[Lekkoksung]

อันดับแรกตอนนี้ทราบรึยังครับว่ามีทฤษฎีนี้เกิดขึ้นก่อน

A semigroup $S$ is a group if and only if for all $a,b \in S$ the equations $ax=b$ and $ya=b$ have solution in $S$ for $x$ and $y$.

Let $S$ be a finite semigroup satisfying the cancellative laws. Let $a,b \in S$.
Consider the equation $ax=b$. We show that this equation has a solution in $S$.
Now, $S= \{ a_{1}, \ldots, a_{n} \}$, where the $a_{i}$'s are all distinct element of $S$.
Since $S$ is a semigroup, $aa_{i} \in S$ for all $i \in \{ 1,\ldots,n \}$.
Thus, $\{aa_{1}, \ldots, aa_{n}\} \subseteq S$. Suppose $aa_{i}=aa_{j}$
for some $i \not = j$. Then by the cancellative laws, $a_{i}=a_{j}$, which is
a contradiction since $a_{i} \not a_{j}$. Hence all elements in $\{aa_{1}, \ldots, aa_{n}\}$
are distinct. Thus $S = \{ aa_{1}, \ldots , aa_{n} \}$. Then $b=aa_{k}$ for some $a_{k} \in S$.
Therefore the equation $ax=b$ has solution in $S$. Similarly, we can show that the
equation $ya=b$ has solution in $S$. As a consequently, $S$ is a group.