Prove that the given group is not cyclic

Question

Let us consider a group $G=\{m/2^n:m,n∈Z\}$. Prove that this is not cyclic.

If it is given to show that for cyclic then we will form a isomorphism With $(Z,+)$ to see it. But how will I show for this? Please help me.

Answer 1

Any cyclic subgroup of the group of rational numbers is of the form $q\mathbb{Z}$, hence has bounded denominators. It follows that the dyadic rationals $$\left\{\frac{m}{2^n} : m, n \in \mathbb{Z}\right\}$$ are not cyclic as subgroup. More directly, assume that the group were cyclic with generator $r$. Then what would happen with $\frac{r}{2}$?

Answer 2

Suppose this group is generated by $\frac{a}{2^b}$ where $a,b$ are integers, $a$ not being equal to $0$. Then for any integers $m$ and $n$ there exists an integer $c_{m,n}$ such that $\frac{m}{2^n}=c_{m,n}\frac{a}{2^b}$, take $m=a$ and $n=b+1$ then $\frac{1}{2}=c_{a,b+1}$ which is absurd, so the group is not cyclic.