Non-Abelian Group has Order Greater than 4/Proof 1

Theorem
Let $\left({G, \circ}\right)$ be a non-abelian group.

Then the order of $\left({G, \circ}\right)$ is greater than $4$.

Proof
Let $\left({G, \circ}\right)$ be a non-abelian group whose identity is $e$.

For a group $\left({G, \circ}\right)$ to be non-abelian, we require:
 * $\exists x, y \in G: x \circ y \ne y \circ x$

From the definition of the identity:
 * $x \circ y = x \implies y = e$

and:
 * $x \circ y = y \implies x = e$

So, if $x \circ y \ne y \circ x$, then both $x \circ y$ and $y \circ x$ must be different elements, both different from $e, x$ and $y$.

Thus, in a non-abelian group, there needs to be at least $5$ elements:


 * $e, x, y, x \circ y, y \circ x$