Intersection theorems for group divisible difference sets

Let 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,d₂?D,d₁?d₂=g}? is equal to λ₁ if g is in H, and equal to λ₂ if g is not in H. Let H₁,H₂,…,H_m be distinct cosets of H in G and S_i=D∩H_i for all i=1,2,…,m. Some properties of S₁,S₂,…,S_m are studied here. Table 1 shows all possible cardinalities of S_i'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,$\text{1}\text{2}$n?1,$\text{1}\text{4}$n?1)-difference set.