Isomorphism between Gaussian Integer Units and Rotation Matrices Order 4

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 $1$