Actions of symmetry groups

This paper studies maps which are invariant under the action of the symmetry group Sk. The problem originates in social choice theory: there are k individuals each with a space of preferences X, and a social choice map : Xk->X which is anonymous i.e. invariant under the action of a group of symmetries. Theorem 1 proves that a full range map : Xk->X exists which is invariant under the action of Sk only if, for all i\geq1, the elements of the homotopy group i (X) have orders relatively prime with k. Theorem 2 derives a similar results for actions of subgroups of the group Sk. Theorem 3 proves necessary and sufficient condition for a parafinite CW complex X to admit full range invariant maps for any prime number k : X must be contractible.