Construction of relative difference sets in p-groups

7 Davis, J.A., Construction of relative difference sets in p-groups, Discrete Mathematics 103 (1992) 7-15. Jungnickel (1982) and Elliot and Butson (1966) have shown that (pi+ , p, pi+•, pi) 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 pi+Z and exponent less than or equal to p has a relative difference set. If j is even, we show that every abelian group of order '2}+ 2 and exponent less than or equal to 2U+> has a relative difference set except the elementary abelian group. Finally, Jungnickel (1982) found (pHi, p', pHi, pi) relative difference sets for all i, j in elementary abelian groups when p is an odd prime and in £'~ x ~ when p = 2. This paper also provides a construction for i + j even and i ,,;;,j in many group with a special subgroup. This is a generalization of the construction found in a submitted paper.