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.