 = (ab)^2$ then $\displaystyle aabb = abab$
since G is a group, $\displaystyle a^{-1}, b^{-1} \in G$
consider $\displaystyle a^{-1}(aabb)b^{-1} = a^{-1}(abab)b^{-1}$ and use the properties of groups and you'll see why G is abelian.