Ambiguous definition of generators in dihedral groups In Dummit and Foote, the dihedral groups $D_{2n}$ are defined first as the groups with rotations (generator $r$) and axial symmetry over the line between the center and the point 1 (generator $s$). The two generators verify $s r = r^{-1} s$.
For $D_{2n}$, one can assign a number in $[1, n]$ each point (which move under transformation), and a number in $[1, n]$ to each positions (which stay fix under transformation). The initial state considered is that the point $k$ in at the position $k$ for each $k \in [1, n]$.
This generators $r$ and $s$ can then be understood in two different ways. And this seems to be related to the "Problem of alias and alibi" as mentioned in the Companion to Lang's Algebra, end of p.8.
Convention A:


*

*$r$ transforms each point to the next point

*$s$ performs the axial symmetry which keep fixed the position 1

*acts by alibi


Convention B:


*

*$r$ transforms the point at each position to become the point at the next position

*$s$ performs the axial symmetry which keep fixed the point 1

*acts by alias


I tried for $D_8$ to perform $r s$ and compare it to $r^{-1} s$ considering both conventions. I have found that:


*

*The equality $s r = r^{-1} s$ is verified for both conventions.

*The final state after applying $s r$ is different in both convention.

*In both conventions you only apply rigid-body motions of the points.


So I have two questions:


*

*What is the convention used in Dummit and Foote (or what is the convention generally implied) ?

*Does it even matters ? In the sense that whatever convention is chosen, you will end up with $s r = r^{-1} s$ in every $D_{2n}$ ? Are both conventions just convenient graphical representations ?

 A: I agree with the commenters that I can't see a difference between your conventions. I'm also not sure what you mean by position. I think you're making a distinction without a difference.
I've written this answer to attempt to clarify things.
A common choice of representation of the dihedral group $D_{2n}$ as rotations and reflections of the plane is the following:
$$r\mapsto R_{2\pi/n} 
= 
\begin{pmatrix}
\cos 2\pi/n & -\sin 2\pi/n \\
\sin 2\pi/n & \cos 2\pi/n
\end{pmatrix}
$$
and 
$$s \mapsto S=
\begin{pmatrix}
1 & 0 \\ 
0 & -1 \\
\end{pmatrix}.
$$
These matrices represent a rotation counterclockwise by $2\pi/n$ and a reflection over the $x$-axis respectively, which appears to be the action you're trying to describe in words.
Edit
Having read your updated question and the wiki article linked in the comments by Calum Gilhooley, 
active and passive transformations, I have a better idea of the question you're trying to ask, I think.
You're asking whether we regard the dihedral group as acting on the points or the coordinate system, right? The answer then is that it doesn't really matter, a representation of a group can be regarded as doing either, since matrices can be regarded as either representing an action on points in a vector space or a change of coordinate system for the vector space. However, in mathematics, and particularly group theory, we almost always regard the representation as acting on the points of the vector space rather than the coordinate system.
