A group G is such than a^3 = e for every a in G. Is it abelian?

[F_idl]

Homework Statement

Let G be a group and e its identity. This group has the property that a^3 = e, for every a in G. What I need to do is verify if this condition is sufficient for G to be abelian.

2. The attempt at a solution
I found a non-trivial group for which this is true, namely the group {e, a, b} with ab = ba = e, a² = b and b² = a. Other than this, I'm having a really hard time with this problem. I also tried to come up with a counter-example, but I couldn't even find another group with the property that a^3 = e for every a.

 

[Dick]

That's actually a pretty tough question. There is a counter-example. I found it by googling. It's a matrix group with matrix entries being integers mod 3. You can probably find it too.