# 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:

 $\displaystyle \map {f_1} z$ $=$ $\displaystyle z$ $\quad$ $\quad$ $\displaystyle \map {f_2} z$ $=$ $\displaystyle -z$ $\quad$ $\quad$ $\displaystyle \map {f_3} z$ $=$ $\displaystyle \dfrac 1 z$ $\quad$ $\quad$ $\displaystyle \map {f_4} z$ $=$ $\displaystyle -\dfrac 1 z$ $\quad$ $\quad$

Let $\circ$ denote composition of functions.

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

### Cayley Table

$\begin{array}{r|rrrr} \circ & f_1 & f_2 & f_3 & f_4 \\ \hline f_1 & f_1 & f_2 & f_3 & f_4 \\ f_2 & f_2 & f_1 & f_4 & f_3 \\ f_3 & f_3 & f_4 & f_1 & f_2 \\ f_4 & f_4 & f_3 & f_2 & f_1 \\ \end{array}$

Expressing the elements in full:

$\begin{array}{c|cccc} \circ & z & -z & \dfrac 1 z & -\dfrac 1 z \\ \hline z & z & -z & \dfrac 1 z & -\dfrac 1 z \\ -z & -z & z & -\dfrac 1 z & \dfrac 1 z \\ \dfrac 1 z & \dfrac 1 z & -\dfrac 1 z & z & -z \\ -\dfrac 1 z & -\dfrac 1 z & \dfrac 1 z & -z & z \\ \end{array}$