Compact complex subgroups - Viewer Submission

I thought I had this one more clearly figured out with my head start on thinking about it, but I don't think I made a convincing argument for my solution. I'm still interested to hear whether I am correct.
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[@__Junioor__]

Hey !
I think you have the right answer, not sure though as i don't have a correction...
You probably already know it but the Gk = {exp((2*i*n*pi)/k) | n in [0, k-1]} are pretty common groups called the k-th roots of the unity as they satisfy the equation z^k = 1 and are usually denoted Uk, k being a non-zero natural number, here in France. I have the confirmation that these groups are a part of the solution.
For the infinite groups, i agree that U (the unity circle or trigonometric circle) is a compact subgroup though i don't have a justification for it being the only one. Your argument with the rational / irrationnal numbers might be enough as if you have a rationnal number in a subgroup, you will end up generating one of the Uk and if you have an irrationnal in the subgroup, you will end up with the whole circle, therefore showing that you cannot generate something else than the Uk or U (as the irrationnal and rationnal numbers form a partition of the real numbers)
I also want to apologize for taking so long to answer i just really wanted to find an answer to the infinite group problem but ended up finding none, so yeah, i'm really sorry D:

[@mathoutloud]

Hi, and thanks again for submitting this problem! I had a good time thinking about it in the time leading up to this video.
I know about roots of unity of course, I thought maybe I called it that at some point, but maybe I did actually forget about the terminology at the time.
I still think about this in the back of my head from time to time, nothing too detailed, and I’m sure if I spent some dedicated time on it then I could flesh out the details.
Let me know or if you have any others you want me to look at! It’s a bit easier if you send them to my email though, that way I don’t spoil the problem for myself before recording a video by reading it here.