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?

The elements of $S$ are complex numbers. The group operation is complex multiplication and the identity element is $1 = e^{2\pi i}$. So $x \in S$ lies in a copy of $\mathbb Z/ n \mathbb Z$ if and only if it is an $n$th root of unity. We have explicit formulas for the roots of unity. This will prove existence. From the fundamental theorem of algebra, we know there are exactly $n$ roots of unity. This will prove uniqueness.
