Divisible not implies rationally powered

This article gives the statement and possibly, proof, of a non-implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., divisible group) need not satisfy the second group property (i.e., rationally powered group)
View a complete list of group property non-implications | View a complete list of group property implications
Get more facts about divisible group|Get more facts about rationally powered group

Statement

It is possible for a divisible group (i.e., a group in which every element has a $n^{th}$ root for every $n$) to not be a rationally powered group (i.e., there is at least one element and one $n$ for which the $n^{th}$ root is not unique).

Proof

The simplest example is the group of rational numbers modulo integers $\mathbb{Q}/\mathbb{Z}$. We can also take the circle group $\mathbb{R}/\mathbb{Z}$. Also, any general linear group over the field of complex numbers (or in general, over any algebraically closed field of characteristic zero) satisfies the condition.