Units of Gaussian Integers form Group/Proof 1

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

Let $S$ be the set defined as:
 * $S = \left\{{1, i, -1, -i}\right\}$

Then $S$ under the operation of complex multiplication forms a group.

Proof
By definition of $i$:
 * $i^2 = -1$
 * $i^3 = -i$
 * $i^4 = 1$

Thus $\left({S, \times}\right)$ is isomorphic to $C_4$, the cyclic group of order $4$.

Hence the result.