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

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:

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:

Group Generated by $1 / z$ and $1 - z$
The Cayley table for $S$ is as follows: