I had never encountered this type of question until a couple of minutes ago during a midterm...
It went something like this
Prove that If x^3=x for all x in G, then G is abelian.
I stated that since x^3=x, the only possibilities for x are 1,0 and -1. In this case I proved that all three combinations of these numbers are abelian.
Looking through some notes, should I have introduced a y with the same properties and then proven something like x^3 * y^3 =xy?
How is this proof done?