A regular polygon can be moved onto itself in exactly $2n$ ways, and these motions form a group. It is the main example of a nonabelian group you can hold in your hands, and it connects the symmetry you met in geometry with permutations.
Definition. $D_{n}$ ($n \ge 3$) is the group of symmetries of a regular $n$-gon, under composition. Number the vertices $1, \dots, n$ counterclockwise. Let $r$ be the rotation by $360^\circ/n$ counterclockwise, and let $s$ be the reflection in the line through vertex $1$ and the center. (Some books call this group $D_{2n}$, after its order.)
The elements. A symmetry is fixed by where it sends vertex 1 (one of $n$ places) and whether it keeps or reverses the order of the labels around the polygon (2 choices), so $|D_{n}| = 2n$. They are the $n$ rotations $e, r, r^{2}, \dots, r^{n-1}$ and the $n$ reflections $s, rs, r^{2}s, \dots, r^{n-1}s$.
Why it works
| Product | Rule | Result |
|---|---|---|
| $r^{a}r^{b}$ | rotations add | $r^{a+b}$ |
| $r^{a}\,r^{b}s$ | rotations add | $r^{a+b}s$ |
| $r^{a}s\,r^{b}$ | $sr^{b} = r^{-b}s$ | $r^{a-b}s$ |
| $r^{a}s\,r^{b}s$ | $sr^{b} = r^{-b}s$, $s^{2} = e$ | $r^{a-b}$ |
Worked example: Products in $D_{4}$
- $(rs)(r^{2}s) = r\,(sr^{2})\,s = r\,r^{-2}s\,s = r^{-1} = r^{3}$.
- $(r^{3})(rs) = r^{4}s = s$, since $r^{4} = e$.
Answer: $r^{3}$ and $s$.
Worked example: Orders
- Rotations: $|r^{k}| = 6/\gcd(6, k)$, so $|r^{2}| = 3$ and $|r^{3}| = 2$.
- Every reflection has order $2$: $(r^{k}s)^{2} = r^{k}(sr^{k})s = r^{k}r^{-k}s^{2} = e$.
Answer: $3$, $2$ and $2$.
Worked example: A symmetry as a permutation
- $r$: $1 \to 2 \to 3 \to 4 \to 1$, the cycle $(1\;2\;3\;4)$.
- $s$ fixes $1$ and the opposite vertex $3$, and swaps $2$ and $4$: $(2\;4)$.
- $rs$: apply $s$ first. $1 \to 1 \to 2$, $2 \to 4 \to 1$, $3 \to 3 \to 4$, $4 \to 2 \to 3$: $(1\;2)(3\;4)$, the mirror through the midpoints of two edges.
Answer: $(1\;2\;3\;4)$, $(2\;4)$ and $(1\;2)(3\;4)$.
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Key idea