Group Generated by Reciprocal of z and Minus z

Definition
Let:
 * $S = \set {f_1, f_2, f_3, f_4}$

where $f_1, f_2, f_3, f_4$ are complex functions defined for all $z \in \C \setminus \set 0$ as:

Let $\circ$ denote composition of functions.

Then $\struct {S, \circ}$ is the group generated by $\dfrac 1 z$ and $-z$.

Also see

 * Group Generated by Reciprocal of z and Minus z is Klein Four-Group, which demonstrates that this is a (finite) group.