Defining sets for Latin squares given that they are based on groups (Q1869037)

A partial Latin square is an \(n\) by \(n\) array with elements from a set of size \(n\), such that no element appears more than once in any row or column. If a partial Latin square can be completed to a unique Latin square within the main class of a Latin square, denoted \(M(L)\), then it is said to be a defining set in \(M(L).\) The density of a defining set is the ratio of the size of the defining set and the number of entries in the Latin square, \(n^2\). The authors show that the density of the smallest defining sets in \(M(G)\), where \(G\) is a group, approaches \(0\) as \(n\) approaches infinity. Additionally, they disprove a conjecture of Keedwell by constructing a defining set of density \({7}\over{16}\) for each member of an infinite class of non-cyclic groups.