A large order asymptotic existence theorem for group divisible 3-designs with index one

In this paper we prove that group divisible 33-designs exist for sufficiently large order with a fixed number of groups, fixed block size and index one, assuming that the necessary arithmetic conditions are satisfied. Let kk and uu be positive integers, 3≤k≤u3≤k≤u. Then there exists an integer m0=m0(k,u)m0=m0(k,u) such that there exists a group divisible 33-design of group type mumu with block size kk and index one for any integer m≥m0m≥m0 satisfying the necessary arithmetic conditions 1.m(u−2)≡0mod(k−2),2.m2(u−1)(u−2)≡0mod(k−1)(k−2),3.m3u(u−1)(u−2)≡0modk(k−1)(k−2).