Cocyclic Generalised Hadamard Matrices and Central Relative Difference Sets

Abstract

Cocyclic matrices have the form $M = [\psi (g,h)]_{g,h} \in G,$ where G is a finite group, C is a finite abelian group and ψ : G × G → C is a (two-dimensional) cocycle; that is,

$$\psi (g,h)\psi (gh,k) = \psi (g,hk)\psi (h,k),\forall g,h,k \in G.$$

This expression of the cocycle equation for finite groups as a square matrix allows us to link group cohomology, divisible designs with regular automorphism groups and relative difference sets. Let G have order v and C have order w, with w|v. We show that the existence of a G-cocyclic generalised Hadamard matrix GH (w, v/w) with entries in C is equivalent to the existence of a relative ( v, w, v, v/w)-difference set in a central extension E of C by G relative to the central subgroup C and, consequently, is equivalent to the existence of a (square) divisible ( v, w, v, v/w)-design, class regular with respect to C, with a central extension E of C as regular group of automorphisms. This provides a new technique for the construction of semiregular relative difference sets and transversal designs, and generalises several known results.