In a vector space, linear transforms can act on points of the space by the usual matrix multiplication rule, but in this note I am reading they use a different action (The Möbius transformation). It's easy to multiply everything out and see that $A(Bx) = (AB)x$ holds but that doesn't explain why this works.
So my question is this, Is there a classification of all possible actions?
edit: just to clarify - I'm more interested in the general case than just the Möbius one but I do appreciate the answers so far!
Edit II: I think I have realized what's really happening here is that the matrices are just representations of, for example rationals, So they act on rationals in a way predetermined by what they represent.