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 an unsolved problem posed by Kelly in 1991, we investigate the problem of recognizing which permutation groups may arise as anonymity, neutrality and symmetry groups of a social preference function. A complete description is provided for neutrality groups. In the case of anonymity groups, we derive a sufficient condition, which largely captures the desired class of objects. Our approach also is of relevance for the notion of representability by Boolean functions and, therefore, the results of this paper also shed some light on this field of study.