Group with Zero Element is Trivial

Theorem
If a group $\struct {G, \circ}$ has a zero element, then $\struct {G, \circ}$ is the trivial group.

Proof
Let $z \in G$ be a zero element, and $e \in G$ be the identity element of $G$.

Let $x \in G$ be any arbitrary element of $\struct {G, \circ}$.

Then:

So whatever $x \in G$ is, it has to be the identity element of $G$.

So $G$ can contain only that one element, and is therefore the trivial group.