I am currently studying Mobius geometry from the book [1]. I found a group in Mobius geometry called Mobius group which contains Mobius transformations. I have the following doubt.

Dose this group contain transformations of the form $f(z)=\frac{az+b}{cz+d}$ only? or the group contains $f(z)=\frac{1}{\bar{z}}$ type functions also.

Kindly help. It will be great if anyone suggests some references to understand it.

Reference: 1. W. Benz, Classical Geometries in Modern Context, Birkhauser Basel, 2012.

I am currently studying Möbius geometry from the book [1]. I found a group in Möbius geometry called Möbius group which contains Möbius transformations. I have the following doubt.

Dose this group contain transformations of the form $f(z)=\frac{az+b}{cz+d}$ only? or the group contains $f(z)=\frac{1}{\bar{z}}$ type functions also.

Kindly help. It will be great if anyone suggests some references to understand it.

Reference: [1] W. Benz, Classical Geometries in Modern Context, Birkhauser Basel, 2012.