Units of Gaussian Integers form Group/Proof 2

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
We have that:
 * $1^4 = 1$
 * $i^4 = \left({-1}\right)^2 = 1$
 * $\left({-1}\right)^4 = 1^2 = 1$
 * $\left({-i}\right)^4 = \left({-1}\right)^2 = 1$

Thus $\left\{{1, i, -1, -i}\right\}$ constitutes the set of the $4$th roots of unity.

The result follows from Roots of Unity under Multiplication form Cyclic Group.