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List all subgroups of the symmetry group of the regular $n$-gon.

If $n$ is prime, there are only $n+1$ subgroups: subgroup of all rotations and $n$ subgroups with $2$ elements (one reflection and rotation by zero degrees).

But if $n$ is composite, there are additional subgroups in the group of rotations, namely rotations by angle $\frac{\pi}{k},\ k|n$. Also there are subgroups containing rotations and reflections, number of which I can't found. I know that $s_\alpha s_\beta=r_{2(\alpha-\beta)}$, where $s_\alpha$ is a reflection by the line inlcined at an angle $\alpha$ to the horizontal axis. So if a subgroup contains $r_\alpha$ and $s_\beta$, it contains their product $s_{\beta - \frac{\alpha}{2}}$.

What is the best way to count all subgroups for each $n$?

Update: I have found an group-theoretic answer at http://ysharifi.wordpress.com/2011/02/17/subgroups-of-dihedral-groups-1/

Later I will post the answer below.

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    Do it for a few small values of $n$, then look it up in the Online Encyclopedia of Integer Sequences.2012-02-12

2 Answers 2