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 m0=m0(k,u) such that there exists a group divisible design of group type mu with block size k and index one for any integer m⩾m0 satisfying the necessary arithmetic conditions1.,2..This paper 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.