The shift action on 2-cocycles

This paper introduces the shift action, whereby each group G acts as a group of automorphisms of Z2(G,C), the abelian group of cocyclesG×G→C, for each choice of abelian group C. Fundamental properties of the shift action—fixed points, orbits and stabilisers—are described in terms of particular types of cocycle: multiplicative, symmetric, skew-symmetric and coboundary. The orbit structure in the simplest case, for G cyclic, is analysed in detail. The shift action preserves frequencies of the values a cocycle takes in C. The idea of differentially uniformcocycles is introduced, for application to the design of highly nonlinear digital sequences.