You can define the action of a Mobius transformation on a point in (the Riemann sphere) to be the image of that point under the Mobius transformation.

Consider where . Since it means and we simply have . Since it means where . Thus, this set is the set of all non-zero dilations and this isomorphic to the complex group under multiplication under the isomorphism as .Show that the subgroup of the mobius group consisting of transformations which fix 0 and infinity is isomorphic to C\{0}

It is similar to what was done above.Now show that the subgroup of the mobius group consisting of transformations which fix 0 and 1 is also isomorphic to C\{0}