The Existence Of(4,λ) -SCGDDs Of Type G~n

Let λ be a positive integer. A group divisible design(GDD) of index λ is a triple(X, G, B), where X is a finite set of points, G is a partition of X into subsets called groups, B is a collection of subsets of X(called blocks), such that each pair of points from different groups occurs in exactly λ blocks and any pair of points from the same group does not occur in any block. Let π be an automorphism of a(k, λ)-GDD. Ifπ permutes any point in each group G ∈ G in a |G|-cycle, the(k, λ)-GDD is called semi-cyclic, denoted by(k, λ)-SCGDD.For the existence problem of a(4, λ)-SCGDD of type gn, the case where λ = 1has been almost solved by J. Wang, J. Yin and K. Wang. In this paper we deal with the case where λ ≥ 2, and apply some direct and recursive constructions to prove the following main result:Let λ ≥ 2 and n ≥ 4. The necessary and sufficient conditions for the existence of a(4, λ)-SCGDD of type gnare that(i) λg(n- 1) ≡ 0(mod 3);(ii) λgn(n- 1) ≡ 0(mod 12).except for the case where n = 4, λ is odd and g is even, no(4, λ)-SCGDD of type gnexist.