What is the difference between a divisible group and a uniquely divisible group?

Question

I have been looking a cohomology where it is known that uniquely divisible modules have trivial cohomology. But in the case of $\mathbb{Z}$-modules I know $\mathbb{Q}$ has trivial cohomology since its "uniquely divisible" but $\mathbb{Q}/\mathbb{Z}$ is not cohomologically trivial but they are both divisible groups, so what exactly is the definition of a divisible group? Since I want to see if $Hom(L,\mathbb{R})$ is uniquely divisible (L some abelian group) but im not quite sure how to do this since I dont know a good definition of uniquely divisible

Thank you

Answer 1

We have an exercise in the book of J.J.Rotman about theory of groups saying:

If $x\in G$, then any two solutions to the equation $ny=x$ differ by an element $z$ with $nz=0$. $y$ is unique if $G$ is torsion-free group.

$\mathbb Q$ is torsion-free and divisible so it is uniquely dividable but $\mathbb Q/\mathbb Z$ is not torsion-free. In fact it is torsion group.