Group G is abelian iff (ab) ^2=a^2b^2 for all a, b in G

Group G is abelian iff (ab) ^2=a^2b^2 for all a, b in G

Solution: 

Since G is abelian so

ab=ba

Now, (ab)^2    = (ab)(ab)

                        = a(ba) b [ by associativity]

                        = a(ab)b    [ ab=ba,as group is abelian]

                        = (aa)(bb)

                        =a^2 b^2

Hence    (ab) ^2=a^2b^2

Conversely,

Let,    (ab) ^2     =    a^2b^2

         (ab)(ab)     =    (aa)(bb)

          a(ba)b       =    a(ab)b [by associativity]

    a^-1 a(ba)bb^-1=a^1(ab)b^b-1 [multiplying by a^-1 from left and b^-1 from right]

            e ba e  =     e ab e

            ba       =    ab

Hence G is abelian.