The Asymptotic Existence of Group Divisible math formula-Designs of Large Order with Index One

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Abstract

The following result gives the answer to the question of whether for a fixed number of groups u and fixed block size k a group divisible t-design of group type math formula with block size k and index one exists for sufficiently large m if the necessary arithmetic conditions are satisfied. Let k and u be positive integers, math formula. Then there exists an integer math formula such that there exists a group divisible t-design of group type math formula with block size k and index one for any integer math formula satisfying the necessary arithmetic conditions

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The math formula case of this theorem gives an existence theorem for transversal t-designs of large order, which was previously proved by J. L. Blanchard in an unpublished manuscript.

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