Gaussian Integer Units form Multiplicative Subgroup of Complex Numbers

Theorem
The Group of Gaussian Integer Units under complex multiplication:
 * $\left({U_\C, \times}\right) = \left({\left\{{1, i, -1, -i}\right\}, \times}\right)$

forms a subgroup of the multiplicative group of complex numbers.

Proof
By Group of Gaussian Integer Units, $\left({U_\C, \times}\right)$ forms a group.

Each of the elements of $U_\C$ is a complex number, and non-zero, and therefore $U_\C \subseteq \C \setminus \left\{{0}\right\}$.

The result follows by definition of subgroup.