Dihedral Groups; what exactly are the elements of the set?

Question

I am reading Dummit and Foote. We have:

[Definition 1:]For each $n \in \mathbb{Z}^+, n \ge 3$ let $D_{2n}$ be the set of symmetries of a regular $n-$gon, where a symmetry is any rigid motion of the $n-$gon which can be effected by taking a copy of the $n-$gon, moving this copy in any fashion in $3-$space and then placing the copy back on he original $n-$gon so it exactly covers it.

$\vdots$

Let $r$ be the rotation clockwise about the origin through $\dfrac {2 \pi}{n}$ radians. Let $s$ be the reflectio about the line of symmetry through vertex $1$ and the origin.

$\vdots$

[Definition 2:] $D_{2n}=\{1, r, r^2, ..., r^{n-1}, s, sr, sr^2, ..., sr^{n-1} \}$

(I added the names Definition $1$ and $2$ so we can talk easier about the text)

In light of Definition $1$, wouldn't it make more sense to define

$$D_{2n}=\{\overline{1}, \overline{r}, \overline{r^2}, ...,\overline{ r^{n-1}}, \overline{s}, \overline{sr}, \overline{sr^2}, ..., \overline {sr^{n-1}} \}$$

where (for exapple) $\overline {r}$ is the equivalence class of all rigid motions in $3$ space which moves vertex $1$ to vertex $2$, vertex $2$ to vertex $3$, etc.

My second question is, what exactly is $r$? Is it the permutation $(12...n)$? Or is $r$ a different object, and the permutation is just associated with it?

Answer 1

First question: your proposed definition is adding some context that isn't present in the definition of $D_{2n}$ -- namely 3-space, which is unnecessary for the purpose of defining $D_{2n}$. I would say it makes more sense not to introduce the extra definitions etc.

Edited to add: Sorry, looks like D&F added the stuff about 3-space. I still think it's unnecessary.

For the second question, $r$ is not equal to the permutation, but $D_{2n}$ is isomorphic to a subset of $S_n$ and one can give an explicit form of that isomorphism $\phi$ such that $\phi(r)=(12\ldots n)$. In my experience, most algebraists ignore the distinction in favor of being succint.

Answer 2

Definition 1 clearly tells you that r is the rotation of a regular n-gon by 2π/n radians which is a single element . To understand it precisely you draw a square ,mark the edges and then try to rotate it by π/2 rad about centre . This rotation is nothing but your r . r is totally different from a permutation but the corresponding n-gon can be linked to group of permutations

Answer 3

For your definition 2 one can take for $r$ the permutation $(1,2,\ldots,n)$ and for $s$ the permutation $(1,n)(2,n-2)\ldots$. There is still a third definition that permits to calculate with $r$ and $s$ by applying the rules: $$ \begin{array}{l} r^n=1 \\ s^2 = 1 \\rs = sr^{n-1} \end{array}$$