A general representation theory for constructing groups of permutation polynomials

Using the left regular action of a group on itself, we develop a general representation theory for constructing groups of permutation polynomials. As an application of the method, we compute polynomial representations of several abelian and nonabelian groups, and we determine the equivalence classes of the groups of polynomials we construct. In particular, when the size of the group is equal to the size of the field in which the group is represented, all non-identity representation polynomials are necessarily fixed-point free permutation polynomials.