Units of Gaussian Integers form Group/Proof 3

Theorem
Let $U_\C$ be the set of units of the Gaussian integers:
 * $U_\C = \left\{{1, i, -1, -i}\right\}$

where $i$ is the imaginary unit: $i = \sqrt {-1}$.

Let $\left({U_\C, \times}\right)$ be the algebraic structure formed by $U_\C$ under the operation of complex multiplication.

Then $\left({U_\C, \times}\right)$ forms a cyclic group under complex multiplication.

Proof
From Units of Gaussian Integers, $U_\C$ is the set of units of the ring of Gaussian integers.

From Group of Units is Group, $\left({U_\C, \times}\right)$ forms a group.

It remains to note that:

thus demonstrating that $U_\C$ is cyclic.