For convenience, consider the symmetries of a square. When finding/specifying the reflection generator of a dihedral group does it matter which relfection I choose? Dummit/Foote say to choose the reflection about the line through vertex 1 (the first labeled vertex) but is that always the fixed line (y=x in case of the square), or does the line my reflection generator obeys rotate with the square? I.e would my generator change to reflection about the line y=-x after one rotation since 1 moved clockwise, or would I keep the line y=x for my reflection?
[Math] Dihedral group generators.
abstract-algebragroup-theory
Best Answer
No, it does not matter what reflection you choose in order to generate the dihedral group. This is because a dihedral group $Dih_n$ is isomorphic to the semidirect product of the cyclic group $C_n$ and the group $\mathbb{Z}_2$, which becomes the subgroup generated by a particular reflection, combined with the fact that all subgroups of $Dih_n$ generated by reflections are conjugate to each other, and so the semidirect product construction works for any such subgroup.