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We present a recursive construction for difference sets which unifies the Hadamard, McFarland, and Spence parameter families and deals with all abelian groups known to contain such difference sets. The construction yields a new family of difference sets with parameters (v, k, *, n)=(2 2d+4 (2 2d+2 &1)Â3, 2 2d+1 (2 2d+3 +1)Â3, 2 2d+1 (2 2d+1 +1)Â3, 2 4d+2 ) for d 0. The construction establishes that a McFarland difference set exists in an abelian group of order 2 2d+3 (2 2d+1 +1)Â3 if and only if the Sylow 2-subgroup has exponent at most 4. The results depend on a second recursive construction, for semi-regular relative difference sets with an elementary abelian forbidden subgroup of order p r . This second construction deals with all abelian groups known to contain such relative difference sets and significantly improves on previous results, particularly for r>1. We show that the group order need not be a prime power when the forbidden subgroup has order 2. We also show that the group order can grow without bound while its Sylow p-subgroup has fixed rank and that this rank can be as small as 2r. Both of the recursive constructions generalise to nonabelian groups. Academic Press

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