Unfortunately, it is not generally possible to determine whether or not an operation is associative simply by glancing at its Cayley table, as it is with commutativity.
Because the cancellation property holds for groups (and indeed even quasigroups), no row or column of a Cayley table may contain the same element twice.
This greatly restricts which Cayley tables could conceivably define a valid group operation.
But because the cancellation law holds, we can conclude that if ax = ay, then x = y, a contradiction.
This property implies that for each x in the group, the one variable function of y f(x,y)= xy must be a one-to-one map.
This particular example lets us create six permutation matrices (all elements 1 or 0, exactly one 1 in each row and column).
(Note that e is in every position down the main diagonal, which gives us the identity matrix for 6x6 matrices in this case, as we would expect.)
It is natural to consider Cayley tables for other algebraic structures, such as for semigroups, quasigroups, and magmas, but some of the properties above do not hold.