The action of GL2(Fq) on irreducible polynomials over Fq, revisited

Let Fq be the finite field with q elements, p=char(Fq). The group GL2(Fq) acts naturally in the set of irreducible polynomials over Fq of degree at least 2. In this paper we are interested in the characterization and number of the irreducible polynomials that are fixed by the elements of a subgroup H of GL2(Fq). We make a complete characterization of the fixed polynomials in the case when H has only elements of the form (1b01), corresponding to translations x↦x+b and, as a consequence, the case when H is a p-subgroup of GL2(Fq). This paper also contains alternative solutions for the cases when H is generated by an element of the form (a001), obtained by Garefalakis (2010) and H=PGL2(Fq), obtained by Stichtenoth and Topuzoglu (2011).