Reconstruction of group multiplication tables by quadrangle criterion (Q5952160)

A matrix \(M\) is said to satisfy the quadrangle criterion if for all \(r,s,t,u,r',s',t',u'\), \(m_{rt}=m_{r't'}\), \(m_{ru}=m_{r'u'}\) and \(m_{st}=m_{s't'}\) implies \(m_{su}=m_{s'u'}\). A Cayley matrix is a Latin square which satisfies the quadrangle criterion. As is well-known, it is also the matrix obtained when the headline and sideline are removed from the multiplication table of a finite group. One question of interest about Cayley tables is how many entries are nescessary for the unique reconstruction of the original matrix. In this paper the author considers reconstruction using only the quadrangle criterion, and shows that, for \(n>3\), every Cayley matrix of order \(n\) with at most \(n-1\) holes (missing entries) can be reconstructed by the quadrangle criterion. Moreover, the order in which the holes are to be filled can be chosen in advance.