List all proper, non-trivial subgroups of $U_6$ where $U_6$ is the group of $6$th roots of unity. Also, list the generators.

Question

A) List all proper, non-trivial subgroups of $U_6$ where $U_6$ is the group of $6$th roots of unity:

So 6th roots of unity are $\pm1, \pm \frac{1}{2} \pm \frac{\sqrt{3}}{{2}}i$ How do these relate to the subgroups?

B) List the generators:

If I am not mistaken the generators of the 6th roots of unity are $\frac{1}{2} \pm \frac{\sqrt{3}}{2}i$. Is that correct?

Answer 1

It's isomorphic to $\Bbb Z_6$, so has $\varphi(6)=2$ generators: $e^{2\pi ki/6},\, k=1,5$. These are the so-called primitive roots.

Since $U_6$ is cyclic, it has a unique subgroup of each order dividing $6$. For instance, $\{1,-1\}$ is the one of order $2$, generated by $-1$.

Answer 2

Hint: Note that $\varphi:\Bbb Z_6\to U_6$ given by $[n]_6\mapsto e^{\frac{2\pi n i}{6}}$ defines an isomorphism. Generators map to generators under isomorphisms. Isomorphisms preserve subgroup structure.

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Here $[n]_6:=\{m\in\Bbb Z: 6\mid n-m\}$.