I am trying to understand hos one can define the dihedral groups $D_n$. I have seen the "definition" that just says this is the group of symmetries of an $n$-polygon. So you have rotations and reflections. But I feel this definition is a bit vague. I asked around and heard that one can define this using generators and relations. I don't know about that.
I know that one can "realize" for example $D_4$ as a subgroup of $S_4$. For example
$$
D_4 =\{(1), (13), (24), (14)(23), (1234), (12)(34), (1432), (13)(24)\}.
$$
I do understand that I get these elements from labelling the vertices of the $4$-gon. This is very concrete for me.
Therefore my question is: Is there a nice way to actually define the general dihedral group $D_n$ as a specific concrete subgroup of $S_n$?
So, for example, I am looking for something like
$$D_n = \{\sigma \in S_n : \text{something} \}.$$
I am not looking for a vague algorithmic way of defining the groups.
From the example of $D_4$ I am thinking that it should always have $2$-cycles, but I don't think that this is always true. For example with $D_5$.
Best Answer
Sure. For instance, you can use the fact that $D_n$ is generated by a rotation and a reflection. In terms of permuting the vertices, this means you can define $D_n$ as the subgroup of $S_n$ generated by the $n$-cycle $(1\ 2\ 3\ 4\ \dots\ n)$ and the permutation $(2\ n)(3\ n-1)(4\ n-2)\dots$ (that is, the permutation that sends $k$ to $n-k+2$ mod $n$). If the vertices are labelled $1,\dots, n$ going around the $n$-gon counterclockwise, the first generator is the rotation by an angle of $2\pi/n$ and the second generator is the reflection through the line of symmetry passing through vertex $1$.
You can in fact explicitly describe all the elements of $D_n$ similarly. The rotations are permutations of the form $k\mapsto k+m$ mod $n$ for some fixed $m$ (that's a rotation by an angle of $2\pi m/n$). The reflections are permutations of the form $k\mapsto m-k$ mod $n$ for some fixed $m$ (that's the reflection across the line of symmetry passing through vertex $\frac{m}{2}$ if $m$ is even and the line of symmetry passing through the edge between vertices $\frac{m-1}{2}$ and $\frac{m+1}{2}$ if $m$ is odd). So you could explicitly define $D_n$ as the set of permutations $\sigma\in S_n$ which have the form $\sigma(k)=ak+m$ mod $n$ for $a\in\{-1,1\}$ and $m\in\{0,\dots,n-1\}$.