Definition:Group of Gaussian Integer Units

Definition

Let $i$ be the imaginary unit: $i = \sqrt {-1}$.

Let $U_\C$ be the set of complex numbers defined as:

$U_\C = \set {1, i, -1, -i}$

Let $\times$ denote the operation of complex multiplication.

The algebraic structure $\struct {U_\C, \times}$ is the group of units of the ring of Gaussian integers.

Cayley Table

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}$

Also see

• Units of Gaussian Integers, where it is shown that $U_\C$ is the set of units of the ring of Gaussian integers

• Units of Gaussian Integers form Group

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 $1 \ (\alpha)$