# Why are Transpositions and rotations in dihedral groups the same operation?

A dihedral group seems to have 2 operations, rotation, and transposition.

These transformations seem so distinct, no composition of rotations may ever lead to a transposition and vice versa (right?).

so why can we use the same operation symbol (like $\circ$) for these 2 seemingly different operations? are they connected in some deeper way?

• If $r$ is the rotation and $s$ is the reflexion, in the dihedral group of order $2n$ we have $$rs=sr^{n-1}.$$ Your question about the operation shows that you should study the notion of abstract group a little better. – Francesco Polizzi Aug 18 '17 at 17:04

No. The dihedral group has two types of elements (not operations), which are rotations and reflections (not transpositions). You are right about one thing, though: no composition of rotations may ever lead to a reflection. And... ? We also have two types of integers: odd and even. Furthermore, no sum of even integers may ever lead to an odd integer. Do you have any problem with this? It seems exactly the same situation.

• In the dihedral group of order $4n$, the composition of a rotation $n$ times with itself is a reflection. In fact, in this case $\langle r^n \rangle \simeq \mathbb{Z}/2 \mathbb{Z}$ is the center of the group. – Francesco Polizzi Aug 18 '17 at 17:05
• @FrancescoPolizzi No, $r^n$ is not a reflection on a line, which is the kind of reflections that I meant here (although it is a point reflection). – José Carlos Santos Aug 18 '17 at 17:13
• Ok, $r^n$ is an involution but it is not a line reflection in the usual identification of the dihedral group as the group of symmetries of a regular polygon. – Francesco Polizzi Aug 18 '17 at 17:17
• the distinction between operation and element is what caused my confusion, you are right, the composition operation is the same, the elements are what represents the different rotations or reflections and they are indeed different. – Joaquin Brandan Aug 18 '17 at 17:38