On the action of GL2(Fq) on irreducible polynomials over Fq

We show that GL(2)(F-q) acts on the set of irreducible polynomials over F-q. We study this action and compute the cardinality of the set of fixed points of certain subgroups of GL(2)(F-q). Our results include enumeration of irreducible polynomials that are invariant under the substitution X bar right arrow X + b and under the substitution X bar right arrow 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. (C) 2010 Elsevier B.V. All rights reserved.