Group Generated by Reciprocal of z and Minus z/Cayley Table

Cayley Table for Group Generated by Reciprocal of $1 / z$ and $-z$

We have:

$\map {f_1} z = z$

$\map {f_2} z = -z$

$\map {f_3} z = \dfrac 1 z$

$\map {f_4} z = -\dfrac 1 z$

Hence from Group Generated by Reciprocal of z and Minus z:

$\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}$