A result on Dillon's conjecture in difference sets

A difference set in a finite group is a subset D c G so that every nonidentity element of G can be written as a difference of elements of D in precisely 2 ways. The order of G is v and the size of D is k. These can be considered in groups of any order, but this paper will be concerned with groups of order a power of 2. Dillon [2] provided a construction for a difference set in groups of order 22d+2 if the group has a normal subgroup isomorphic to Zt+ ‘. He was able to show that the construction works if the subgroup is central, but difficulties arose in the general case. This paper explores cases when the subgroup is not central, and we provide a sufficient condition for the construction to work. It is helpful to consider the ring Z[G]. If A c G, we will abuse notation by writing A = C,, E ,,, a’ as an element of Z[G]. Also, A(-l)=&eA (a’)-‘. By the definition of a difference set, D c G is a difference set iff DD’-“= (k-2) 1 +IG.