On the action of on irreducible polynomials over Fq

We show that acts on the set of irreducible polynomials over Fq. We study this action and compute the cardinality of the set of fixed points of certain subgroups of . Our results include enumeration of irreducible polynomials that are invariant under the substitution X↦X+b and under the substitution X↦aX. From those results enumeration formulas follow for the set of fixed points of any element of order r, where r is a prime divisor of q or q−1. The results are combined to provide enumeration formulas for the fixed points of upper triangular matrices.