the group of Mobius transformations

Once you have one family of maps, the next question is how they fit together: can you undo a Mobius map, and is doing two of them in a row again one of them? The answers are yes and yes — and that double 'yes' is exactly what makes the Mobius transformations a group. They are closed under composition (do two, get a third), every one has an inverse (which is itself a Mobius map), and the identity map w = z sits among them. This group structure is not a decoration; it is the source of nearly every clean property the family enjoys.

The structure becomes transparent when you attach to each map w = (a z + b)/(c z + d) the 2-by-2 matrix [a, b; c, d]. Composing two Mobius maps corresponds to multiplying their matrices, and the inverse map corresponds to the inverse matrix. The nondegeneracy condition a*d - b*c not equal to 0 is just saying the matrix is invertible. Because scaling all four coefficients by the same nonzero number gives the SAME transformation, the Mobius group is really the group of invertible 2-by-2 complex matrices with that scaling divided out — written PGL(2, C), or equivalently PSL(2, C) once you normalize the determinant to 1. So a problem about composing fractional-linear maps becomes ordinary matrix multiplication, which is far easier to track.

Geometrically, this is the full group of conformal bijections (automorphisms) of the Riemann sphere — every angle-preserving, orientation-preserving self-map of the sphere is a Mobius transformation, and there are no others. That is a strong rigidity statement. The big standard subgroups live inside it: the rotations of the sphere, the automorphisms of the unit disk, and the automorphisms of the upper half-plane (which turn out to be the real Mobius maps, those with real coefficients and positive determinant). Knowing the group lets you generate every map you need by composing a few simple building blocks: translations, rotations, scalings, and the single inversion w = 1/z.