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:

\(\ds \map {f_1} z\)

\(=\)

\(\ds z\)

\(\ds \map {f_2} z\)

\(=\)

\(\ds -z\)

\(\ds \map {f_3} z\)

\(=\)

\(\ds \dfrac 1 z\)

\(\ds \map {f_4} z\)

\(=\)

\(\ds -\dfrac 1 z\)

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

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

Sources

• 1964: Walter Ledermann: Introduction to the Theory of Finite Groups (5th ed.) ... (previous) ... (next): Chapter $\text {I}$: The Group Concept: $\S 7$: Isomorphic Groups: Example $2 \ \text{(a)}$