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Let $D_n$ be the dihedral group of a regular $n-$polygon. I'd like to write $D_n$ as a subgroup of $S_n$, the set of all permutations of $n$ objects and therefore show why the order of $D_n$ is $2n$.

Clearly, I need to only consider the permutations of the vertices that preserve distances. However, I'm not quite sure how to go about listing these permutations.

Any advice would be helpful!

zeta1203
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  • Are you fixing $n$ and explicitly listing out all $2n$ elements? It sounds like you want to write them all out, but you can't do that as $n$ gets bigger and bigger. – AHusain Oct 09 '21 at 02:39
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    Try it for $n=3,4,5$ first. – Thomas Andrews Oct 09 '21 at 02:42
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    Try to understand how the cycle representation of a reflection is and how the representation of a rotation is. You can then use the fact that the elements of the Dihedral group are either rotations or a composition between a reflection and a rotation. – Oscar Oct 09 '21 at 18:20

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You can embed $D_n$ into $S_n$ by considering the action of $D_n$ by left multiplication on the left quotient set $X:=D_n/H$, where $H:=\{1,s\}\le D_n$ ($s$ is a reflection). This action is faithful, because the normal core of $H$ is trivial. If you compose this embedding after the isomorphism from $S_X$ to $S_n$ arising from any bijection $f\colon \{1,\dots, n\}\to X$, then the image of every $g\in D_n$ is a permutation on $n$ elements and the image of $D_n$ is an isomorphic copy of it in $S_n$.

citadel
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By considering $\langle a, b\rangle$, where $a=(1\: 2\: 3\: ... \: n)$ and $b=(1\; \; n)(2 \; \; n-1)(3\; \: n-2)...$,($b$ is product of $[n/2]$ transpositsions), we can see that $D_{n}\cong \: \langle a, b\rangle\: \le \: S_{n}$.

Ash
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