While searching online, I've come across two ways to define the elements of the dihedral group. Both ways are internally consistent and are fine as far as I can tell, but they are mutually exclusive, so I was wondering which of the two ways is more standard or commonly used.
The two ways are as follows:
Way 1. Elements are defined as transformations against a fixed set of axes.
In this way of defining the elements, each element of $D_n$ is either a reflectional symmetry or a rotational symmetry of the polygon being considered. In an $n$-gon, there are $n$ reflectional symmetries and $n$ rotational symmetries.
The reflectional symmetries are described as follows: An $n$-gon has $n$ axes of symmetry. For each axis of symmetry in the $n$-gon there is an element in the dihedral group that reflects the $n$-gon along that axis. It is important to note that these axes stay fixed, even after the n-gon itself undergoes a rotational symmetry.
Similarly, there are $n$ rotational symmetries in the $n$-gon being discussed, and each of these rotations is a member of $D_n$. It is important to note that here, the rotations are always in the same direction, even if the shape undergoes reflectional symmetry. So, for example, an element $r$ that rotates a square $90^{\circ}$ clockwise will always rotate the square $90^{\circ}$ clockwise, even after the square is reflected.
Way 2. Elements are defined as permutations of vertices.
The second way of defining the elements of $D_n$ is that each element is defined as a permutation of vertices of the $n$-gon. (It should be noted that there are in total $n!$ permutations of vertices, yet only $2n$ elements in $D_n$; this discrepancy is explained by the fact that a permutation must also preserve the structure of the $n$-gon in order to be included in $D_n$.) Like all permutations on a finite set, these permutations can be written out as cycles such as $(1234)$, where $1$, $2$, $3$, and $4$ are the names of vertices of a square, for example.
Way 2 is actually distinct from Way 1, both geometrically and algebraically (as far as I can tell).
There are still $n$ rotations and $n$ reflections, but unlike in Way 1 where the transformations are defined in terms of a fixed "background" which doesn't move as the $n$-gon undergoes rotation or reflection, the transformations in Way 2 are now defined in terms of the vertices of the $n$-gon, which DO move as the $n$-gon undergoes rotation or reflection. What this means is that in Way 2, the transformations change depending on the current orientation of the $n$-gon. For example, consider the rotation of a square. In Way 1, we can let $r$ be the element that rotates the square $90^{\circ}$ clockwise. The direction of rotation ($90^{\circ}$ clockwise or, equivalently, $270{\circ}$ counterclockwise) never changes, even after the square is reflected across one of its axes of symmetry. On the other hand, the roughly corresponding rotation in Way 2 would be something like the cycle $(1234)$. Unlike $r$, which always rotates the square $90^{\circ}$ clockwise, $(1234)$ may rotate the square $90^{\circ}$ either clockwise or counterclockwise, depending on whether the square has been reflected or not. Similarly, the axes of reflection in Way 2 move along with the square, while those of Way 1 remain fixed.
The fact that the two ways are non-equivalent can also be verified algebraically (I think...). No isomorphism exists between the two Ways (at least as far as I can tell; I tried constructing an isomorphism by making a bijection between the elements as defined in Way 1 with the elements as defined in Way 2, matching the rotations and reflections in Way 1 with the corresponding ones in Way 2, but the bijection did not satisfy the criterion that $f(a)f(b) = f(ab)$ for all $a$, $b$ in Way 1. The problem arose when $a$ was a rotation and $b$ was a reflection. However, if there actually is an isomorphism that I overlooked, please correct me.)
My main question is: Which of these two ways is standard, or used more often by working mathematicians? Are both acceptable?
It seems to me that Way 2 is just nicer all-around, mainly because all of the elements are permutations and can thus be written as cycles, which are easy to work with algebraically (by Cayley's Theorem, Way 1 is isomorphic to some group of permutations anyway, but then it seems like kind of a hassle finding a way to write it as cycles and whatnot.)
If there are some benefits to Way 1, then I would like to learn about those too. Thanks in advance for those who bothered to read through all this!
