Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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

share|improve this question

1 Answer 1

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.

share|improve this answer
    
You know your groups... ;-) +1 –  amWhy Apr 9 '13 at 0:33

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.