Jungnickel (1982) and Elliot and Butson (1966) have shown that (pj+1,p,pj+1,pj) relative difference sets exist in the elementary abelian p-group case (p an odd prime) and many 2-groups for the case p = 2. This paper provides two new constructions of relative difference sets with these parameters; the first handles any p-group (including non-abelian) with a special subgroup if j is odd, and any 2-group with that subgroup if j is even. The second construction shows that if j is odd, every abelian group of order pj+2 and exponent less than or equal to p(j+3)/2 has a relative difference set. If j is even, we show that every abelian group of order 2j+2 and exponent less than or equal to 2(j+4)/2 has a relative difference set except the elementary abelian group. Finally, Jungnickel (1982) found (pi+j,pi,pi+j,pj) relative difference sets for all i, j in elementary abelian groups when pis an odd prime and in i4×j2 when p = 2. This paper also provides a construction for i+j even and ij in many group with a special subgroup. This is a generalization of the construction found in a submitted paper.

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Copyright © 1992, Elsevier. This article first appeared in Discrete Mathematics: 103:1 (1992), 7-15.

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