Proving G is Abelian: A Group Theory Case

[mansi]

seeing lots of group theory here after a really long time...
let G be a finite group of order n, where n is not divisible by 3. suppose
(ab)^3 = a^3 b^3 ,for a, b in G . prove that G is abelian.

 

[mansi]

i guess i must( or rather i am supposed to) add my thought process...
since 3 does not divide n (n,3)=1 so we have un +3v=1...now for any x in G, x = x^(un+3v)= x^3v.
so G ={y^3 / y is in G}...using this i think we're supposed to show that ab=ba for all a in G. here's where I'm stuck.

 

[Hurkyl]

It's frequently useful to reverse what's inside an exponent. For example:

b * (ab)^3 * a = b(ababab)a = (ba)^4