1.6.84 Fields, Groups, Rings

condition: Show that the set of fractional linear mappings of a complex sphere into itself is a group with respect to composition. Let us associate with each matrix \( \left(\begin{array}{ll}a & b \\ c & d\end{array}\right) \in G L_{2}(\mathbb{C}) \) a fractional linear mapping onto the complex sphere \( f(z)=\frac{a z+b}{c z+d} \) Show that this is a group homomorphism and find the kernel.