I need help to find the family of Fractional-Linear (Mobius) transformations that map the unit circle onto itself

a tranformation includes a translation, inversion, magnification, rotation and translation. and takes the form w(z)=(az+b)/(cz+d)

Whenever i work through these i get {a,b,c,d}=1 so this isn't helpful to me. any help would be great

cheers