Differences of Bijections and Applications

 Let G={g_1, g_2, ... , g_n} be a finite group of order n and let D={d_1, d_2, ..., d_n} be a list of n elements -- not necessarily distinct -- of G. Under what conditions on G and D does there exist a permutation phi of G such that phi(g_i) - g_i = d_i for every i=1,2,..., n?
 
Although this problem remains open in general, partial solutions are known when G is abelian, or for an arbitrary group G but when D=G (in this case phi is called a complete mapping).
In this talk, I will discuss some generalizations of the above problem  and  present some applications to construct combinatorial designs.