Rationally powered group

Definition
A group $$G$$ is termed rationally powered or uniquely divisible if it satisfies the following equivalent conditions:


 * 1) For every $$u \in G$$ and every natural number $$n$$, there is a unique $$v \in G$$ such that $$v^n = u$$.
 * 2) $$G$$ is a powered group for all prime numbers.
 * 3) For any integers $$m,n$$ with $$n \ne 0$$, and for any $$g \in G$$, there exists a unique $$h \in G$$ such that $$g^m = h^n$$.

More generally, we can talk of a powered group for a set of primes.

Facts

 * A nilpotent group is rationally powered iff it is divisible and torsion-free. This follows from the more general fact that in a nilpotent torsion-free group, if roots exist, they are unique.