Isomorphism between Gaussian Integer Units and Reduced Residue System Modulo 5 under Multiplication
Jump to navigation
Jump to search
Theorem
Let $\struct {U_\C, \times}$ be the group of Gaussian integer units under complex multiplication.
Let $\struct {\Z'_5, \times_5}$ be the multiplicative group of reduced residues modulo $5$.
Then $\struct {U_\C, \times}$ and $\struct {\Z'_5, \times_5}$ are isomorphic algebraic structures.
Proof
Establish the mapping $f: U_C \to \Z'_5$ as follows:
\(\ds \map f 1\) | \(=\) | \(\ds \eqclass 1 5\) | ||||||||||||
\(\ds \map f i\) | \(=\) | \(\ds \eqclass 2 5\) | ||||||||||||
\(\ds \map f {-1}\) | \(=\) | \(\ds \eqclass 4 5\) | ||||||||||||
\(\ds \map f {-i}\) | \(=\) | \(\ds \eqclass 3 5\) |
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:
- $\begin{array}{r|rrrr} \times & 1 & i & -1 & -i \\ \hline 1 & 1 & i & -1 & -i \\ i & i & -1 & -i & 1 \\ -1 & -1 & -i & 1 & i \\ -i & -i & 1 & i & -1 \\ \end{array}$
Multiplicative Group of Reduced Residues Modulo $5$
The Cayley table for $\struct {\Z'_5, \times_5}$ is as follows:
- $\begin{array}{r|rrrr} \times_5 & \eqclass 1 5 & \eqclass 2 5 & \eqclass 3 5 & \eqclass 4 5 \\ \hline \eqclass 1 5 & \eqclass 1 5 & \eqclass 2 5 & \eqclass 3 5 & \eqclass 4 5 \\ \eqclass 2 5 & \eqclass 2 5 & \eqclass 4 5 & \eqclass 1 5 & \eqclass 3 5 \\ \eqclass 3 5 & \eqclass 3 5 & \eqclass 1 5 & \eqclass 4 5 & \eqclass 2 5 \\ \eqclass 4 5 & \eqclass 4 5 & \eqclass 3 5 & \eqclass 2 5 & \eqclass 1 5 \\ \end{array}$
By arranging the rows and columns into a different order, its cyclic nature becomes clear:
- $\begin{array}{r|rrrr} \times_5 & \eqclass 1 5 & \eqclass 2 5 & \eqclass 4 5 & \eqclass 3 5 \\ \hline \eqclass 1 5 & \eqclass 1 5 & \eqclass 2 5 & \eqclass 4 5 & \eqclass 3 5 \\ \eqclass 2 5 & \eqclass 2 5 & \eqclass 4 5 & \eqclass 3 5 & \eqclass 1 5 \\ \eqclass 4 5 & \eqclass 4 5 & \eqclass 3 5 & \eqclass 1 5 & \eqclass 2 5 \\ \eqclass 3 5 & \eqclass 3 5 & \eqclass 1 5 & \eqclass 2 5 & \eqclass 4 5 \\ \end{array}$
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 $1$