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:
| \(\ds \map \phi e\) | \(=\) | \(\ds z\) | ||||||||||||
| \(\ds \map \phi a\) | \(=\) | \(\ds -z\) | ||||||||||||
| \(\ds \map \phi b\) | \(=\) | \(\ds \dfrac 1 z\) | ||||||||||||
| \(\ds \map \phi c\) | \(=\) | \(\ds -\dfrac 1 z\) |
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:
- $\begin{array}{c|cccc}
& e & a & b & c \\
\hline e & e & a & b & c \\ a & a & e & c & b \\ b & b & c & e & a \\ c & c & b & a & e \\ \end{array}$
Group Generated by $1 / z$ and $-z$
The Cayley table for $S$ is as follows:
- $\begin{array}{r|rrrr}
\circ & f_1 & f_2 & f_3 & f_4 \\ \hline f_1 & f_1 & f_2 & f_3 & f_4 \\ f_2 & f_2 & f_1 & f_4 & f_3 \\ f_3 & f_3 & f_4 & f_1 & f_2 \\ f_4 & f_4 & f_3 & f_2 & f_1 \\ \end{array}$
Expressing the elements in full:
- $\begin{array}{c|cccc}
\circ & z & -z & \dfrac 1 z & -\dfrac 1 z \\ \hline z & z & -z & \dfrac 1 z & -\dfrac 1 z \\ -z & -z & z & -\dfrac 1 z & \dfrac 1 z \\ \dfrac 1 z & \dfrac 1 z & -\dfrac 1 z & z & -z \\ -\dfrac 1 z & -\dfrac 1 z & \dfrac 1 z & -z & z \\ \end{array}$
Sources
- 1964: Walter Ledermann: Introduction to the Theory of Finite Groups (5th ed.) ... (previous) ... (next): Chapter $\text {I}$: The Group Concept: $\S 7$: Isomorphic Groups: Example $2$