Group Generated by Reciprocal of z and Minus z

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

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


 * $f_1 \left({z}\right) = z$


 * $f_2 \left({z}\right) = -z$


 * $f_3 \left({z}\right) = \dfrac 1 z$


 * $f_4 \left({z}\right) = -\dfrac 1 z$

Let $\circ$ denote composition of functions.

Then $\left({S, \circ}\right)$ is the group generated by $\dfrac 1 z$ and $-z$.

Also see

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