Rationally powered not implies nilpotent

Statement
It is possible to construct a rationally powered group (i.e., a group that is uniquely $$p$$-divisible for all primes $$p$$) that is not a nilpotent group.

Proof
The easiest example is the following group $$G = GA^+(1,\R)$$: $$G$$ is the group of all (affine) linear maps from $$\R$$ to $$\R$$ with positive leading coefficient, under composition. Explicitly:

$$G = \{ x \mapsto ax + b \mid a,b \in \R, a > 0 \}$$

Then, we have that:


 * $$G$$ is rationally powered: See GAPlus(1,R) is rationally powered
 * $$G$$ is not nilpotent: In fact, GAPlus(1,R) is centerless