Are these factor groups cyclic? а) $\mathbb{Q}/2\mathbb{Z}$
b) $\mathbb{R}^*/\mathbb{Q}^*$
How to find out (and prove) if the factor groups above are cyclic or not? 
Thanks in advance!
 A: Suppose that $\mathbb{Q}/ 2 \mathbb{Z}$ is cyclic $\Rightarrow \exists a=m/s \in \mathbb{Q}: a^k + 2 \mathbb{Z} = 1/2 + 2 \mathbb{Z} \Rightarrow ka - 1/2 \in 2 \mathbb{Z} \Rightarrow \exists n \in \mathbb{Z} : ka-1/2=2n \Rightarrow 2ka-1=4n \Rightarrow (2km)/s -1 = 4n \Rightarrow 2km -s = 4ns$ which seems to me to be clearly impossible .
[I think I'll have to edit this demonstration]
For the second problem consider $I := <\pi,e>$. $\pi_{\mathbb{Q}}(I)=<\pi,e> \mathbb{Q}  .$ 
$ \pi_{\mathbb{Q}}(I) < \mathbb{R}^*/\mathbb{Q}^*$ because it's the imagine of a subgroup.
If $\mathbb{R}^*/\mathbb{Q}^*$ is cyclic $\Rightarrow \pi_{\mathbb{Q}}(I)$ is cyclic but if it's cyclic it's generate by  a $x\mathbb{Q}$ for such $x \in \mathbb{R}^*$. So $x^n\mathbb{Q} = e\mathbb{Q} \Rightarrow x^n/e \in Q \Rightarrow e|x.$ Same idea proves that $\pi|x$ So $ x=\pi e r$ (with $r \in \mathbb{R}$) but it's clear that $ \forall n,$ $ (\pi e r)^n\mathbb{Q}\not=e \mathbb{Q} $ , becuase $\pi \not\in \mathbb{Q}$. 
A: I think the first group is not cyclic, because it is divisible, torsion group and so:
$$\frac{\mathbb Q}{2\mathbb Z}\cong \sum_p\mathbb Z(p^{\infty})$$
A: Hint: a group which contains torsion elements of arbitrarily large order cannot be cyclic. (Proof: in order to get a contradiction, suppose $t$ is a generator. Show that $t$ must be torsion. What is its order?)
