Isomorphism between Gaussian Integer Units and Rotation Matrices Order 4
Jump to navigation
Jump to search
Theorem
Let $\struct {U_\C, \times}$ be the group of Gaussian integer units under complex multiplication.
Let $\struct {R_4, \times}$ be the group of rotation matrices of order $4$ under modulo addition.
Then $\struct {U_\C, \times}$ and $\struct {R_4, \times}$ are isomorphic algebraic structures.
Proof
Establish the mapping $f: U_C \to R_4$ as follows:
\(\ds \map f 1\) | \(=\) | \(\ds r_0\) | ||||||||||||
\(\ds \map f i\) | \(=\) | \(\ds r_1\) | ||||||||||||
\(\ds \map f {-1}\) | \(=\) | \(\ds r_2\) | ||||||||||||
\(\ds \map f {-i}\) | \(=\) | \(\ds r_3\) |
From Isomorphism by Cayley Table, the two Cayley tables can be compared by eye to ascertain that $f$ is an isomorphism:
Cayley Table of Gaussian Integer Units
The Cayley table for $\struct {U_\C, \times}$ is as follows:
- $\begin{array}{r|rrrr} \times & 1 & i & -1 & -i \\ \hline 1 & 1 & i & -1 & -i \\ i & i & -1 & -i & 1 \\ -1 & -1 & -i & 1 & i \\ -i & -i & 1 & i & -1 \\ \end{array}$
Group of Rotation Matrices Order $4$
The Cayley table for $\struct {R_4, \times}$ is as follows:
- $\begin{array}{r|rrrr} \times & r_0 & r_1 & r_2 & r_3 \\ \hline r_0 & r_0 & r_1 & r_2 & r_3 \\ r_1 & r_1 & r_2 & r_3 & r_0 \\ r_2 & r_2 & r_3 & r_0 & r_1 \\ r_3 & r_3 & r_0 & r_1 & r_2 \\ \end{array}$
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 $1$