Group Generated by Reciprocal of z and 1 minus z

Theorem
Let $\struct {S, \circ}$ denote the group generated by $\dfrac 1 z$ and $1 - z$.

Then $\struct {S, \circ}$ is a finite group of order $6$.

Proof
By definition:
 * $S = \set {f_1, f_2, f_3, f_4, f_5, f_6}$

where $f_1, f_2, \ldots, f_6$ are complex functions defined for all $z \in \C \setminus \set {0, 1}$ as:

Also by definition, $\circ$ denotes composition of functions.

First it is necessary to establish the Cayley table for $\struct {S, \circ}$.

First note that it is apparent by definition of composition of functions that:
 * $\forall n: f_1 \circ f_n = f_n = f_n \circ f_1$

It remains to establish the rest of the compositions

Hence the Cayley table is established:

Taking the group axioms in turn:

G0: Closure
By inspection it can be seen that $\struct {S, \circ}$ is closed.

G1: Associativity
From Composition of Mappings is Associative, it follows that $\struct {S, \circ}$ is associative.

G2: Identity
We have that:
 * $\forall n: f_1 \circ f_n = f_n = f_n \circ f_1$

Thus $\struct {S, \circ}$ has an identity element.

G3: Inverses
From the above analysis:

Thus every element of $\struct {S, \circ}$ has an inverse.

All the group axioms are thus seen to be fulfilled, and so $\struct {S, \circ}$ is a group.

$\struct {S, \circ}$ has $6$ elements and so is a finite group of order $6$.