Is $\frac{1}{\overline{z}}$ a Möbius transformation in the unit circle?

Question

Basically, I have been studying about Möbius inversions recently and was a bit confused. I know that inversion in the unit circle can be represented as the map $z \rightarrow \frac{1}{\overline{z}}$, but I don't know if this is a Möbius transformation? And if it is not, is there a difference between this inversion and inversion in Möbius transformations?

Answer 1

$z \mapsto 1/\overline{z}$ is the composition of a Möbius transformation and a reflection about the real axis.

Möbius transformations are all orientation-preserving, so they cannot model inversions directly. Instead, we have the Möbius transformation $z \mapsto \frac1z$, which is an inversion followed by a reflection (to restore orientation). This transformation keeps many of the properties of an inversion, such as mapping clines (circles or lines) to other clines.

All Möbius transformations can be expressed as compositions of translations, rotations, dilations, and the $z \mapsto \frac1z$ combo-inversion-reflection map.