how does this look:
if xy=e, then x=y^-1, so |x|= 2 contradiction.
if xy=e, then y=x^-1, so |y|=3 contradiction.
if xy=x, then y=e contradiction, if x=x(y^-1), then (y^-1)=e contradicition.
suppose xy=x^2, y=x^3=e contradicition,
xy=y, then x=e (by right hand cancellation) contradicition.
I am not sure, what to do if xy=yx, is that a contradicion due to the fact that G is a non-abelian group?