After proving that unique solvability of equations turns a semigroup into a group, we now study a very useful finite version of that idea. In a finite semigroup, cancellation laws are strong enough to force unique solvability of equations. The reason is that an injective map from a finite set to itself must also be onto. Thus cancellation gives one-to-one left and right multiplication maps, finiteness gives onto maps, and the equation criterion then gives a group. In this lesson, students will prove that every finite cancellative semigroup is a group.

Let $(S,\circ)$ be a semigroup. Then $(S,\circ)$ is called $\textbf{cancellative}$ if both cancellation laws hold in $S$, that is, for all $a,b,c\in S$: (i) $a\circ b=a\circ c \implies b=c$. (ii) $b\circ a=c\circ a \implies b=c$.