Group Generated by Reciprocal of z and Minus z/Cayley Table
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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:
$\quad \begin {array} {r|rrrr} \circ & f_1 & f_2 & f_3 & f_4 \\ \hline f_1 & f_1 & f_2 & f_3 & f_4 \\ f_2 & f_2 & f_1 & f_4 & f_3 \\ f_3 & f_3 & f_4 & f_1 & f_2 \\ f_4 & f_4 & f_3 & f_2 & f_1 \\ \end{array}$
Expressing the elements in full:
$\quad \begin {array} {c|cccc} \circ & z & -z & \dfrac 1 z & -\dfrac 1 z \\ \hline z & z & -z & \dfrac 1 z & -\dfrac 1 z \\ -z & -z & z & -\dfrac 1 z & \dfrac 1 z \\ \dfrac 1 z & \dfrac 1 z & -\dfrac 1 z & z & -z \\ -\dfrac 1 z & -\dfrac 1 z & \dfrac 1 z & -z & z \\ \end{array}$