Group Generated by Reciprocal of z and 1 minus z/Cayley Table
Cayley Table for Group Generated by Reciprocal of $1 / z$ and $1 - z$
We have:
| \(\ds \map {f_1} z\) | \(=\) | \(\ds z\) | ||||||||||||
| \(\ds \map {f_2} z\) | \(=\) | \(\ds \dfrac 1 {1 - z}\) | ||||||||||||
| \(\ds \map {f_3} z\) | \(=\) | \(\ds \dfrac {z - 1} z\) | ||||||||||||
| \(\ds \map {f_4} z\) | \(=\) | \(\ds \dfrac 1 z\) | ||||||||||||
| \(\ds \map {f_5} z\) | \(=\) | \(\ds 1 - z\) | ||||||||||||
| \(\ds \map {f_6} z\) | \(=\) | \(\ds \dfrac z {z - 1}\) |
Hence from Group Generated by Reciprocal of z and 1 minus z:
- $\begin{array}{r|rrrrrr}
\circ & f_1 & f_2 & f_3 & f_4 & f_5 & f_6 \\ \hline f_1 & f_1 & f_2 & f_3 & f_4 & f_5 & f_6 \\ f_2 & f_2 & f_3 & f_1 & f_6 & f_4 & f_5 \\ f_3 & f_3 & f_1 & f_2 & f_5 & f_6 & f_4 \\ f_4 & f_4 & f_5 & f_6 & f_1 & f_2 & f_3 \\ f_5 & f_5 & f_6 & f_4 & f_3 & f_1 & f_2 \\ f_6 & f_6 & f_4 & f_5 & f_2 & f_3 & f_1 \\ \end{array}$
Expressing the elements in full:
- $\begin{array}{c|cccccc}
\circ & z & \dfrac 1 {1 - z} & \dfrac {z - 1} z & \dfrac 1 z & 1 - z & \dfrac z {z - 1} \\ \hline z & z & \dfrac 1 {1 - z} & \dfrac {z - 1} z & \dfrac 1 z & 1 - z & \dfrac z {z - 1} \\ \dfrac 1 {1 - z} & \dfrac 1 {1 - z} & \dfrac {z - 1} z & z & \dfrac z {z - 1} & \dfrac 1 z & 1 - z \\ \dfrac {z - 1} z & \dfrac {z - 1} z & z & \dfrac 1 {1 - z} & 1 - z & \dfrac z {z - 1} & \dfrac 1 z \\ \dfrac 1 z & \dfrac 1 z & 1 - z & \dfrac z {z - 1} & z & \dfrac 1 {1 - z} & \dfrac {z - 1} z \\ 1 - z & 1 - z & \dfrac z {z - 1} & \dfrac 1 z & \dfrac {z - 1} z & z & \dfrac 1 {1 - z} \\ \dfrac z {z - 1} & \dfrac z {z - 1} & \dfrac 1 z & 1 - z & \dfrac 1 {1 - z} & \dfrac {z - 1} z & z \\ \end{array}$
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