On (p(a), p(b) p(a) p(a-b))-relative difference sets

This paper provides new exponent and rank conditions for the existence of abelian relative (p(a), p(b), p(a), p(a-b))-difference sets. It is also shown that no splitting relative (2(2c), 2(d), 2(2c), 2(2c-d))-difference set exists if d > c and the forbidden subgroup is abelian. Furthermore, abelian relative (16, 4, 16, 4)-difference sets are studied in detail; in particular, it is shown that a relative (16, 4, 16, 4)-difference set in an abelian group G not congruent to Z(8) x Z(4) x Z(2) exists if and only if exp(G) less than or equal to 4 or G = Z(8) x (Z(2))(3) with N congruent to Z(2) x Z(2).