Symmetry groups for social preference functions

We introduce the anonymity group, the neutrality group and the symmetry group of a social preference function. Inspired by a problem posed by Kelly in 1991 and remained unsolved, we investigate the problem of recognizing which permutation groups may arise as anonymity, neutrality and symmetry group of a social preference function. A complete description is found for the neutrality groups and a sufficient condition, which largely encompasses the problem, is found for the anonymity groups. Using the concept of orbit extension of a group $U$, we formulate manageable necessary conditions for being $U$ an anonymity or a symmetry group. Our research deeply interacts with problems of representability by Boolean functions shedding light on them.