Isomorphism between Group Generated by Reciprocal of z and 1 minus z and Symmetric Group on 3 Letters

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $S_3$ denote the symmetric group on $3$ letters.

Let $G$ be the group generated by $1 / z$ and $1 - z$.


Then $S_3$ and $G$ are isomorphic algebraic structures.


Proof

Establish the mapping $\phi: S_3 \to G$ as follows:

\(\ds \map \phi e\) \(=\) \(\ds z\)
\(\ds \map \phi {\paren {123} }\) \(=\) \(\ds \dfrac 1 {1 - z}\)
\(\ds \map \phi {\paren {132} }\) \(=\) \(\ds \dfrac {z - 1} z\)
\(\ds \map \phi {\paren {23} }\) \(=\) \(\ds \dfrac 1 z\)
\(\ds \map \phi {\paren {13} }\) \(=\) \(\ds 1 - z\)
\(\ds \map \phi {\paren {12} }\) \(=\) \(\ds \dfrac z {z - 1}\)


From Isomorphism by Cayley Table, the two Cayley tables can be compared by eye to ascertain that $\phi$ is an isomorphism:


Cayley Table of Symmetric Group on $3$ Letters

The Cayley table for $S_3$ is as follows:

$\begin{array}{c|cccccc}

\circ & e & (123) & (132) & (23) & (13) & (12) \\ \hline e & e & (123) & (132) & (23) & (13) & (12) \\ (123) & (123) & (132) & e & (13) & (12) & (23) \\ (132) & (132) & e & (123) & (12) & (23) & (13) \\ (23) & (23) & (12) & (13) & e & (132) & (123) \\ (13) & (13) & (23) & (12) & (123) & e & (132) \\ (12) & (12) & (13) & (23) & (132) & (123) & e \\ \end{array}$


Group Generated by $1 / z$ and $1 - z$

The Cayley table for $S$ is as follows:

$\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

Retrieved from "https://proofwiki.org/w/index.php?title=Isomorphism_between_Group_Generated_by_Reciprocal_of_z_and_1_minus_z_and_Symmetric_Group_on_3_Letters&oldid=372382"