A (v,k,lambda) difference set D in a group G is a subset of G such that every nonidentity element of G is covered exactly lambda times by quotients d1d2-1 where d1 and d2 are in D. In the group ring, this means that D obeys the equation DD(-1) = k·1 + lambda(G - 1). An (m,n,k,lambda) relative difference set R is a difference set relative to a normal subgroup N of G satisfying the similar equation RR(-1) = k·1 + lambda(G - N).
We will describe various search techniques for relative difference sets (RDS), including the exhaustive search method for small groups using the computer program GAP, as well as the multiplier theorem and group representations methods used for larger groups. We will provide a catalog of RDS found, as well as those eliminated, using these methods. Next, a proof is presented of the non-existence of (2m,2,2m,m) relative difference sets in quaternion groups of order 4m where m is odd. In conclusion, we will state several interesting results found for specific parameters, including (12,2,12,6) and (12,3,12,4).