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

• 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