Isomorphism between Gaussian Integer Units and Integers Modulo 4 under Addition/Proof 2

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Theorem

Let $\struct {U_\C, \times}$ be the group of Gaussian integer units under complex multiplication.

Let $\struct {\Z_n, +_4}$ be the integers modulo $4$ under modulo addition.


Then $\struct {U_\C, \times}$ and $\struct {\Z_4, +_4}$ are isomorphic algebraic structures.


Proof

Let the mapping $f: \Z_4 \to U_\C$ be defined as:

\(\ds \map f 0\) \(=\) \(\ds 1\)
\(\ds \map f 1\) \(=\) \(\ds i\)
\(\ds \map f 2\) \(=\) \(\ds -1\)
\(\ds \map f 3\) \(=\) \(\ds -i\)


From Isomorphism by Cayley Table, the two Cayley tables can be compared by eye to ascertain that $f$ is an isomorphism:


Cayley Table of Integers Modulo $4$

The Cayley table for $\struct {\Z_4, +_4}$ is as follows:

$\begin{array}{r|rrrr}

\struct {\Z_4, +_4} & \eqclass 0 4 & \eqclass 1 4 & \eqclass 2 4 & \eqclass 3 4 \\ \hline \eqclass 0 4 & \eqclass 0 4 & \eqclass 1 4 & \eqclass 2 4 & \eqclass 3 4 \\ \eqclass 1 4 & \eqclass 1 4 & \eqclass 2 4 & \eqclass 3 4 & \eqclass 0 4 \\ \eqclass 2 4 & \eqclass 2 4 & \eqclass 3 4 & \eqclass 0 4 & \eqclass 1 4 \\ \eqclass 3 4 & \eqclass 3 4 & \eqclass 0 4 & \eqclass 1 4 & \eqclass 2 4 \\ \end{array}$


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}$


Sources

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