# $D_{2n}$ Acting on Opposite Pairs of Vertices

I have been asked to show that $D_{2n}$ acts on the set consisting of pairs of opposite vertices of a regular $n$-gon, where $n$ is taken to be a positive even integer.

The first thing that came to my mind was to represent $r$ and $s$ in cycle notation, i.e., to show that $D_{2n}$ is isomorphic to a certain subgroup of $S_n$, and then I can show that permutations acting the two element sets of opposite vertices is a group action of $D_{2n}$ on them. More specifically, if $\sigma$ is a permutation and $\{v_1,v_2\}$ is a set consisting of opposite vertices, then I would show $\sigma\cdot \{v_1,v_2\} := \{\sigma(v_1), \sigma(v_2) \}$ is a group action.

So, $r \mapsto (1~2~...~n)$ and $s \mapsto (2~n)(3~(n-1))~...?$.

There are a few things I am having difficulty with: (1) will this argument succeed; (2) how do I make the arguments more rigorous; (3) how do I determine exactly what $s$ gets mapped to; and (4), given an arbitrary vertex, what will its opposite vertex look like (in other words, what does the set $\{v_1,v_2\}$ look like)?

Note, more questions may come.

• See the first part of the answer here. – Dietrich Burde Oct 2 '16 at 15:53
• @DietrichBurde I actually already saw this, but I didn't like the answer. The author uses a homomorphism between $D_{2n}$ and certain matrices. I want to figure out my idea, as this solution came most naturally to me – user193319 Oct 2 '16 at 15:56