Does $\Bbb Q/ \Bbb Z$ have a proper subgroup that is not finite? 
Does $\Bbb Q/ \Bbb Z$ have a proper subgroup that is not finite?

I suspect it does not. However since we could take a subgroup of all $p$ sets $\{\frac{1}{p} + \Bbb Z\}$ if we consider $p$ to be arbitrarily large is this the same thing as a non-finite subgroup?

The set of elements of the form $$\left\{\frac{k}{2^n} + \mathbb{Z} : k, n \in \mathbb{Z}\right\}$$
gives a proper subgroup which is certainly infinite. Replace $2$ by any prime for a similar result.
