I need to find an isomorphism between $D_n$ (all symmetries of an $n-gon$) and a subgroup of $S_n$.
I know that Cayley's theorem gives a nice isomorphism that shows that $D_n$ is isomorphic to a subgroup of $S_{2n}$, but this isn't the case here, since I need an isomorphism to $S_n$.
Now, I have a general idea on how to accomplish this; numbering all vertices of the $n-gon$ by $1,2,…,n$, and assign each symmetry in $D_n$ with the permutation of the vertices in respect to the initial symmetry $e$.
For example, if $n=4$, then the element $r^2$ will correspond to the permutation: $$\begin{pmatrix}
1 & 2 & 3 & 4\\
3 & 4 & 1 & 2
\end{pmatrix}$$
My problem is, I'm having difficulty defining the map itself.
I know that it suffices to define a homomorphism $\varphi$ on the $D_n$ generators, $r$ and $s$, and I noticed that $\varphi(r)$ always equals to the permutation $(n (n-1) (n-2)…321)$ (written in cycle nonation), but I have no idea how to define $\varphi$ on $s$ and since I also need to prove that $\varphi$ is one-to-one homomorphism, I need to reconsider whether this is the optimal way of defining it, or maybe there is better way to define it, in such that it'll be easier for me to prove it.
Any suggestions?
Best Answer
The solution, i.e., an explicit embedding of $D_n$ into $S_n$ is given on the first page here. An explicit definition of a one-to-one map is given. It needs perhaps filling in some details, but should be very helpful.
Edit: In the meantime the link was removed, but also there appeared new posts with an answer, e.g., here:
Defining dihedral groups $\{\sigma \in S_n: $ something $\}$