 Finite cancellative semigroup

A nonempty finite set with an associative binary operation satisfying both cancellation laws is a group. For fixed $a$, cancellation makes $x\mapsto ax$ and $x\mapsto xa$ injective, hence surjective. Choose $e$ with $ae=a$. Then $a(ex)=ax$ gives $ex=x$, and $(xe)x=xx$ gives $xe=x$. Choose $ab=e=ca$; associativity gives $c=c(ab)=(ca)b=b$, so $a$ has a two-sided inverse element.