Intersection theorems for group divisible difference sets

AbstractLet D be an (m,n;k;λ1,λ2)-group divisible difference set (GDDS) of a group G, written additively, relative to H, i.e. D is a k-element subset of G, H is a normal subgroup of G of index m and order n and for every nonzero element g of G,⨍{(d1,d2)⨍,d1,d2ϵD,d 1−d2=g}⨍ is equal to λ1 if g is in H, and equal to λ2 if g is not in H. Let H1,H2,…,Hm be distinct cosets of H in G and Si=D∩Hi for all i=1,2,…,m. Some properties of S1,S2,…,Sm are studied here. Table 1 shows all possible cardinalities of Si's when the order of G is not greater than 50 and not a prime. A matrix characterization of cyclic GDDS's with λ1=0 implies that there exists a cyclic affine plane of even order, say n, only if n is divisible by 4 and there exists a cyclic (n−1,12n−1,14n−1)-difference set