Gaussian Integer Units form Multiplicative Subgroup of Complex Numbers

Theorem
The group of Gaussian integer units under complex multiplication:
 * $\struct {U_\C, \times} = \struct {\set {1, i, -1, -i}, \times}$

forms a subgroup of the multiplicative group of complex numbers.

Proof
By Units of Gaussian Integers form Group, $\struct {U_\C, \times}$ forms a group.

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

The result follows by definition of subgroup.