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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

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up vote 3 down vote accepted

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.

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You know your groups... ;-) +1 – amWhy Apr 9 '13 at 0:33
It's exercise 10.2 in Rotman. See also the remark after Example 10.8 on p. 320. Neither the term "uniquely divisible" nor "uniquely dividable" is formally defined there, though. But as Babak writes, it just means that a group is divisible and torsion-free. – eins6180 Nov 20 '15 at 10:52

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