Group Generated by Reciprocal of z and Minus z is Klein Four-Group

Theorem
Let $K_4$ denote the Klein $4$-group.

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

Then $K_4$ and $S$ are isomorphic algebraic structures.

Proof
Establish the mapping $\phi: K_4 \to S$ 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 Klein $4$-Group
The Cayley table for $K_4$ is as follows:

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