The shift action on 2-cocycles

This paper introduces the shift action, whereby each group G acts as a group of automorphisms of Z(2)(G, C), the abelian group of cocycles G x 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 uniform cocycles is introduced, for application to the design of highly nonlinear digital sequences.