Demonstrate the statement in section 3.I.3 on the decomposition of the Möbius transformations.

Question

I'm wondering how to do this question. I believe the question wants to decompose the following statements together from the text book to find the Mobius function

Thank you in advance (:

Answer 1

Compose the 4 transformations you'll get $(az+b)/(cz+d)$ where $a,b,c,d$ were (almost) arbitrary. For the calculation it might help you to notice that for $c\ne 0$ $$\frac{az+b}{cz+d} = \frac{a}{c}+\frac{a(-d/c)+b}{cz+d}$$