A composition theorem for generalized Bhaskar Rao designs

Let H be a normal subgroup of a finite group G. We show that: If a GBRD(v, k, A.; G IH) exists and a GBRD(k,j, JA.; H) exists then a GBRD(v,j, AJA.; G) exists. We apply this result to show that: i) If k does not exceed the least prime factor of IG I, then a GBRD(k, k, A; G) exists for all A EO (mod IG I); ii) If G is of order IG Ie lor 5 (mod 6) then a GBRD(v, 3, A = t IG I; G), v>3, exists if and only if a BIBD(v, 3, t) exists; iii) If G is a nilpotent group of odd order then the necessary conditions are sufficient for the existence of a GBRD(v, 3, A; G); and, iv) If G is ap -group,p ;f2, then the necessary conditions are sufficient for the existence of a GBRD(v, 3, A.; G).