A group must have a Cayley table in which each row and column has one and only one of each element. This can be proved by considering the opposite: suppose one row of a set’s Cayley table did not contain a particular element. That is, let AX not equal B, whatever X is. Let C be A’s inverse, such that CA = I. We easily reach a contradiction of associativity with ACB. (A*C)B = IB = B. On the other hand, A*(CB) = AX, does not equal B. Hence, each row and column of a group's Cayley table must contain exactly one of each element.
Two questions then arise:
Does a Cayley table in which each row and column has exactly one of each element guarantee that it is a group?
For a given number of elements, how many possible non-isomorphic groups are there? For instance, there is only one group with three elements. I think there are 4 groups with four elements.