Let $S$ be a finite, cancellative semigroup under a binary operation $ \theta $. Then $(S,\theta)$ is a group.
The proof of this statement is already given here, but I would like to check if this is an equivalence relationship? I.e. I want to check if the reciprocal is also a true statement?
Well, no: there are groups that are not finite. If you are asking if all finite groups are cancellative, then yes (and, indeed, all groups are cancellative): if $G$ is a group and $a, b, c \in G$ such that $ab = ac$, we have $b = 1b = (a^{-1}a)b = a^{-1}(ab) = a^{-1}(ac) = (a^{-1}a)c = 1c = c$, so $G$ is cancellative (where those equalities are the identity, inverse, and associativity axioms of a group, then the assumption that $ab = ac$, and finally those same axioms in reverse order).