The definitions of a divisible group that I have seen all seem to assume abelian is an a priori property of the group. My question is as to whether or not it is known that--given a non-torsion element $\tau\in\mathbf{G}$* and $n\in\mathbb{N}$, do we know if $\exists \tau'\in\mathbf{G}_\mathbb{Q}$ such that $\tau=\tau'^n$?

The motivation is that--if this never happens--then the fields $\overline{\mathbb{Q}}^\tau$ should be sort of "maximal" subfields of $\overline{\mathbb{Q}}$ since fixing by the generator would be equivalent to fixed by any power of it.

*My TeX was not rendering properly when I wrote $\mathbf{G}_\mathbb{Q}$ here, so I just included it as a sidenote rather than have it look like a mess.