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Let $G$ be a compact Lie group. What can we say about subgroups of $G$ isomorphic to $Z_n$? For example, $G$ is $SU(2)$, or $SO(3)$ and $n$ is $2$ or $3$ and so on.

A naive idea is that if $x\in G$ is the image of the generator, then such subgroups are in one to one correspondence with the solution of equation $x^3=1$. What can we say about this equation? Are there solutions? how many? how to parametrize if there is a smooth family?

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  • $\begingroup$ A tag 'lie-groups' is also needed here. $\endgroup$
    – Jim Humphreys
    Commented Apr 26, 2013 at 20:42

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It's clear that you might start by looking inside a maximal torus of the given compact (say connected) Lie group. But given the long history of such problems, naive methods are unlikely to get very far with this type of question. Two useful sources are (1) a Bourbaki seminar report by Serre here and (2) the detailed discussion with references given earlier on MO here.

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