Group Generated by Reciprocal of z and Minus z

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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:

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


Also see


Sources