TITLE: On the Factor group $\Bbb Q/\Bbb Z$
QUESTION [11 upvotes]: Possible Duplicate:
$\mathbb{Q}/\mathbb{Z}$ has a unique subgroup of order $n$ for any positive integer $n$? 

I have the factor group $\Bbb Q/\Bbb Z$, where $\Bbb Q$ is group of rational numbers and $\Bbb Z$ is group of integers (operation in both of them is addition). 

It's necessary to prove that for every natural number $n$ there exists one and only one subgroup of $\Bbb Q/\Bbb Z$ with order equal to $n$.

I've proved that such group exists (cyclic group suits this condition), but have almost no idea about how to prove its uniqueness.

REPLY [3 votes]: Hint $\rm\ \ (a,b) = 1\: \Rightarrow\ \left<\dfrac{a}b\right>\equiv \left<\dfrac{1}b\right>\:\ (mod\ \mathbb Z)\: $ by $\rm\:b^{-1}\:\! (j\:a + k\: b = 1)\ $ by Bezout.
Remark $\ $ This Bezout reduction of fractions $\rm\:\! (mod\ 1)\:\! $ often proves handy, e.g. see here.