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

In this paper we prove that group divisible 3 -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 k and u be positive integers, 3 ≤ k ≤ u . Then there exists an integer m 0 = m 0 ( k , u ) such that there exists a group divisible 3 -design of group type m u with block size k and index one for any integer m ≥ m 0 satisfying the necessary arithmetic conditions 1. − 2 ) ≡ 0 mod ( k − 2 ) , u − 1 ) ( u − 2 ) ≡ 0 mod ( k − 1 ) ( k − 2 ) , ( u − 1 ) ( u − 2 ) ≡ 0 mod k ( k − 1 ) ( k − 2 ) .