The asymptotic existence of group divisible designs of large order with index one

This paper gives the answer to a question of R.M. Wilson regarding the existence of group divisible designs of large order. Let k and u be positive integers such that 2 ⩽ k ⩽ u . Then there exists an integer m 0 = m 0 ( k , u ) such that there exists a group divisible design of group type m u with block size k and index one for any integer m ⩾ m 0 satisfying the necessary arithmetic conditions1. − 1 ) ≡ 0 mod ( k − 1 ) , ( u − 1 ) ≡ 0 mod k ( k − 1 ) . aper also presents a large-index asymptotic existence theorem for group divisible t-designs with a fixed number of groups, fixed group size and fixed block size.