Definition:Group Generated by Reciprocal of z and 1 minus z
Definition
Let:
- $S = \set {f_1, f_2, f_3, f_4, f_5, f_6}$
where $f_1, f_2, \ldots, f_6$ are complex functions defined for all $z \in \C \setminus \set {0, 1}$ as:
\(\ds \map {f_1} z\) | \(=\) | \(\ds z\) | ||||||||||||
\(\ds \map {f_2} z\) | \(=\) | \(\ds \dfrac 1 {1 - z}\) | ||||||||||||
\(\ds \map {f_3} z\) | \(=\) | \(\ds \dfrac {z - 1} z\) | ||||||||||||
\(\ds \map {f_4} z\) | \(=\) | \(\ds \dfrac 1 z\) | ||||||||||||
\(\ds \map {f_5} z\) | \(=\) | \(\ds 1 - z\) | ||||||||||||
\(\ds \map {f_6} z\) | \(=\) | \(\ds \dfrac z {z - 1}\) |
Let $\circ$ denote composition of functions.
Then $\struct {S, \circ}$ is the group generated by $\dfrac 1 z$ and $1 - z$.
Cayley Table
- $\begin{array}{r|rrrrrr}
\circ & f_1 & f_2 & f_3 & f_4 & f_5 & f_6 \\ \hline f_1 & f_1 & f_2 & f_3 & f_4 & f_5 & f_6 \\ f_2 & f_2 & f_3 & f_1 & f_6 & f_4 & f_5 \\ f_3 & f_3 & f_1 & f_2 & f_5 & f_6 & f_4 \\ f_4 & f_4 & f_5 & f_6 & f_1 & f_2 & f_3 \\ f_5 & f_5 & f_6 & f_4 & f_3 & f_1 & f_2 \\ f_6 & f_6 & f_4 & f_5 & f_2 & f_3 & f_1 \\ \end{array}$
Expressing the elements in full:
- $\begin{array}{c|cccccc}
\circ & z & \dfrac 1 {1 - z} & \dfrac {z - 1} z & \dfrac 1 z & 1 - z & \dfrac z {z - 1} \\ \hline z & z & \dfrac 1 {1 - z} & \dfrac {z - 1} z & \dfrac 1 z & 1 - z & \dfrac z {z - 1} \\ \dfrac 1 {1 - z} & \dfrac 1 {1 - z} & \dfrac {z - 1} z & z & \dfrac z {z - 1} & \dfrac 1 z & 1 - z \\ \dfrac {z - 1} z & \dfrac {z - 1} z & z & \dfrac 1 {1 - z} & 1 - z & \dfrac z {z - 1} & \dfrac 1 z \\ \dfrac 1 z & \dfrac 1 z & 1 - z & \dfrac z {z - 1} & z & \dfrac 1 {1 - z} & \dfrac {z - 1} z \\ 1 - z & 1 - z & \dfrac z {z - 1} & \dfrac 1 z & \dfrac {z - 1} z & z & \dfrac 1 {1 - z} \\ \dfrac z {z - 1} & \dfrac z {z - 1} & \dfrac 1 z & 1 - z & \dfrac 1 {1 - z} & \dfrac {z - 1} z & z \\ \end{array}$
Also see
- Group Generated by Reciprocal of z and 1 minus z, 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: $\S 6$: Examples of Finite Groups: $\text{(ii)}$
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Introduction