Group has Latin Square Property

Theorem
For any elements $a$ and $b$ in a group $G$, there exists a unique $g \in G$ such that $a g = b$.

Similarly, there exists a unique $h \in G$ such that $h a = b$.

Corollary
The Cayley table for any finite group is a Latin square.

Proof
Thus, such a $g$ exists.

Suppose $x \in G$ where $a x = b$.

Then:

Thus, $x$ is uniquely of the form $a^{-1} b$.

To prove the second part of the theorem, let $h = b a^{-1}$.

The remainder of the proof follows a similar procedure to the above.

Proof of Corollary
Follows directly from the definition of both a Cayley table and a Latin square.