| Abstract: | A semiregular relative difference set (RDS) in a finite group E which avoids a central subgroup C is equivalent to a cocycle which satisfies an additional condition, called orthogonality. However the basic equivalence relation, cohomology, on cocycles, does not preserve orthogonality, leading to the perception that orthogonality is essentially a combinatorial property. We show this perception is false by discovering a natural atomic structure within cohomology classes, which discriminates between orthogonal and non‐orthogonal cocycles. This atomic structure is determined by an action we term the shift action of the group G = E/C on cocycles, which defines a stronger equivalence relation on cocycles than cohomology. We prove that for each triple (C, E, G), the set of equivalence classes of semiregular RDS in E relative to C is in one to one correspondence with the set of shift‐orbits of the (Aut(C) × Aut(G))‐orbits of orthogonal cocycles. This determines a new algorithm for detecting and classifying central semiregular RDS. We demonstrate it, and propose a 7‐parameter classification scheme for equivalence classes of central semiregular relative difference sets. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 330–346, 2000 |
