Week 8. Let p and q be finite points in C+. Let G denote the set of hyperbolic Mobius transformations fixing p and q (the definition of hyperbolic Mobius transformation appears in Hitchman section 3.5). For any complex number a, let Ha(z) = az. Note that if a e R+, then Ho is a dilation centered at the origin. (a) (2 points) Let S(z) = = P. Prove that if T is a Mobius transformation fixing p and q, then SoToS 1 = Ha for some complex number a. (b) (1 point) If T E G, which hypothesis above allows you to conclude that a is a positive real number in part (a)? (c) (1 point) Parts (a) and (b) together imply that G = S-1KS, where K = (Ha |a e R*} is the group of dilations centered at 0. Prove that G is a transformation group. (d) (3 points) Let C be a Type I cline for p and q. Compute the orbit of C under the action of G. State and prove a claim that solves this problem. (e) (3 points) Let C be a Type II cline for p and q. Compute the orbit of C under the action of G. State and prove a claim that solves this