consider the group $G=\mathbb Q/\mathbb Z$. For $n>0$, is there a cyclic subgroup of order n consider the group $G=\mathbb Q/\mathbb Z$. Let $n$ be a positive integer. Then is there a cyclic subgroup of order $n$?


*

*not necessarily.  

*yes, a unique one. 

*yes, but not necessarily a unique one. 

*never


Another hint to prove unicity: suppose $\overline {a/b}$ has order $n$, $0 < a < b$, and $gcd(a,b)=1$, then $na=kb$ for some $k \in \mathbb{Z}$, hence $a$ divides $k$ and $n=bk'$. It follows that $\overline {a/b}$ = $\overline {ak'/n}$, so ...
