Unique Cyclic Subgroups If we let $$S = \{ e^{q\pi i} : q\in Q   \} $$
Prove that for each $ n \ge 1$ there is a unique cyclic subgroup of order $n$ in $S$ and the union of these cyclic subgroups is $S$. Any help on this?

$S_n = \{ e^{2 \pi i \frac{m}{n}} : m\in \{0...n-1\}    \}$
$(S_n,\cdot)$ is cyclic of order n (you can show the isomorphism with $\mathbb Z_n$)
For the uniqueness use an argument on the generator of the subgroup (show that if $x\in S \ x^n=1$ then $x \in S_n$ )
1)$ \forall n \ S_n\subset S  \implies \bigcup_{n=1}^{\infty}S_n \subset S$
2)$ x \in S\implies \ x=e^{i \pi q},  q \in Q \implies x=e^{i \pi \frac{m}{n}} \ x\in S_n$ for some n $\implies S\subset \bigcup_{n=1}^{\infty}S_n$
$1)+2)\implies S=\bigcup_{n=1}^{\infty}S_n$
